Выявление динамических эффектов в движении спутников планет и астероидов на основе наблюдений покрытий и видимых тесных сближений со звездами Gaia тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Бикулова Динара Александровна

• Бикулова Динара Александровна
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Бикулова Динара Александровна

3.1 Наблюдения в ГАО РАН

3.2 Наблюдения в КрАО РАН

4 Обзор методики анализа изображений звезд и тел Солнечной системы, представленных на ПЗС-кадрах

4.1 Отождествление объектов на снимках

4.2 Определение пиксельных координат фотоцентров изображений звездообразных объектов

4.3 Астрометрическая редукция

5 Результаты наблюдений покрытия звезды астероидом 87 Sylvia

6 Результаты наблюдений видимых тесных сближений избранных АСЗ со звездами Gaia

7 Результаты наблюдений видимых тесных сближений спутников планет-гигантов и со звездами Gaia

7.1 Явления для спутников Урана и Тритона - спутника Нептуна

7.2 Результаты определений положений спутников планет «традиционным» методом

7.3 Явления для главных спутников Сатурна за 35 лет пулковских фотографических наблюдений

8 Заключение

9 Благодарности Литература

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Введение диссертации (часть автореферата) на тему «Выявление динамических эффектов в движении спутников планет и астероидов на основе наблюдений покрытий и видимых тесных сближений со звездами Gaia»

1 Введение

Изучение строения, формирования и эволюции как всей Солнечной системы, так и отдельных составных частей является одной из фундаментальных задач астрономии. В последние десятилетия наблюдается колоссальный приток информации о телах Солнечной системы благодаря космическим миссиям и высокотехнологичным астрономическим наблюдениям во всех диапазонах электромагнитного спектра. Для понимания динамических процессов в системах спутников планет и популяциях астероидов колоссальную роль играет изучение сложных систем резонансов, взаимных приливов, влияния эффектов светового давления и анизотропного переизлучения. Как и ранее, остается актуальной задача совершенствования эфемерид небесных тел как для фундаментальных исследований, так и для прикладных (обеспечение навигации и надежности космических полетов). Довольно остро стоит проблема изучения астероидно-кометной опасности.

Именно необходимость изучения комплекса перечисленных динамических эффектов стала основным мотивирующим фактором для выполнения представленной работы. На данном этапе представляется важным выявить наличие эффектов, не описываемых моделями, лежащими в основе современных эфемерид. Мы сосредоточили свои усилия на анализе движений спутников Сатурна, Урана и Нептуна. Изучение популяции астероидов, сближающихся с Землей, тоже ставит сложные динамические вопросы. Например, как отделить эффекты, обусловленные резонансами от проявлений УОЯР-эффекта, столкновений, двойственности и т.п. Поэтому часть данного исследования посвящена анализу результатов наблюдений астероидов.

Наблюдательной основой для прогресса в изучении динамики тел Солнечной системы являются их астрометрические наблюдения. Речь идет обо всех типах позиционных наблюдений от радиолокации, измерений координат и скоростей с борта космических аппаратов, до традиционных наземных оптических наблюдений. Роль высокотехнологичных измерений постоянно растет, но до сих пор вклад наземных наблюдений остается значимым. Есть ряд причин, обеспечивающих их востребованность. Эти измерения на порядки дешевле космических и радиолокационных. Поэтому есть возможность обеспечить требуемую

массовость наблюдений (в этом отношении наземные астрометрические наблюдения многократно превосходят космические). К тому же в ряде случаев они позволяют достичь конкурентоспособной точности относительно космических и радарных измерений.

Изложенное позволяет говорить о том, что развитие методов наземных аст-рометрических наблюдений востребовано в современной астрономии. Наше исследование в значительной мере мотивировано необходимостью полнее использовать потенциал наземных телескопов ради повышения качества наземных астрометрических наблюдений. Проведение исследований в этом направлении невозможно без учета того, что развитие астрометрии наших дней в значительной мере определяет успех космической миссии Gaia. Помимо беспрецедентной по точности реализации опорной системы, этот проект вносит весомый вклад в изучение динамики тел Солнечной системы, в том числе астероидов и спутников больших планет. Однако число отдельных наблюдений данных объектов в проекте Gaia относительно невелико (порядка 50 положений на уровне точности около 1 mas) для примерно 15-ти тысяч астероидов 1. Поэтому, прежде всего, вклад Gaia в изучение динамики астероидов (особенно сближающихся с Землей) и спутников больших планет связывают с использованием релизов Gaia как опорных каталогов для калибровки ПЗС-кадров в ходе наземных оптических наблюдений. Хорошо известно, что систематические ошибки наземных ПЗС-наблюдений при таком подходе достаточно велики — десятки mas. Поэтому востребованы разнообразные методы, позволяющие выявить и/или учесть разнообразные систематические эффекты, влияющие на качество астрометрических наблюдений.

1 Приведенный анализ вклада миссии Gaia в изучение динамики Солнечной системы справедлив на момент написания этого текста (осень 2021 года). Сейчас ситуация серьезно изменилась. В июне 2022 года был представлен полный третий релиз миссии Gaia. В нем доступны как данные наблюдений (координаты, оценки блеска), так и параметры орбит для 156801 астероида [1]. Правда доминируют здесь объекты главного пояса (144975 астероидов). Доля астероидов, сближающихся с Землей (NEAs - Near Earth Asteroids), в силу особенностей сканирования небесной сферы Gaia и относительно больших скоростей движения данных объектов невелика (менее 2000 объектов. Учитывая это, можно говорить о том, что качественная привязка астрометрических измерений NEAs не потеряла актуальности, а скорее наоборот. Ведь кроме решения уталитарных задач, исследования именно этих астероидов важны для понимания вклада негравитационных эффектов в динамическую эволюцию астероидов. Доля данных для спутников больших планет в Gaia DR3 микроскопическая. Опубликованы результаты измерений для 31 спутника.

В этой связи серьезная ставка делается на наблюдения покрытий звезд астероидами и спутниками планет. Действительно, анализируя кривую блеска покрытия, можно очень надежно определить многие параметры астероида и самой звезды, включая астрометрические (координаты, скорость перемещения по небесной сфере). Прецизионное качество координат звезд в релизах Gaia гарантирует недостижимую в обычных наземных наблюдениях точность определения положений тел Солнечной системы во время таких событий. Недостаток этого метода в том, что покрытия сравнительно редки и не обеспечивают требуемого потока позиционных данных.

В качестве возможной альтернативы можно рассматривать явления, которые состоят в том, что астероид или спутник планеты в своем видимом движении по небесной сфере проходит на очень малых угловых расстояниях от какой-либо звезды Gaia. Учитывая высокую плотность распределения звезд Gaia по небесной сфере (от нескольких до сотен тысяч на квадратный градус до 19 -23 звездной величины), можно ожидать что такие события будут наблюдаться с необходимой регулярностью по всему небу. Ограничения могут быть связаны с тем, что большая разность блеска между звездами Gaia и телами Солнечной системы может свести на нет преимущества метода. Поэтому, в зависимости от скорости перемещения астероида или спутника планеты по небесной сфере, может наблюдаться от нескольких событий в час до одного на несколько ночей.

К сожалению, в русскоязычной научной литературе нет лаконичного термина, однозначно определяющего это явление. Принято говорить так: видимое тесное сближение астероида или спутника планеты со звездой. В англоязычной научной периодике сложился термин appulse, использованный Перриманом в его книге [2].

Необходимость наблюдений видимых тесных сближений тел Солнечной системы со звездами должна обосновываться не только сравнительно высокой частотой событий, но и преимуществами в точности определения координат по сравнению с традиционной астрометрической редукцией по опорным звездам.

Несмотря на то, что звезды Gaia характеризуются точностью координат порядка десятков микросекунд дуги или, в худших случаях, нескольких миллисекунд дуги, и при точности определения пиксельных координат составляющей сотые и тысячные доли пикселя (при характерных угловых масштабах совре-

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Рис. 1: Зависимость разностей относительных координат вида «наземные наблюдения минус для 451 визуально-двойной звезды из работы И.С. Измайлова [3].

менных телескопов это соответствует величине 10-20 mas) привязка к Gaia автоматически сохранит качество измерений, достигнутое в ходе аппроксимации изображений звезд и тел Солнечной системы на ПЗС-кадре, также на результат значительно влияют трудноучитываемые искажения проекции в фокальной плоскости телескопа. Поэтому основной смысл наблюдения сближений опирается на следующую идею (иллюстрацией для нее служит рис. 2). При тесном сближении систематические смещения, вызванные атмосферными эффектами и оптикой телескопа, одинаковы для объекта Солнечной системы и звезды Gaia. При этом вопрос, при каких угловых расстояниях такой подход эффективен не является простым. Несложно показать, что это сильно зависит от того при каком угловом разделении систематические искажения значимо различаются. А это, в свою очередь, сильно зависит от конкретного инструмента, зенитного расстояния и других параметров.

Некоторую оценку можно получить, погружаясь в тематику изучения визуально-двойных звезд астрометрическими методами. Точность воспроизведения опорной системы наземными астрометрическими каталогами, построенными на основе фотографических наблюдений в XX веке, в хорошем случае, составляла 0.2 угловой секунды (например, AGK3 или PPM). А ведь при построении таких каталогов производилась та же операция, что и сейчас выполняется в ходе наземных наблюдений астероидов и спутников планет-гигантов. Речь идет об астрометрической редукции, реализованной каким-либо способом: от метода Тернера до моделей, учитывающих наклонность, дисторсию и другие эффекты проекции. Одновременно с этим относительные координаты двойных звезд определялись из тех же фотографических наблюдений на уровне точности лучше 0.05 arcsec (об этом свидетельствуют данные рис. 1). Подобное расхождение вызвано тем, что при определении относительных положений двойных звезд не нужны опорные звезды уже обремененные систематическими ошибками разного вида. Точность измерения в таком случае зависит в первую очередь от определения углового масштаба изображения. Близость компонент визуально-двойных звезд друг к другу (обычно угловые разделения не превосходят 10 arcsec) минимизирует вклады различных систематических эффектов, так как они проявляются не в полную величину, а в виде разностей. Кроме того, исследуемая звездная пара обычно распологается вблизи оптического цен-

тра, где малы смещения, вызванные оптическими аберрациями, приводящими к перераспределению энергии излучения в пучке (например, комой). Логично предположить, что и при наблюдении явлений видимых тесных сближений тел Солнечной системы со звездами Сша можжно получить кардинальное улучшение точности относительных координат вида «звезда - астероид» или «звезда - спутник планеты».

В представленной работе изложены вопросы методики наблюдений видимых тесных сближений тел Солнечной системы со звездами Сша, продемонстрированы возможности этого подхода. С опорой на данные, в том числе, собственных наблюдений предпринята попытка доказать, что развиваемый метод обеспечивает значимое улучшение точности астрометрических наблюдений астероидов и спутников планет по сравнению с традиционным подходом. Обсуждается возможность выявления значимых отклонений в движении изучаемых небесных тел от эфемерид из-за неучтенных динамических эффектов.

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Рис. 2: Иллюстрация базовой идеи данного исследования. Векторное поле на рисунке демонстрирует искажения проекции телескопа в фокальной плоскости. Черные квадраты показывают два положения некоего астероида (серая линия - трек астероида в рабочем поле телескопа). Для каждого положения астероида показан вектор, отвечающей отклонению его наблюдаемого положения от истинного. Черный кружок - звезда Са1а. В момент максимального сближения астероида со звездой векторы отклонений для обоих объектов примерно одинаковы, что и обеспечивает повышение точности позиционных наблюдений.

2 Динамика спутников планет и астероидов в контексте наземных астрометрических наблюдений

2.1 Новые результаты в области изучения динамики тел Солнечной системы, полученные на основе астрометрических наблюдений

Последние два десятилетия ознаменовались заметным прогрессом в изучении динамики малых тел Солнечной системы. Реализация космических миссий к карликовым планетам («Новые горизонты», NASA JPL), к астероидам (например, Hayabusa, JAXA), к кометам («Розетта», ESA) позволила не только получить новые данные о физической природе этих небесных тел, но и пролить свет на современные вопросы небесной. механики [4].

Аналогичные вещи можно сказать о семействах спутников планет-гигантов, где с помощью космических аппаратов подтверждены механизмы приливного разогрева недр спутников [5], выявлена сложная система взаимных резонансов и миграций в системе главных спутников Сатурна (миссия «Кассини», NASA [6]).

За сравнительно короткое время открыто колоссальное количество астероидов, определены их орбитальные, а для многих тел и физические параметры, что позволило увидеть сложную картину динамической эволюции разных популяций (семейств) астероидов под действием двухтельных и трехтельных резонансов [7], эффектов фотогравитационной небесной механики (например, YORP-эффект [8]. В последние годы активно изучаются явления кометоподоб-ной активности у астероидов и их динамические следствия, проявляющиеся в движении и вращении тел с переменной массой [9].

Существенный вклад в решение рассматриваемых проблем вносят радарные наблюдения астероидов и спутников планет-гигантов. Отличительная черта таких исследований — высочайшая точность. В результате координаты тел получаются с точностью от метра до десятков метров, а точности определения скоростей составляют менее метра в секунду. Но решение небесномеханических

Рис. 3: Положения спутников Юпитера, для которых были выполнены радарные измерения. Рисунок взят из работы [10].

проблем на основе этих данных затрудняется небольшим числом измерений. Например, для Галилеевых спутников Юпитера за два последних десятилетия было сделано 22 радарных измерения (расположение точек показано на рис. 3) [10].

Развитие современных теорий движения спутников планет-гигантов сталкивается с нетривиальными вещами: физическое строение недр (вязкость вещества [11]) проявляется в орбитальной динамике. Например, в движении галиле-евых спутников Юпитера наблюдается вековое изменение среднего движения. Исследователи связывают его с приливной диссипацией энергии в системе спутников Юпитера. Обширная статья Лэни и коллег на эту тему представлена в Nature [5].

Поразительные результаты были получены командой специалистов под руководством Лэни для миграций главных спутников Сатурна [12]. Как видно из рис. 4, вследствие приливных и резонансных явлений большие орбиты этих небесных тел заметно изменяются на масштабе сотен миллионов лет. В отличие от Ио, который мигрирует по направлению к Юпитеру, спутники Сатурна наоборот непрерывно удаляются от своей планеты.

Рис. 4: Миграции главных спутников Сатурна на масштабе в сотни миллионов лет, полученные на основе векового ряда фотографических наблюдений спутников и данных аппарата Кассини. Рисунок взят из работы [12].

Перечисленные успехи — лишь небольшой набор примеров, демонстрирующих развитие приложений небесной механики к изучению Солнечной системы. В значительной мере данные исследования стали возможными благодаря развитию методов астрометрических наблюдений (от наземной ПЗС-астрометрии до радиолокации). Появление таких астрометрических каталогов как Tycho2 [13] позволило поднять уровень точности определения координат астероидов до уровня 50 - 100 mas в зависимости от блеска и скорости перемещения астероида по небесной сфере. Однако до сих пор значительное число наблюдений астероидов (порядка 80%) в базе MPC рассматривается как «астрометрический брак» ввиду того, что для привязки к ICRF использовались такие каталоги как USNO B1.0. Поэтому до сих пор развиваются работы, направленные на изучение систематических ошибок координат звезд в подобных каталогах, с целью внесения поправок в «старые» координаты астероидов с целью улучшения орбит данных небесных тел [14]. Такие усилия можно оправдать не только необходимостью изучения эволюции орбит на значительных временных интервалах, но и просто недостатком высокоточных наблюдений астероидов, отнесенных к современной опорной системе.

Рис. 5: Распределение 14 тысяч астероидов, положения которых определялись в рамках миссии Gaia и представлены во втором релизе каталога, в зависимости от величины большой полуоси орбиты. Рисунок взят из работы [17].

Развитие космической астрометрии и появление второго релиза миссии Gaia [15] заметно улучшили качество орбитальных параметров астероидов, исследованных с борта космических аппаратов [16]. В частности, в базе данных Gaia востребован каталог орбит более 14 тысяч астероидов [17]. Однако следует иметь в виду, что данная выборка хоть и большая по сравнению с предыдущими, но все равно не охватывает и малой доли всех известных астеро-идов2. Рис. 5 помогает лучше представить себе сложившуюся ситуацию. Подавляющее число астероидов, данные о которых содержатся во втором релизе Gaia, относится к главному поясу, в то время как количество астероидов, сближающихся с Землей (АСЗ или КЕЛэ) в данной выборке весьма невелико (несколько сотен). Поэтому изучение динамики популяции АСЗ невозможно без применения новых методов наземной оптической астрометрии, обеспечивающих более высокую точность.

Стоит отметить, что в силу специфики сканирования небесной сферы в миссии Gaia, наблюдения астероидов распределены неравномерно. Подавляющее

2Выше отмечалось, что данный текст был написан до выхода третьего релиза миссии Са1а. В Са1а БИЗ общее число изученных астероидов возросло на порядок [1]. Но для МЕЛв ситуация сильно не улучшилась. В этом смысле выводы, представленные в тексте, сохраняют силу.

500 400

300

200

30 20

10

Рис. 6: Распределение наблюдений астероидов во втором релизе Са1а по небесной сфере (в экваториальных координатах). Рисунок взят из работы [17].

большинство орбит астероидов, естественным образом, имеет плоскости орбит не слишком далёкие от плоскости эклиптики, но на рис. 6 отчётливо видны области со значительно большей плотностью наблюдений, распределённые с некоторым примерно равным интервалом. Такая неравномерность тоже сказывается на полноте выборки и итоговых выводах.

Еще одна трудность, обусловленная методикой наблюдений на борту космического телескопа, приводит к тому, что эллипс ошибок получается чрезвычайно узким (соотношение полуосей 100 к 1). Это проистекает из того, что Са1а очень точно измеряет дуги между звездами в направлении сканирования. В перпендикулярном направлении точность определяется только процедурой измерения пиксельных координат объектов. Более наглядно эта ситуация представлена на рис. 7. Звезды (основные объекты Са1а) двигаются очень медленно. Поэтому можно гарантировать высокую точность определения астрометриче-

Рис. 7: Различие точности определения координат небесных объектов с помощью телескопов Са1а вдоль сканирования (ЛЬ) и поперек сканирования (АС). Рисунок взят из работы [17].

ских параметров звезд за счет того, что каждый раз измерения происходят при разных значениях позиционного угла (РА на рис. 7). В случае астероидов это не получается. Выход из положения состоит в том, что можно определить орбиту, используя только АЬ-измерения. Вычисление эфемерид астероида по такой орбите обеспечит высокую точность координат данных объектов.

Наблюдения покрытий звезд астероидами и спутниками планет - еще один метод, вносящий серьезный вклад в появление результатов очень высокой точности. До некоторой степени похожий подход реализуется при наблюдении взаимных покрытий и затмений в системах спутников планет-гигантов. Неслучайно многие обсерватории участвуют в международных кампаниях по наблюдению подобных явлений. Например, в программе РНЕМШ5 по наблюдению явлений в системе галилеевых спутников участвовало более 100 обсерваторий [18]. Пример кривой блеска такого явления представлен на рис. 8.

Стоит обратить внимание на то, что высокоточные космические и радиолокационные наблюдения астероидов и спутников планет, как и данные, полученные в ходе реализации кампаний по наблюдению покрытий и взаимных затме-

Рис. 8: Пример кривой блеска, полученной на 26-дюймовом рефракторе Пулковской обсерватории в рамках международной программы РНЕМШ5, для затмения спутника Юпитера Европы спутником Ио 13 марта 2015 года.

ний охватывают сравнительно короткий интервал времени, в течение которого велись наблюдения. Это либо одна точка, либо короткая серия наблюдений. На данный момент этого недостаточно для дальнейшего прогресса в понимании динамики Солнечной системы. Наиболее плотные по времени ряды наблюдений возможны только с помощью наземных астрометрических наблюдений. Но для повышения эффективности этих наблюдений крайне важно повысить их точность до уровня 10 - 20 mas.

Резюмируя сказанное выше, отметим, что недостаточная точность наземных астрометрических наблюдений спутников планет и астероидов в настоящее время не позволяет «различить» детали динамической структуры комплекса астероидов, разделить вклады разных физических процессов, влияющих на движение и вращение спутников больших планет Солнечной системы. Стандартный подход астрометрической редукции ПЗС-кадра сталкивается с громадными трудностями на пути существенного повышения точности даже при наличии такого каталога, как Gaia DR2. Таким образом, на сегодняшний день существует необходимость поиска новых подходов к решению этой проблемы.

Чисто астрометрическая проблема - анализ взаимной ориентации опорных систем, построение которых основано на разных принципах. Gaia-CRF2 - первая система отсчета в оптическом диапазоне, основанная на абсолютных определениях углов между внегалактическими объектами (квазарами и AGN). Системы отсчета, реализуемые эфемеридами тел Солнечной системы (например, EPM, DE, INPOP) определяются тем набором положений и скоростей тел Солнечной системы, которые выбраны в качестве начальных условий интегрирования, знанием масс тел Солнечной системы и физическими моделями движения тел Солнечной системы (соответствующим набором астрономических постоянных). В идеале направления осей Gaia-CRF2 и динамических эфемерид должны совпадать на уровне десятков микросекунд дуги, а взаимное вращение систем должно характеризоваться скоростями порядка нескольких микросекунд дуги в год. Именно такими величинами характеризуется связь ICRF3 и Gaia-CRF2 [19]. Вопрос о связи динамических эфемерид и Gaia-CRF2 пока обсуждается с осторожностью. По крайней мере в фундаментальной работе по наблюдению астероидов в ходе миссии Gaia эта проблема прямо не упоминается [17]. Поэтому проблема оценки хотя бы доверительного интервала, в котором заключены

значения углов ориентации становится актуальной, аналогично тому, как это происходило после публикации каталога миссии HIPPARCOS [20]. Миссия Gaia продолжается. Появляются новые релизы. На момент написания этой работы наболее свежие данные были представлены в так называемом «раннем» третьем релизе Gaia EDR3, но новые наблюдения астероидов в нём представлены не были3.

2.2 Недостатки «стандартных» подходов к определению положении тел Солнечной системы в наземной оптической астрометрии

Современные телескопы, оснащенные ПЗС-камерами и используемые для массовой астрометрии тел Солнечной системы, имеют масштаб порядка 200 -500 mas/pix. Аппроксимация изображения астероида с помощью хорошо отработанных методик дает точность определения пиксельных координат фотоцентра близкую к 10-50 mas. Но после привязки к опорным звездам из каталога Gaia DR2 или Gaia DR3 сходимость положений по серии ПЗС-кадров обычно характеризуется ошибкой порядка 50-100 mas и хуже. Это явно говорит о наличии потенциала для существенного повышения точности астрометрических наблюдений астероидов и спутников больших планет.

Существует масса причин, ведущих с снижению точности наземных аст-рометрических ПЗС-наблюдений. Чаще всего опорные звезды распределены по ПЗС-кадру в области неба размером в десятки угловых минут. Систематические ошибки координат звезд, обусловленные атмосферной дисперсией и свойствами оптики телескопа для различных участков ПЗС-кадра могут различаться на десятки миллисекунд дуги. При недостатке опорных звезд и применении моделей

3Как указано выше в аналогичных сносках, текст данной работы был написан до выхода Gaia DR3, в котором содержатся данные наблюдений и орбитальные параметры для 156801 астероида [1]. Но, астероиды, сближающиеся с Землей, и спутники планет почти не представлены в нем. Для этих объектов высокоточные астрометрические наблюдения с Земли сохраняют свою значимость. Новая реализация опорной системы теперь именуется Gaia-CRF3. Проблематика существования возможной систематики между системами отсчета, определяемыми эфемеридами тел Солнечной системы, и Gaia-CRF3 остается актуальной. То есть выход нового релиза не привел к кардинальному изменению выводов, которые приведены в резюмирующей части этого раздела.

высокого порядка это приводит к появлению систематических ошибок, которые в массе выглядят как случайные.

Упоминание проблемы недостатка опорных звезд в эпоху Gaia, на первый взгляд, выглядит необоснованно. Но более глубокий анализ показывает, что никакого противоречия нет. Например, астероиды, сближающиеся с Землей, (АСЗ) имеют довольно большие скорости видимого движения по небесной сфере (от нескольких секунд до градусов в час). В результате при временах накопления, требующихся для достижения достаточного для измерений отношения сигнала к шуму для слабых звезд 18-ой - 20-ой звездной величины, изображение астероида «растягивается». Или, говоря более строго, изображение астероида становится суммой множества мгновенных изображений, фотоцентры которых сильно смещены относительно друг друга. То есть нарушаются условия панорамности снимка. Такие изображения довольно сложно измерять и точность определения положения фотоцентра на центральный момент экспозиции значимо падает. Уменьшение времени накопления обеспечивает более «звездные» характеристики изображения астероида, но при этом не удается накопить достаточный сигнал для слабых опорных звезд и их число и плотность покрытия рабочего поля опорными звездами пропорционально уменьшаются. Для наглядности на рис. 9 показан трек астероида 2005 VY17, накопленный в ходе наблюдений видимого тесного сближения данного астероида со звездой Gala source id = 3436496968115120512.

Список литературы диссертационного исследования кандидат наук Бикулова Динара Александровна, 2023 год

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The Central Astronomical Observatory of the Russian Academy of Sciences at Pulkovo

Manuscript

Bikulova Dinara Aleksandrovna

The detection of dynamic effects in the motion of the planetary satellites and asteroids based on observations of occultations and apparent close approaches to the Gaia stars

Candidate of Physical and Mathematical sciences

Speciality 1.3.1. Space physics, astronomy (Translation from Russian)

Scientific advisor: Candidate of Physical and Mathematical Sciences M.Yu. Khovritchev

Saint Petersburg - 2022

Contents

1 Introduction 92

2 Dynamics of satellites of planets and of asteroids in the context of ground-based astrometric observations 99

2.1 New results in the field of studying the dynamics of the bodies of the Solar System, obtained on the basis of astrometric observations ... 99

2.2 Disadvantages of "standard" approaches to determining the positions

of bodies in the Solar System in ground-based optical astrometry . . 107

2.3 Relevance of observations of visible close encounters of bodies of the Solar System with Gaia stars......................113

3 Used tools and equipment, observation process 116

3.1 Observations at the Pulkovo Astronomical Observatory RAS.....116

3.2 Observations at the Crimean Astrophysical Observatory RAS .... 117

4 The review of the methods of analysis of stars and Solar system bodies images in the CCD frame. 118

4.1 Identification of the objects in the images...............118

4.2 The calculation of pixel coordinates of the photocenters of star-like object images .............................. 118

4.3 Astrometric reduction..........................119

5 The results of the observations of stellar occultation by the asteroid 87 Sylvia 123

6 The results of appulse observations of selected NEAs and Gaia's stars 129

7 The report of the observations of giant planets' natural satellites and Gaia's stars appulses 132

7.1 Phenomena of Uranus's moons and Triton Neptune's natural satellite 132

7.2 The results of natural satellites position measurements using the "traditional" methods............................144

7.3 Saturnian satellite close approaches to the Gaia stars during 35 years

of the Pulkovo photographic observations...............151

8 Conclusion 156

9 Acknowledgements 159 Bibliography 169

1 Introduction

The understanding of the Solar system, its structure, formation and evolution, as well as the study of its distinct regions are one of the most important tasks in astronomy. In recent decades, a considerable amount of data has been accumulated due to space missions and high-tech astronomical observations in all ranges of the electromagnetic spectrum. The study of complex resonance systems, mutual tides, the impact of light-pressure effect and anisotropic reradiation plays a huge role in understanding the dynamic processes that take place in the systems of natural satellites and asteroid populations. As before, the improvement of ephemerides remains relevant for both fundamental and applied research (to ensure navigation and spaceflight reliability). Asteroid and comet hazard is another major issue in the modern astronomy.

This research paper has emerged from the need to study the named set of dynamic effects. At the current stage, it is considered important to identify the effects which haven't not been described by the models that underlie the modern ephemerides. Therefore, the motion of natural satellites of Saturn, Uranus, and Neptune are the main focus of this research. The study of near-Earth asteroids population also raises difficult dynamic questions. For example: how does one differentiate the effects caused by resonance from YORP-effect, collisions, duality, etc. Thus, a part of this research paper focuses on the analysis of steroid observations.

Astrometric observations provide the observational basis for further studies of Solar system bodies dynamics. By astrometric observations we mean all types of positional observations: radar, coordinates and speed measurements obtained from spacecrafts, traditional ground-based optical observations. The significance of high-tech measurements is constantly growing but the scientific contribution of ground-based observations remains relevant. There are several reasons for their importance and relevance. These measurements are much cheaper than the space and radar ones. Therefore, it is possible to provide a large set of observations (in this regard, ground-based astrometric observations surpass the space ones). Moreover, in certain cases they enable to achieve accuracy comparable to space and radar measurements.

It suggests, therefore, that the development of ground-based astrometric observation methods is essential for modern astronomy. The research is largely motivated by the need to fully exploit the potential of the telescopes with the aim to improve the quality of ground-based astrometric observations. The research in this area cannot be complete without acknowledging the success of a global space astrometry mission Gaia which greatly contributes to the development of modern astrometry. Besides its reference system which is unprecedented in terms of implementation accuracy, this project also makes a significant contribution to the study of the Solar system bodies dynamics. Nevertheless, the number of individual observations within the Gaia project is relatively low (around 50 positions on the precision level of 1 mas) for around 15 thousand asteroids 15. Therefore, Gaia's contribution to the study of asteroids (especially Near Earth Asteroids) and natural satellites of giant planets is primarily linked to the use of Gaia's releases which serve as reference catalogs for CCD image calibration during ground-based optical observations. It is well known that the systematic errors of ground-based CCD observations when using this approach is considerably high and amounts to tens of mas. Thus, various methods that allow to identify and take into account different systematic effects which affect the quality of astrometric observations are required.

In this regards, we pin our high hopes on the observations of (stellar) occultation. Indeed, by analysing the light curve during occultation it is possible to clearly determine many properties of an asteroid and a star including the astrometric ones (coordinates, speed of movement across the celestial sphere). Precision quality of stellar coordinates in Gaia 's releases ensures the accuracy of Solar system bodies position measurements during such events which is unattainable in conventional

15The above analysis of the contribution of the Gaia mission to the study of the dynamics of the Solar system is correct at the time of writing this text (autumn 2021). Now the situation has seriously changed. In June 2022, the full third release of the Gaia mission was presented. It contains both observational data (coordinates, brightness estimates) and orbital parameters for 156801 asteroid [1]. True, objects of the main belt (144975 asteroids) dominate here. The proportion of asteroids approaching the Earth (NEAs - Near Earth Asteroids), due to the peculiarities of scanning the Gaia celestial sphere and the relatively high speeds of movement of these objects, is small (less than 2000 objects. Taking this into account, we can say that the qualitative binding of NEAs astrometric measurements has not lost actuality, but rather vice versa.After all, in addition to purely solving utilitarian problems, studies of precisely these asteroids are important for understanding the contribution of non-gravitational effects to the dynamic evolution of asteroids. The share of data for satellites of large planets in Gaia DR3 is microscopic. Measurement results for 31 satellites have been published.

ground-based observations. The disadvantage of this method is that occultations are relatively rare and do not provide an adequate amount of positional data.

Alternatively, one can observe the events during which an asteroid or a natural satellite in their apparent motion through the celestial sphere passes at very small angular distances from any Gaia star. Given high density distribution (from several to hundreds of thousands per square degree to 19-23 magnitude) of Gaia's stars, one can expect that such phenomena will be observed throughout the sky with sufficient regularity. One of the limiting factors is that large magnitude difference between Gaia's stars and other Solar system bodies can outweigh the advantages of this method. Therefore, depending on an asteroid's or a natural satellite's speed movement one can expect to observe from several events an hour to one event during several nights.

Unfortunately, in Russian scientific literature there is no appropriate scientific term to describe that phehomenon. It is commonly referred to as (literally) "apparent close approach of an asteroid or a natural satellite to a star". In English scientific literature there is a term "appulse" used by Perryman in his book [2].

The relevance of the observations of Solar system bodies close approach to the stars is substantiated by relatively high frequency of events and higher accuracy of position measurements compared to traditional astrometric reduction using reference stars.

Given that the accuracy of Gaia's stars coordinates is around tens of microseconds of arc or, at worse, several milliseconds of arc and that accuracy of pixel coordinates measurements amounts to hundredths and thousandths of a pixel (with a typical angular scales of modern telescopes it equals the magnitude of 10-20 mas), the reference to Gaia will automatically preserve measurement quality that has been achieved during the approximation of images of stars and Solar system bodies in a CCD frame. The results are substantially affected by the projection distortions in the focal plane of the telescope which can hardly be addressed. Therefore, the purpose of appulse observations is predicated on the following idea (see figure . 2). During the appulse systematic drifts caused by atmospheric effects and telescope optics are the same for a Solar system object and a Gaia's star. At the same time, it is difficult to say at which angular distances such approach is efficient. It is easy to show that this largely depends on angular separation at which systematic distor-

arcsec

Figure 1: The dependence of the differences in relative coordinates «ground-based observations minus Gaia» for 451 visual binary stars taken from the I.S. Izmailov's paper [3].

tion are detected. And this, in turn, is largely determined by the use of a specific instrument, zenith distance and other parameters.

Some estimations can be made if one delves into the study of visually binary stars with the use of astrometric methods. The accuracy of the reference system reproduced by ground-based astrometric catalogs, which were constructed on the basis of photographic observations in XX century, accounted for 0.2 arcseconds at best (for example, AGK3 or PPM). Interestingly though, catalogs construction involved the same operation which is now performed during ground-based observations of asteroids and natural satellites of the giant planets. This refers to an astrometric reduction which is performed by any method starting from Turner method to models that take into account inclination, distortion and other projection effects. At the same time, relative coordinates of binary stars were derived from the same photographic observations at the precision level higher than 0.05 arcsec (This is evidenced by the data in figure 1). The cause of this inconsistency is the fact that there is no need for reference stars to determine the relative position of the binary stars as the former are already burdened with systematic errors of different kinds In this case, the accuracy of measurements depends primarily on the image angular diameter. The proximity of the components of visually binary stars (usually angular separations do not exceed 10 arcsec) minimizes the contribution of different systematic effects since they are expressed not in absolute value but in the form of differences. In addition, the studied pair of stars is usually located close to the optical center, where drifts caused by optical aberrations which lead to redistribution of radiation energy in the beam (for example, coma) are insignificant. It is reasonable to assume that significant improvement in relative coordinates of the type "star-asteroid" or "star-natural satellite" can be achieved during the observation of appulse of Solar system bodies and Gaia's stars.

The paper discusses the methodological issues surrounding the observations of the appulse of Solar system bodies and Gaia's stars and highlights the potential of this approach. On the basis of data and personal observations we have made an attempt to prove that the developed method significantly improves the accuracy of astrometric observations of asteroids and natural satellites compared with the traditional method. The possibility of identifying significant deviations in the movement

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Figure 2: An illustration of the basic idea of this study. The vector field in the figure shows distortions of the telescope projection in the focal plane. The black squares show two positions of a certain asteroid (the gray line is the track of the asteroid in the telescope's field of view). A vector that demonstrates the deviation between the observed and true position of the asteroid is shown. The black dot is the Gaia star. At the moment of the asteroid's closest approach to the star, the deviation vectors for both objects are approximately the same, which ensures an increase in the accuracy of positional observations.

of the studied celestial bodies from ephemerides due to unaccounted dynamic effects is discussed.

2 Dynamics of satellites of planets and of asteroids in the context of ground-based astrometric observations

2.1 New results in the field of studying the dynamics of the bodies of the Solar System, obtained on the basis of astrometric observations

The last two decades have been marked by significant progress in the study of the dynamics of small bodies in the Solar System. The implementation of space missions to dwarf planets (New Horizons, NASA JPL), to asteroids (for example, Hayabusa, JAXA), to comets (Rosetta, ESA) made it possible not only to obtain new data about the physical nature of these celestial bodies, but also shed light on modern issues of celestial mechanics [4].

Similar can be said about the families of satellites of giant planets, where with the help of spacecraft the mechanisms of tidal heating of the interiors of satellites were confirmed [5], a complex system of mutual resonances and migrations was revealed in the system of the main satellites of Saturn (the Cassini mission, NASA [6]).

In a relatively short time, a colossal number of asteroids was discovered, their orbital and, for many bodies, their physical parameters were determined, which made it possible to see a complex picture of the dynamic evolution of different populations (families) of asteroids under the influence of two-body and three-body resonances [7], the effects of photogravitational celestial mechanics (for example, the YORP effect [8]. In recent years, the phenomena of comet-like activity in asteroids and their dynamic consequences, manifested in the motion and rotation of bodies with variable mass, have been actively studied. [9].

A significant contribution to the solution of the problems under consideration is made by radar observations of asteroids and satellites of giant planets. A distinctive feature of such studies is the highest accuracy. As a result, the coordinates of bodies are obtained with an accuracy from a meter to tens of meters, and the accuracies of determining velocities are less than a meter per second. But the solution of celestial mechanical problems based on these data is hampered by a small number of

Figure 3: Positions of Jupiter's satellites for which radar measurements have been made. Figure is taken from [10].

measurements. For example, for the Galilean satellites of Jupiter, 22 radar measurements have been made over the past two decades (the location of points is shown in Fig. 3) [10].

The development of modern theories of the motion of satellites of giant planets is faced with non-trivial things: the physical structure of the bowels (substance viscosity [11]) is reflected in the orbital dynamics. For example, in the motion of the Galilean satellites of Jupiter, a secular change in the mean motion is observed. Researchers attribute it to tidal energy dissipation in Jupiter's satellite system. An extensive paper by Lainey and colleagues on this topic is presented in Nature [5].

Striking results have been obtained by a team of specialists led by Lainey for the migrations of the main moons of Saturn [12]. As can be seen from fig. 4, due to tidal and resonance phenomena, the large orbits of these celestial bodies change noticeably on a scale of hundreds of millions of years. Unlike Io, which migrates towards Jupiter, Saturn's moons, on the contrary, are continuously moving away from their planet.

These successes are just a small selection of examples demonstrating the development of applications of celestial mechanics to the study of the Solar system. To a large extent, these studies have become possible due to the development of astro-

Figure 4: Migrations of the main moons of Saturn on a scale of hundreds of millions of years, obtained on the basis of a century-long series of photographic observations of satellites and data from the Cassini orbiter. Figure is taken from [12].

metric observation methods (from ground-based CCD astrometry to radar). The appearance of such astrometric catalogs as Tycho2 [13] made it possible to raise the level of accuracy in determining the coordinates of asteroids to the level of 50 -100 mas, depending on the brightness and speed of the asteroid moving across the celestial sphere. However, until now a significant number of observations of asteroids (about 80%) in the MPC database are considered as "astrometric defect" due to the fact that catalogs such as USNO B1.0 were used to link to ICRF. Therefore, works are still being developed aimed at studying systematic errors in the coordinates of stars in such catalogs, in order to amend the "old" coordinates of asteroids in order to improve the orbits of these celestial bodies [14]. Such efforts can be justified not only by the need to study the evolution of orbits over significant time intervals, but also simply by the lack of high-precision observations of asteroids related to the modern reference system.

The development of space astrometry and the appearance of the second release of the Gaia mission [15] have significantly improved the quality of the orbital parameters of asteroids studied from spacecraft [16]. In particular, the catalog of orbits of more than 14 thousand asteroids is in demand in the Gaia DR2 database [17]

Figure 5: Distribution of 14,000 asteroids, whose positions were determined within the framework of the Gaia mission and presented in the second release of the catalog, depending on the magnitude of the semi-major axis of the orbit. Figure is taken from [17].

However, it should be borne in mind that this sample, although large compared to the previous ones, still does not cover even a small fraction of all known asteroids16. Fig. 5 helps to better imagine the current situation. The vast majority of asteroids, data on which are contained in the second release of Gaia, belongs to the main belt, while the number of near-Earth asteroids (NEAs) in this sample is very small (a few hundred). Therefore, the study of the dynamics of the NEA population is impossible without the use of new methods of ground-based optical astrometry, which provide higher accuracy.

It should be noted that due to the specifics of scanning the celestial sphere in the Gaia mission, observations of asteroids are unevenly distributed. The vast majority of asteroid orbits naturally has orbital planes not too far from the ecliptic plane, but fig. 6 clearly shows regions with a much higher density of observations, distributed with some approximately equal interval. Such an unevenness also affects the completeness of the sample and the final conclusions.

16It was noted above that this text was written before the release of the third release of the Gaia mission. In Gaia DR3, the total number of studied asteroids has increased by an order of magnitude [1]. But for NEAs, the situation has not improved much. In this sense, the conclusions presented in the text remain valid.

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Figure 6: Distribution of asteroid observations in the second release of Gaia over the celestial sphere (in equatorial coordinates). Figure is taken from [17].

Figure 7: The difference in the accuracy of determining the coordinates of celestial objects using Gaia telescopes along the scan (AL) and across the scan (AC). Figure is taken from [17].

Another difficulty caused by the technique of observations on board of the space telescope leads to the fact that the error ellipse is extremely narrow (the ratio of the semi-axes is 100 to 1). This stems from the fact that Gaia very accurately measures the arcs between stars in the direction of scanning. In the perpendicular direction, the accuracy is determined only by the procedure for measuring the pixel coordinates of objects. This situation is shown more clearly in fig. 7. Stars (the main objects of Gaia) move very slowly. Therefore, it is possible to guarantee high accuracy in determining the astrometric parameters of stars due to the fact that each time the measurements are made at different values of the position angle (PA in Fig. 7). In the case of asteroids, this does not work. The way out is that the orbit can be determined using only AL measurements. Calculation of the asteroid ephemeris in such an orbit will provide high accuracy of the coordinates of these objects.

Observations of occultations of stars by asteroids and satellites of planets are another method that makes a serious contribution to the appearance of very high accuracy results. To some extent, a similar approach is implemented when observ-

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Figure 8: An example of the light curve obtained with the 26-inch refractor of the Pulkovo Observatory within the framework of the international PHEMU15 program for the eclipse of Jupiter's moon Europa by Io's moon on March 13, 2015.

ing mutual occultations and eclipses in satellite systems of giant planets. It is no coincidence that many observatories participate in international campaigns to observe such phenomena. For example, more than 100 observatories participated in the PHEMU15 program for observing phenomena in the system of Galilean satellites [18]. An example of the light curve of such a phenomenon is shown in fig. 8.

It is worth paying attention to the fact that high-precision space and radar observations of asteroids and satellites of planets, as well as data obtained during the implementation of campaigns for observing occultations and mutual eclipses, cover a relatively short time interval during which observations were made. This is either a single point or a short series of observations. At the moment, this is not enough for further progress in understanding the dynamics of the Solar System. The most dense time series of observations are possible only with the help of ground-based astrometric observations. But to improve the efficiency of these observations, it is extremely important to increase their accuracy to the level of 10 - 20 mas.

Summarizing the above, we note that the insufficient accuracy of ground-based astrometric observations of satellites of planets and asteroids at present does not allow us to "distinguish" the details of the dynamic structure of the asteroid complex,

to separate the contributions of various physical processes that affect the motion and rotation of satellites of the major planets of the Solar System. The standard approach of astrometric reduction of a CCD frame faces enormous difficulties on the way to a significant increase in accuracy even with a catalog such as Gaia DR2. Thus, today there is a need to find new approaches to solving this problem.

A purely astrometric problem is the analysis of the mutual orientation of reference systems, the construction of which is based on different principles. Gaia-CRF2 is the first reference frame in the optical range based on absolute definitions of angles between extragalactic objects (quasars and AGNs). The reference systems implemented by the ephemerides of the Solar System bodies (for example, EPM, DE, INPOP) are determined by the set of positions and velocities of the Solar System bodies that are chosen as the initial integration conditions, by the knowledge of the masses of the Solar System bodies and by physical models of the motion of the Solar System bodies (the corresponding set of astronomical constants). Ideally, the directions of the Gaia-CRF2 and dynamic ephemeris axes should coincide at the level of tens of microarcseconds, and the mutual rotation of the systems should be characterized by velocities of the order of several microarcseconds per year. It is these values that characterize the relationship between ICRF3 and Gaia-CRF2 [19]. The question of the relationship between dynamic ephemeris and Gaia-CRF2 is still being discussed with caution. At least in the fundamental work on observing asteroids during the Gaia mission, this problem is not directly mentioned [17]. Therefore, the problem of estimating at least the confidence interval in which the values of orientation angles are contained becomes relevant, similarly to how it happened after the publication of the HIPPARCOS mission catalog [20]. The Gaia mission continues. New releases appear. At the time of this writing, the most recent data was presented in the so-called "early" third release of Gaia EDR3, but new observations of asteroids were not presented in it17.

17As mentioned above in similar sources, the text of this work was written before the release of Gaia DR3, which contains data on the expected and orbital parameters for the 156801 asteroid [1]. But, near-Earth asteroids and planetary satellites are almost not represented in it. For objects of high-precision astrometric observations, observations retain their memory. The new reference system implementation is now called Gaia-CRF3. The problem of a possible consistency between the origin of the reference, the observed ephemerides of the bodies of the Solar System, and Gaia-CRF3 remains relevant. That is, the release of a new release that did not arrive at the representative of the collection of conclusions, which are presented in the summary of this section.

2.2 Disadvantages of "standard" approaches to determining the positions of bodies in the Solar System in ground-based optical astrometry

Modern telescopes equipped with CCD cameras and used for mass astrometry of the bodies of the Solar System have a scale of about 200 - 500 mas/pix. Approximation of the asteroid image using well-established techniques gives the accuracy of determining the pixel coordinates of the photocenter close to 10 - 50 mas. But after reference to the reference stars from the Gaia DR2 or Gaia DR3 catalog, the convergence of positions from a series of CCD frames is usually characterized by an error of the order of 50-100 mas and worse. This clearly indicates the potential for a significant increase in the accuracy of astrometric observations of asteroids and satellites of major planets.

There are many reasons leading to a decrease in the accuracy of ground-based astrometric CCD observations. Most often, reference stars are distributed over a CCD frame in a region of the sky with a size of tens of arc minutes. Systematic errors in stellar coordinates due to atmospheric dispersion and the properties of telescope optics for different parts of the CCD frame can differ by tens of milliarcseconds. With a lack of reference stars and the use of high-order models, this leads to the appearance of systematic errors, which in the mass look like random ones.

Mentioning the problem of the lack of reference stars in the Gaia era, at first glance, looks unreasonable. But a deeper analysis shows that there is no contradiction. For example, near-Earth asteroids (NEA) have rather high speeds of apparent movement in the celestial sphere (from several seconds to degrees per hour). As a result, at the accumulation times required to achieve a signal-to-noise ratio sufficient for measurements for faint stars of the 18th - 20th magnitude, the image of the asteroid "stretches". Or, speaking more strictly, the image of an asteroid becomes the sum of many instantaneous images, the photocenters of which are strongly displaced relative to each other. I.e., the conditions of the panoramic image are violated. Such images are rather difficult to measure, and the accuracy of determining the position of the photocenter at the central moment of exposure decreases significantly. Reducing the accumulation time provides more "stellar" characteristics of the asteroid image, but at the same time it is not possible to accumulate a sufficient signal for

Figure 9: An illustration of a typical phenomenon of a visible close approach of a Solar System body to the star Gaia. The left panel of the figure shows the result of adding 60 CCD frames centered relative to the star Gaia source id = 3436496968115120512, Gmag = 16.223. Two tracks near the center are the corresponding images of asteroid 2005 VY17 (mag = 16.4) projected onto one frame. The accumulation time of each frame is 5 seconds. The break in the track is due to weather conditions. The right panel shows the star image structure (in the center) and examples of asteroid images from three separate CCD frames. The survey was carried out using the Pulkovo 1-meter telescope «Saturn» on November 24, 2019. More detailed results of observations of this phenomenon are given in Table 4 and Table 5.

weak reference stars, and their number and the coverage density of the working field by reference stars decrease proportionally. For clarity, fig. 9 shows the track of asteroid 2005 VY17 accumulated during observations of the apparent close approach of this asteroid to the star Gaia source id = 3436496968115120512.

For the main satellites of the giant planets, the accumulation time is limited by the presence of the image of the planet itself in the working field. A halo of scattered light from the planet creates gradients that make it difficult to measure satellite positions. Here, reducing the accumulation time also becomes a natural solution to the problem, but at the cost of losing the quantity and quality of reference stars. Fig. 10 illustrates this situation. To reduce the effect of background gradients on astrometric measurements, different techniques are used. Median filtering was used in the paper [21], where the figures 10 and 11 were taken from.

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Figure 10: An illustration of the influence of scattered light from a planet on the quality of satellite images. On the left there is the original image, on the right there is the result of applying the median filtering algorithm. The observations were made with the Faulkes Telescope North on September 3, 2007. Figure is taken from [21].

Under such conditions, astrometric reduction using high-order models that take into account various effects is inappropriate. Increasing the order of the model, of course, reduces the magnitude of the unit weight error, but at the cost of reducing the number of independent rows in the redundant system of equations. The consequences of the unjustified application of such models are truly terrible. They are easy to see if we calculate the residuals of recent NEA observations published in the database of the International Minor Planet Center 18. All kinds of observations of asteroids are published in this database, from those obtained in specialized observatories to amateur ones. In the following, only Gaia data provided by reliable sources, such as the asteroid-specializing PanSTARRS system, are considered. Tables 1 and 2 clearly illustrate the situation. The error values for modern NEA observations often turn out to be close to 0.5 arc seconds.

As can be seen from the tables 1 and 2 the individual (O-C) for individual observatories are large and differ significantly. Large values (-) for individual observation points could be attributed to the imperfection of the NEA ephemeris. But then it is impossible to explain the differences (O-C) of hundreds of mas for parallel observations of asteroids from different observatories performed simultaneously. If comparisons are made with the ephemeris for dozens of observatories, the mean values (O-C) decrease markedly, but the variance remains too large. Thus, serious dynamic studies based on observational data are impossible. They are suitable only

18https://minorplanetcenter.net

Figure 11: An illustration of the influence of scattered light from a planet on the quality of satellite images. Image profile of Uranus and its satellites from [21] before and after taking into account the background gradient.

Table 1: Values (O-C) for asteroid 65690 (1991 DG) according to the results of observations by observatories with MPC code L18 and 558.

MPC code (O-C)a (O-C), N

L18 0.158 0.468 -0.137 0.396 143

557 0.131 0.113 0.132 0.161 20

The moments of time of observations for these points differ from the times of our observations of visible close encounters within seven days. N is the number of observations.

Table 2: Mean (O-C) for five near-Earth asteroids observed by different observatories in 2019-2020 and published in MPC.

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arcsec

243025 (2006 UM216) -0.073 0.263 0.088 0.117

162723 (2000 VM2) 0.092 0.559 0.087 0.312

333555 (2005 VY17) 0.021 0.084 -0.015 0.067

35107 (1991 VH) 0.006 0.217 0.098 0.426

65690 (1991 DG) 0.101 0.443 -0.027 0.490

Figure 12: The field of generalized distortion for the WISE telescope based on the results of the work [23].

for supporting ephemeris at a level that allows not to lose the object. An analysis of asteroid observations using, for example, PanSTARRS, presented in [22], gives a disappointing assessment. The accuracy of observations does not exceed 100 mas even when using Gaia. From this analysis, it is clear that the point is the incorrect accounting for systematic errors in the projection of the telescope, coupled with atmospheric dispersion and anomalous refraction. Below there is a brief description of the main strategy for accounting for systematic errors in stellar coordinates.

During the construction of astrometric catalogs of the first two decades of this century (Tycho-2, UCAC4), the method of constructing the field of residual differences (FDP - Field Distortion Pattern) was widely used. I.e., if it is impossible to adequately determine the coefficients in front of high-order terms from one frame, then it is reasonable to perform the reduction by the linear method and calculate the residual differences in the coordinates of the reference stars. If this procedure is carried out for hundreds of thousands frames, then it is easy to average all the differences that fall into the interval in terms of the brightness of the stars and the position in the working field of the telescope in both coordinates. They can

be considered as components of correction vectors from coordinates obtained by the method of six constants to true coordinates (taking into account distortions of higher orders). Fig. 12 gives an example of such a correction field for the WISE space telescope obtained in [23]. The use of such an approach seems justified if the FDP is relatively constant over time, which is not the often case in reality. Nevertheless, the construction of such distortion maps and their use makes it possible, at least on average, to take into account the influence of a complex set of factors on the results of astrometric observations.

The analysis of "established" methods of ground-based CCD observations of asteroids and satellites of planets and the results of their implementation, presented in this section, shows that they need to be revised in the Gaia era. New approaches must be proposed to realize the potential of telescopes and cameras. And this process in different observatories of the world proceeds quite dynamically. For example, Zhai Chengxing and colleagues [24] proposed the method of synthetic tracking technique based on a series of short (subsecond exposures) and subsequent addition of images with appropriate deformations. Our work corresponds to this line of research, but is based on a slightly different approach.

2.3 Relevance of observations of visible close encounters of bodies of the Solar System with Gaia stars

In the 1990s, in the era of implementation and understanding of the results of the HIPPARCOS mission, the idea of observing not only occultations of stars by asteroids and satellites of planets (phenomena, as noted in the introduction, is relatively rare for a particular object), but also visible close encounters with stars HIPPARCOS has been expressed many times (e.g. [25]). When the images of the star Gaia and the asteroid are close, the systematic changes in the coordinates of both objects are close in magnitude, which makes it possible to obtain relative positions at the level of accuracy up to several milliseconds of arc. However, taking into account the low density and significant brightness of the HIPPARCOS stars, it was not possible to fully realize the potential of this approach.

It is important to pay attention to the fact that the motivation of astronomers who focused the attention of their colleagues on the fact that it is reasonable to

observe visible close encounters is fundamentally different from the current one. The post-HIPPARCOS era is the very small operating fields of CCD cameras for most instruments (typically less than 10 arc minutes). HIPPARCOS stars sparsely fill the celestial sphere (1-3 stars per square degree). Therefore, the asteroid under study did not always fall into the working field of the telescope together with the HIPPARCOS star. The accuracy of the HIPPARCOS star coordinates was 10 to 100 times higher than the accuracy of the positions of stars in other catalogs, which ensured sufficient filling of the celestial sphere.

Similar considerations motivate astrometrists to observe visible close encounters of asteroids with quasars included in the ICRF lists - ICRF3 [26]. The subtlety here is that the study of the dynamics of asteroids based on such observations is presented as a secondary task. At the forefront is the problem of the connection of the dynamic reference system, represented by the ephemerides of asteroids, and the radio system, the ICRF itself.

We adhere to a different goal-setting in our constructions - for us, the visible close encounters of the bodies of the Solar System with the stars of Gaia are a tool for obtaining an exhaustive accuracy of the coordinates of asteroids and satellites of planets, both randomly and systematically.

The appearance of Gaia DR2, which ensured the colossal accuracy of the positions of faint stars and the high density of their distribution over the celestial sphere, makes it possible to turn again to the "apparent close encounters" [27] method. The release - Gaia EDR3 - provides even higher accuracy of faint star positions, further enhancing the potential of the visible close approach method19.

Preliminary calculations have shown that the phenomena of visible close encounters occur quite frequently (sometimes dozens of times during an observational night). Analysis of the results of these observations reduces to the well-developed problem of determining the relative positions of visual binary stars. For example, we have developed a method based on the shapelet decomposition of images of closely spaced images of celestial bodies, which allows us to avoid systematic errors in de-

19 The release of the full-fledged Gaia DR3 did not change the situation, since the astrometric catalog was simply transferred from Gaia EDR3 to Gaia DR3 without changes [28]. The difference between Gaia DR3 and the previous release is the publication of data for the bodies of the Solar System, binary stars, and photometric and spectrometric data for the studied stars.

termining the angular separation and position angle for interacting images [29]. All this makes the proposed approach very promising.

3 Used tools and equipment, observation process

As evidenced in practice, the observations should be made in advance, if possible, an hour (or more) before the phenomenon. They also should be finished accordingly. Phenomena that occur in close proximity (p < 3 * FWHM) require taking into account the distortions caused by the superposition of star and object images. More distant phenomena do not provide sufficient accuracy. Data on the appulse of asteroids and Gaia's stars were retrieved from the internet-page Pulkovo NEO Page20. This service was set up on the basis of software package Ephemerical Program for Solar System Objects created at the Pulkovo Observatory [30].

The moment of the appulse was also computed by using a special code written for this purpose. We used data from the following service MULTISAT21 Sampling procedure was conducted with regard to stellar magnitude of the observed object. The difference between the star and the observed object which approaches it must not exceed 2 stellar magnitudes. Otherwise the shooting settings that enable to obtain high-quality image of an object will not be suitable for the star and vice versa. Asteriods which are approaching the Earth are weak objects, therefore, it is more reasonable to observe them from the orbit point closest to the Earth.

3.1 Observations at the Pulkovo Astronomical Observatory RAS

Observations at the Pulkovo Astronomical Observatory were performed using a meter-long mirror telescope «Saturn» (D = 1 m, F = 4). During the first year of observations we used SBIG ST L-11K installed in the main mirror focus. This created a 30x20 working area that have a scale of 0.460 arcseconds/pixel. Visible distortion of the image caused by the aberrations can be neglected in case of appulse observations which is one of the advantages of the method. Nevertheless, the results improve when the image of an observed object is proximal to the center of the frame since optical distortions are more salient towards the edges. Later, the SBIG camera was replaced by the ZWO ASI1600 Pro camera. Field of view is 13x10, arcminutes,

20http://www.gaoran.ru/personal/neo/rus/general/p01.htm

21http://www.sai.msu.ru/neb/nss/html/multisat/nssima0r.htm [31] and https://ssd.jpl.nasa.gov/ horizons.cgi

the scale equals 172 mas/pix. Long series of images (hundreds of them) with the exposures of several seconds (1-10) were taken. In 2020, the observations were performed from August to December. In total, around 8 thousand of independent observations of Uranus's natural satellites (Ariel, Umbriel, Titania, Oberon) and Triton, Neptune's natural satellite, were made.

3.2 Observations at the Crimean Astrophysical Observatory RAS

Observations at the Crimean Astrophysical Observatory RAS (CrAO RAS) were performed with the Newtonian telescope (D=350 mm, F= 1767 mm). Telescope tube is parallel to the main tube of a meter telescope «Sintez» (it can be reconstructed) and is used for guidance system debugging. It is located in the building with a sliding roof and is equipped with a self-made semi-automatic guidance system. Camera with KAF-8300M sensor was used as a receiver, pixel size was 5.4 micron, scale was 630 mas/pix, working area was 35x26 arcminutes, accumulation time amounted to 45 seconds. The observations of the natural satellites were conducted from mid-November to early December at the CrAO RAS and amounted to 170 observations of Triton and 500 observations of Uranus's natural satellites (Titania, Oberon).

4 The review of the methods of analysis of stars and Solar system bodies images in the CCD frame.

4.1 Identification of the objects in the images

We used Sextractor [32] for automatic detection of the images of star-like objects in frames and offline version astrometry.net [33] to identify the stars (we identified them with Gaia's catalog data). As a result, WCS parameters which enable to turn pixel coordinates of the image photo centers into the equatorial coordinates of the points on the celestial sphere and back were added to each CCD image. HDU with the data from the Gaia EDR3 catalog and ephemerides of asteroids or natural satellites were added to each image to exclude multiple database requests through the Internet during multiple image processing.

4.2 The calculation of pixel coordinates of the photocenters of star-like object images

Preliminary studies showed that in our case (for CCD images obtained with the tool used for this research) the best measurement results enable to construct PSF which reflects all the specific features of star-like object images in certain location of a CCD image. The new approach means the transition of methods applied to the analysis of galaxies images and weak leansing. By this we mean the possibility of using shapelet decomposition of astronomical object images. It enables to represent an image as a set of coefficients that "restore" all the details of an original image with the help of orthogonal functions system - (shapelets). The example of such decomposition is shown in figure 13). Therefore, an image of a star, an asteroid or a natural satellite is defined by a set of shapelet coefficients for a specific model order, background value, the flow and coordinates of the photocenter which have to be determined. As shapelet coefficients for PSF are known, it is easy to vary the last four parameters. Minimization can be done in any convenient way (for example, by using Nelder-Mead or Levenberg-Marquardt methods). As a result of these calculations, data relevant to the future research can be obtained. It includes sets of

Figure 13: Demonstration of the quality of the shaplet decomposition of the typical stellar image. The top panel shows the raw image, the middle panel contains the image constructed using the shapelet coefficients, and the bottom panel is the result of subtraction of the reconstructed image from the raw one.

relative coordinates "asteroid-Gaia star" for the moments of time near phenomenon central moment. We would like to underscore that it is the difference in the pixel coordinates of the objects that is computed. In case of particularly close appulse (which means that photo centers differ less than the tripled value of FWHM) "two-point" approximation algorithm is used. This allows to vary the position of the photo center, the flows and background for both objects at once [29].

4.3 Astrometric reduction

Astrometric alignment traditionally requires to use the coordinates of the stars from the star catalog distributed throughout the CCD image. It is believed that the more evenly the stars are distributed across an area of the sky which is shown in the CCD image, the more accurate the calibration is. As it was noted above, this statement is not always true. Systematic differences between the projection in the focal plane and the central one can be consistent with very complex models the parameters of which are unknown and can vary from image to image.

From this section 4.2 it is clear that at the first stage we determine the locations of the photocenter of the objects of primary interest (asteroids, natural satellites, and a corresponding star) are identified. To identify the parameters of the appulse, the orientation of the images and their scale should be considered. To do this, 10 stars located at the minimum angular distances from the studied Gaia star are selected. Six constants method is used to determine the "local" scale and image rotation.

It enables to obtain tangential coordinates £ = aa cos S, v = aS "asteroid - Gaia star". It means that the equatorial coordinates of the Gaia's star for an epoch of observations (with regard to proper motion) are the center of projection.

Series of CCD images is obtained during the phenomena observations. From each image the differences of the equatorial coordinates "star - a body of the Solar system" are extracted. Commonly, motion linearity of the Solar system body relative to the Gaia's star remains unchanged within a period of several to tens of minutes (this time frame can sometimes be shorter for NEAs). By approximating the movement of an object with the help of a linear model (2) it is easy to determine when the appulse took place (tc - central moment of the event) and obtain the coordinates at this point of time. As the result, instead of a large set of equatorial coordinates we get only 4 parameters (components of relative coordinates £o,vo at the moment of to and velocities Me,Mv) which fully describe the movement near the point of the appulse.

£ = £o + Me(t - to),V = Vo + Mv(t - to). (2)

Exactly these parameters, first of all, £(tc),v(tc), are compared with the ephemeris data. If necessary, these values are easily transformed into the equatorial coordinates of an asteroid or a satellite of the planet at the moment tc.

One of the main goals of this paper is to develop methods of data analysis to determine the parameters of the appulses under the current conditions of ground-based astrometric observations. When proposing this computational schema, it is necessary to make a comparison by processing the images in accordance with the "traditional" method.

As it was noted in the section 2.2, it is usually impossible to take into account all geometric distortions during the astrometric reduction of a sinlge image. Therefore, FDP based on multiple images processing is constructed. These corrections enable to use a simple six constants model at the first stage of calculations and then make corrections to the tangential coordinates of the asteroids.

Figure 14 and Figure. 15 show FDP for the Pulkovo 1-meter telescope «Saturn» and the hydrographic telescope «Sintez» of the CrAO RAS.

Therefore, we have performed the calculations by using not only the appulse method, but also "traditional" models of astrometric reduction with reference stars.

u CD (/) U

ro

300 2001000-100-200-300

Saturnlm FDP

> f ■ -200

w * * V >

< * i V \ > ,

< / i i V » .

\ A

A A 1

T 0

200

x [arcsec]

Figure 14: Field distortion pattern of the Pulkovo 1-meter telescope «Saturn».

Figure 15: Field distortion pattern of the guiding telescope installed on the «Sintez» CrAO RAS telescope.

Table 3: Results of observations of test pairs of stars from the Gaia EDR3 catalog.

p Gmag A Gmag AC Arj ACi AVi AC2 Am av

arcsec mag arcsec

5.8 14.4 1.0 -0.045 -0.017 0.021 0.015 0.066 0.032 0.013 0.013

6.3 15.6 1.5 -0.014 -0.008 0.007 -0.013 0.021 -0.004 0.018 0.018

9.8 15.6 0.3 0.009 0.035 -0.078 0.026 -0.088 -0.008 0.019 0.019

11.3 13.3 1.4 0.009 -0.017 -0.103 -0.022 -0.111 -0.005 0.018 0.018

13.6 14.9 0.9 -0.020 -0.014 0.119 0.016 0.139 0.031 0.023 0.023

It is evident that there is a possibility to test the methods without observing the phenomena. For almost any area of sky in Gaia catalog one can find pairs of stars with the required angular distances and magnitude differences. Thus, test observations were made. Their reports are shown in table 3.

Each row of the table 3 shows the measurement data of a certain pair of stars. In total, 5 pairs with the separation ranging from 5 arcsec to 15 arcsec in a typical magnitude range and magnitude difference were explored. The values a£ a^ are O-C obtained using the appulse method (differences in pixel coordinates of image photo centers transformed into the differences of tangential coordinates with regards to the scale and rotation). The differences a£i5 a^i and a£2, a^2 are "simple" O-C computed the same way as in case with asteroids and natural satellites. It is evident how large the O-C can be (particularly from the last three lines) when they are obtained with traditional method compared with the results of relative measurements. These findings clearly demonstrate that the method of relative coordinates measurements during the appulse is adequate both in terms of accuracy and reliability of accounting for systematic effects. FDP construction requires a substantial amount of images but the "averaging" of distortions makes it impossible to exclude the specific features of a particular image series. The method of differential determinations partially solves this problem since all the calculations are related to specific series of images obtained during the event.

Figure 16: The map of the path of the shadow during the occultation of the Gaia DR2 star source id = 679856115656562048 by 87 Sylvia asteroid, 2019-10-29.

5 The results of the observations of stellar occultation by the asteroid 87 Sylvia

As it was noted in the introduction, observations of stellar occultation by the asteroids are very useful for astronomy in general. Besides an apparent opportunity to accurately determine the location of an asteroid at a definite moment in time and subsequently use this data in astrometric and Solar system bodies dynamics, it is also possible to determine the size and projection of the asteroid body with fair accuracy (to do it, observations from dozens of locations disseminated over the Earth are required). By using the light curve one can make an attempt to identify the duality of an asteroid or a star. By the way, it was the occultation of the star BD +29 1748 by the asteroid 87 Sylvia in 2006 that helped detect its duality [34].

To date, we have collected light curves for several stellar occultation by this asteroid. The event of October 29, 2019 aroused significant interest [35] since occultation

2019-10-29T23:40:32.157 2019-10-29T23:40:35.343 2019-10-29T23:40:38.528 2019-10-29T23:40:41.697 2019-10-29T23:40:44.868

2019-10-29T23:40:48.039 2019-10-29T23:40:51.274 2019-10-29T23:40:54.456 2019-10-29T23:40:57.623 2019-10-29T23:41:00.796

Figure 17: CCD frames near the central moment of the occultation of the Gaia DR2 source id =679856115656562048 by asteroid 87 Sylvia on 2019-10-29.

zone for the phenomenon crossed Western Europe where a relatively large number of observatories are located (occultation zone map is shown in figure. 16). As the result, a great number of light curves 22 including occultation of a natural satellite by an asteroid were accumulated. The observations made with large telescopes show that asteroid 87 Sylvia (385 x 265 x 230 ± 10 km) has two satellites named Romulus

18 km) and Remus (« 7 km) which have large semi-axes of orbits -1365 ± 5 km and 706 ± 5 km respectively.

At the Pulkovo Astronomical Observatory the observations were made with the telescope ZA320-M and the Pulkovo 1-meter telescope «Saturn». Weather conditions at the time were very poor (cloudy). It hindered our ability to carry out a survey at a much higher frequency. Images close to the occultation are shown in figure . 17. To accumulate a sufficient signal in «Saturn» telescope we had to use an accumulation time of 2 seconds. The star Gaia DR2 Source id = 679856115656562048 is fairly bright, Gmag = 9.89. The asteroid during the occultation was weaker than the 13th magnitude. Neighboring stars, which were used as a comparison for the light curve construction, covered the magnitude range from 11 to 13. Thus, we had to substantially reduce the temporal resolution during the shooting to obtain astrometric data which bear the most significance for this research.

The light curve obtained with «Saturn» telescope is shown in figure 18. To determine the difference between the stellar magnitude before the occultation and the magnitude of the star+asteroid pair during the occultation a shapelet decomposition of the images of Gaia DR2 Source id = 679856115656562048 and comparison stars on all CCD frames was performed. This made it possible to calculate the

22http://www.euraster.net/results/2019/index.html#1029-87

87 Sylvia occults Gaia679856115656562048 (Pulkovo, Saturnlm telescope)

fi = 2019-10-29T23.tu.tu.^ou t2 = 2019-10-29T23:40:56.163 t] tmax = 2019-10-29T23:40:48.212 tmax = 58785.9866691 Amag = 3.26 ± 0.08 t0 = 58785.9802636

-l.o

+1.6

E

-1.0

<

+i.6

-1.0

0

0

2

4

6 8

t- t0l min

10 12

14

Figure 18: Light curve of occultations for the Gaia DR2 star with source id = 679856115656562048 by asteroid 87 Sylvia on 29-10-2019.

flow using instrumental scale ((Fs for the occulted star, Fr - comparison stars) and determine the differences of stellar magnitudes Amag = —2.5lg(Fs/Fr). The analysis of the light curve shows that the central phase of the phenomenon occurred at UTC = 2019-10-29T23:40:48.2+j;6cc (MJD=58785.986669). The drop of brightness was Amag = 3.26 ± 0.08. Relatively low level of accuracy was attributed to poor weather conditions, as we mentioned before. It is undoubtedly true that modern telescopes and cameras allow to achieve much better results.

After the occultation some hundred CCD images were obtained to determine the coordinates of the asteroid. 87 Sylvia motion diagram based on the reports of these observations is shown in figure 19. In such a short time interval the motion of the main-belt asteroid can be adequately represented by a linear model. It allows to formally describe the central occultation moment as the moment of time when angular distance from "star-asteroid" reaches its minimum. This moment equals UTC = 2019-10-29T23:40:48.5±1.4 c (MJD = 58785.9866727), which agrees fairly well with the photometric curve data. At this moment, the differences in the position of the photo center of the asteroid and the star amount to a a = 0.003 ± 0.005 arcsec, AS = -0.014 ± 0.004 arcsec. We would like to note that the length of the asteroid chord during the occultation at the Pulkovo Observatory amounted to 0.029 ± 0.007 arcsec. It means that differences in coordinates are significantly smaller than the angular size of the asteroid.

Figure 19: Tracks of the asteroid 87 Sylvia relative to the Gaia DR2 star with source id = 679856115656562048 near the occultation time for «Saturn» telescope (top panel) and MTM-500M (MAS Pulkovo Observatory near Kislovodsk) (bottom panel).

The shooting of 87 Sylvia with the telescope MTM-500M located at Kislovodsk Mountain Astronomical Station was performed in parallel with the observations at the Pulkovo Observatory. Occultation zone was fairly distant from the observatory, therefore, appulse of 87 Sylvia and Gaia's star DR2 Soigse id = 679856115656562048 was observed. A staff member of the Observational Astrometry Laboratory of the Pulkovo Astronomical Observatory D. L. Gorshanov and the head of this division A. V. Devyatkin provided a series of thirty CCD images obtained with 500. The observations were made by a staff member of Kislovodsk Mountain Astronomical Station of the Pulkovo observatory A. H. Aliev.

Asteroid track relative to the star for MAS Pulkovo Observatory (Kislovodsk) is shown at the bottom of the figure 19. The moment of minimum angular separation corresponds to UTC = 2019-10-29T23:39:20.549±1.6 . (MJD = 58785.9856545). For this moment aa = -0.105 ± 0.049 arcsec, = 0.469 ± 0.050 arcsec. The relative position for a central moment of stellar occultation at the Pulkovo Observatory is aa = 0.255 ± 0.031 arcsec, AS = 0.680 ± 0.022 arcsec. It follows that the estimate of the "instantaneous" parallax relative to the Pulkovo Observatory - Kislovodsk Mountain Astronomical Station is $ = 0.727 ± 0.030 arcsec.

The ephemeris of the phenomenon for the Pulkovo Observatory can be easily calculated using the JPL HORIZONS23 service.

For the «instantaneous» parallax of an asteroid relative to the base Pulkovo -MAS(Kislovodsk) , this service gives the value $ = 0.710 arcsec, which provides O-C=0.017 ± 0.030 arcsec. This discrepancy cannot formally be considered significant.

NASA JPL is not the only such web service. Miriade (IMCCE)24 allows for similar calculations. In the case of Miriade «instantaneous» parallax of the asteroid relative to the base Pulkovo - MAS (Kislovodsk) $ = 0.705 arcsec. This corresponds to O-C=0.022 ± 0.030 arcsec.

To supplement the comparison of observations with the ephemeris, we should add that ephemerides improvement requires to use data from various observations which are naturally characterized by random errors. The developers of JPL's HORIZONS tried to take that into account and grade the accuracy of their ephemerides. For a

23https://ssd.jpl.nasa.gov/?horizons

24https://ssp.imcce.fr/webservices/miriade/

studied asteroid estimates of standard errors were 0.014 arcsec 0.004 arcsec for both coordinates. As it is clear from the analysis, occultation observations (in this case) allowed to achieve the accuracy close to the accuracy of current ephemerides which are based on all the available observations.

At the end of this section, it should be noted that MTM-500M telescope is equipped with a camera which has a large pixel size (1.34 arcsec/pix) to increase the working area. As for «Saturn» telescope, this parameter equals 0.172 arcsec/pix which, along with a large amount of points, provides more accurate data.

In total, the reports of baseline observations of occultation of Gaia DR2 star with source id = 679856115656562048 by an asteroid 87 Sylvia 2019-10-29 demonstrated the high accuracy. Coordinates and the distance to asteroid were measured with a high degree of accuracy that matches the accuracy of ephemerides. The methodology looks promising in terms of the study of asteroid dynamics including the asteroids of the main asteroid belt (which are located at fairly large geocentric distances). For NEAs that method can yield promising results including radar measurement data.

The importance of determining «instantaneous» parallaxes by the considered method lies in the fact that in this way it is possible to remove the uncertainty of the topocentric distance to the asteroid, which significantly complicates the determination of the parameters of the asteroid's orbit. As a result, it is necessary to accumulate a fairly large series of observations in order to obtain estimates of the first two derivatives of the equatorial coordinates of the asteroid and use the Laplace or PVD methods. For near-Earth asteroids, this problem is quite significant. The poor quality of their ephemeris (especially immediately after the discovery of the asteroid) most often comes from the insufficient number of observations. Determining the parallax from baseline measurements immediately allows you to estimate the distance and calculate the orbit knowing only the first derivatives of the equatorial coordinates. In this case, it is not necessary that exactly the coverage is observed in one of the points. It is enough to observe a visible close convergence from two points at once.

6 The results of appulse observations of selected NEAs and Gaia's stars

During the observation season of 2019-2020 we performed various observations with the 1-meter telescope «Saturn» ranging from the study of high proper motion stars and lensing events to Uranus's and Neptune's natural satellites. NEO observations are not the main goal of this tool. Yet, we allocated some free time for experimental observations. As the result, eight phenomena of NEO's appulses to Gaia's stars were selected using Pulkovo NEO service Page25. Although this service provides data on appulses to UCAC4 stars, it is very efficient as it enables to select phenomena available from different observatories. We would like to underscore that tables provided by Pulkovo NEO Page contain information about the phenomena when brightness difference "asteroid - UCAC4 star" amounts to less than one stellar magnitude. That is very important as systematic error which depends on the stellar magnitude can be almost excluded

The ephemerides of the eight studied phenomena and the results of the observation processing are summarized in the tables 4 and 5. Ephemerides were updated using JPL's HORIZONS system26 and Gaia's data EDR3. Tc - the ephemeris moment of maximum approach; £(Tc), H(Tc) - the corresponding tangential coordinate differences at the moment of Tc; £(Tc), HT(Tc) are velocities of apparent motion on the celestial sphere (the first derivatives of tangential coordinates differences with respect to time). All other columns of the table 4 do not require any reasoning or explanation (D(Tc) - topocentric distance to the asteroid expressed in astronomical units at the moment of Tc).

The table 5 shows relative tangential coordinates £(tc),^(tc) at the moment of appulse tc (these moments are shown as well) derived from the observations. Data regarding repeatability of the results with ephemerides data are derived from O-C values (tc — Tc - for the moments of the event maximum phase, (O-C)^, (O-C)^ for relative coordinates). It is apparent that real time of the event maximal phase can significantly differ from the ephemeris values. For asteroid 2005 VY17 this difference amounts to 8 seconds in case of both appulses. The O-C differences for

25http://www.gaoran.ru/personal/neo/rus/general/p01.htm

26https://ssd.jpl.nasa.gov/?horizons

Table 4: Ephemerides of apparent close approaches observed in the Pulkovo program framework.

Tc NEA S (Tc) H(Tc) mag D(Tc) s (Tc) H (Tc) Gaia source id Gmag

arcsec/hr A.U. arcsec

2019-11-24T00:51:29.1 243025 (2006 UM216) -13.9 -88.6 16.3 0.47 11.552 -1.793 3365960063080786816 16.150

2019-11-24T01:38:35.6 243025 (2006 UM216) -13.9 -88.7 16.3 0.47 -10.130 1.572 3365959960001541120 14.563

2019-11-24T19:24:44.9 162723 (2000 VM2) -472.8 -82.2 15.7 0.16 1.765 -10.209 1942346874552792320 15.422

2019-12-10T23:40:27.2 333555 (2005 VY17) -31.9 178.8 16.4 0.36 10.033 1.811 3436496968115120512 16.223

2019-12-11T00:53:44.0 333555 (2005 VY17) -31.7 178.6 16.4 0.36 -14.053 -2.524 3436506073445421952 15.441

2020-02-05T01:47:56.1 35107 (1991 VH) -52.0 122.1 15.6 0.31 -5.716 -2.438 3870223892606361856 15.583

2020-03-23T19:27:54.4 65690 (1991 DG) -130.4 315.6 16.0 0.11 4.982 2.070 607072019309754624 13.996

2020-03-23T19:36:07.5 65690 (1991 DG) -130.4 315.6 16.0 0.11 5.646 2.345 607072088029231232 15.972

D(Tc) is the topocentric distance to the NEA at the central moment of approach. The rest of the columns are described in the text.

Table 5: The results of observations of the apparent close approaches performed in the Pulkovo program framework.

NEA tc tc - Tc £ (tc) (O-C)f V(tc) (O-C),

243025 (2006 UM216) 2019-11-24T00:51:25.9 ± 0.6 -3.215 11.552 ± 0.010 -0.000 -1.793 ± 0.013 -0.000

243025 (2006 UM216) 2019-11-24T01:38:34.7 ± 0.9 -0.930 -10.149 ± 0.012 -0.020 1.577 ± 0.019 0.004

162723 (2000 VM2) 2019-11-24T19:24:46.9 ± 0.6 1.986 1.750 ± 0.060 -0.015 -10.177 ± 0.057 0.031

333555 (2005 VY17) 2019-12-10T23:40:19.1 ± 0.9 -8.081 10.045 ± 0.032 0.012 1.807 ± 0.032 -0.004

333555 (2005 VY17) 2019-12-11T00:53:36.3 ± 1.2 -7.742 -13.945 ± 0.042 0.107 -2.510 ± 0.041 0.014

35107 (1991 VH) 2020-02-05T01:47:54.2 ± 1.3 -1.946 -5.770 ± 0.034 -0.054 -2.471 ± 0.032 -0.034

65690 (1991 DG) 2020-03-23T19:27:53.5 ± 0.8 -0.925 4.885 ± 0.044 -0.097 2.022 ± 0.057 -0.049

65690 (1991 DG) 2020-03-23T19:36:06.7 ± 0.4 -0.764 5.536 ± 0.022 -0.110 2.292 ± 0.035 -0.052

the coordinates in many cases are consistent with the ephemeris within the errors of estimate which fall within the range of 10 mas - 60 mas. In most cases, they are less than 40 mas which is close to the desired result. As we noted in the introduction, one of our research goals is to maintain the accuracy which is obtained during image analysis in the final result and not to lose it due to incorrect astrometric alignment. The most noticeable deviations from the ephemeris were found in case of asteroids 2005 VY17 and 1991 DG. To some extent, substantial O-C can be the result of a short period of observation of these objects. But for other phenomena, O-C values are consistent with the ephemeris, thus, we can assume that significant deviations for 2005 VY17 and 1991 DG are dynamic in nature. We would like to emphasize that at the time of the phenomenon 1991 DG was at the shortest topocentric distance of all the studied asteroids.

„ 0.0 -

u

<u

I/)

1 -0.2-

o

g -0.4-0.6- ,

—4 -3-2-10 1 2

t- tc [min]

Figure 20: (O-C)j depending on the time relative to the moment tc=2019-11-24T19:24:44.9 162723 (2000 VM2). The black points correspond to the measurement of positions relative to the Gaia's star source id = 1942346874552792320, G = 15.422 made using the appulse of asteroids to Gaia's stars. (O-C) calculated in the "traditional" way are indicated by gray square markers.

Fig. 20 demonstrates the advantage of the «approaches» method over the «traditional» astrometric reduction for asteroid 162723 (2000 VM2).

The analysis suggests that the observations of the studied phenomena are efficient for obtaining high-quality data relevant for further studies of their dynamics. To establish the significance of this problem and demonstrate the capabilities of the "old" method in new circumstances we have published an article [36].

7 The report of the observations of giant planets' natural satellites and Gaia's stars appulses

7.1 Phenomena of Uranus's moons and Triton Neptune's natural satellite

Positional observations of Jupiter's and Saturn's natural satellites made from spacecrafts as well as their radar observations from Earth with long continuous set of ground-based observations are currently available. A large number of observations and their high accuracy led to the substantial progress in understanding the dynamics and internal structure of the planets and their natural satellites. [12]. The Voyager Interstellar Mission (VIM) did not provide large compilations of astronomical data for such outer planets as Uranus and Neptune. For this reason, high-precision ground-based observations of Uranus and Neptune's natural satellites will continue to be relevant in the upcoming decades. Major natural satellites of Uranus, and Triton, one of Neptune's moons, are, therefore, the focus of the programme based on the observations made with the telescope «Saturn» at the Pulkovo Observatory. A small set of observations (10 nights in 2020) was made by the research associate of the Crimean Astrophysical Observatory (CrAO RAS) S. V. Nazarov with the use of a 350-millimeter Newtonian telescope. The data were acquired for testing purposes. Figure 21 shows fragments of typical CCD images of Uranus and its major natural satellites.

The main objective of this programme is to collect data on the motion of natural satellites over a substantial amount of time with the aim to verify the coherence of the observations with the modern theories of planetary and satellite motion. It is possible to draw conclusions regarding the significance of the research by analyzing the following data: figure 22 and figure 23. The upper regions of the images show what will happen, if ephemerides are computed using the same planetary theory (DE431) but different theories for the natural satellites (the figure illustrates the differences for Oberon). Unlike Saturn's major natural satellites where these differences are much smaller than the standard errors typical for ground-based as-trometric observations in such a short time period, in case of Uranus systems the differences can amount to 20 ms of arc. Under favourable conditions this matches

«

Figure 21: Examples of the typical CCD images of Uranus system taken with the telescope «Saturn» (top panel) and a guiding instrument of the telescope «Sintez» at CrAO RAS (bottom panel).

the accuracy of ground-based observations. The lower regions of the images show that the transition from planetary theory to EPM2017 [38] immediately changed the differences by 40 mas. With a rich set of observations this cannot go unnoticed.

As Triton is Neptune's largest moon (it has over 99% of the total mass of Neptune's moons) and in terms of dynamics the study of tidal interaction between Neptune and Triton is not practically hindered by the influence of other celestial bodies, it is much easier to assess the predictions made by different scientific teams in the field of celestial mechanics. As has been repeatedly noted, the presence of a bright planet creates a halo affect (background gradient, see figure 21. Thus, satellites position measurements need to be treated with caution. Another problem is that it is difficult to collect images of dim stars appulse to the natural satellites with an adequate signal-to-noise ratio. This somewhat limited the number of observable phenomena. Appulses of Triton and Uranus's moons to Gaia's stars are shown in the table 6. It shows ephemeris time of appulse, relative position of a satellite, angular separation (distance), characteristics of a star. Angular separation of Uranus's natural satellites amounts to 10 arcsec - 20 arcsec (sometimes up to 30 arcsec). Since only a small number of stars appear on the celestial sphere along Tritons' trajectory, the phenomena at a large angular distance (up to 75 arcsec) were selected. The natural satellites were brighter than the stars by one or two magnitudes.

To account for the background gradient effect created by the halo, all the pixel counts in the area of 5 • FWHM size relative to the satellite position were selected. Image model consisted of two components: the background which was represented by polynomials of two third-order variables and the image of the satellite itself which was formed with a stellar PSF. The point spread function was expressed in terms of shapelets decomposition coefficient as it was described earlier. PSF model construction involved the selection of the stars which were closest to the satellite images and yet distant enough so that their profiles were not distorted by the halo affect created by the planet. Further, Levenberg-Marquardt algorithm was used to optimize the parameters which describe variations of background and value of flow and serve for the purpose of providing coordinates of the photocenter for the natural satellites.

Figure 22: Comparison of Oberon's right ascensions calculated from combinations of the planetary and satellite ephemerides.

Figure 23: Comparison of Oberon's declensions calculated from combinations of planetary and satellite ephemerides.

Table 6: Apparent close approaches of the satellites of Uranus and Neptune to the Gaia stars, which were observed as a part of this study.

Satellite

JD

Year

P"

a"

S° Gmag BPmag RPmag

Source ID

Ariel 2458745.49324 2019.714 5.187 17.496 18.249 33.642920 12.955470 15.999 16.590 15.212

2459119.58294 2020.738 -26.968 3.620 27.209 37.460952 14.266451 12.495 12.938 11.885

2459146.47534 2020.812 -6.691 -12.218 13.930 36.480523 13.956437 15.782 16.677 14.856

Umbriel 2459120.56524 2020.741 -0.064 10.621 10.621 37.419498 14.252129 16.460 17.062 15.738

2459120.56591 2020.741 29.506 5.057 29.936 37.410991 14.253672 15.102 15.772 14.309

2459146.47842 2020.812 17.647 -13.715 22.350 36.480523 13.956437 15.782 16.677 14.856

Titania 2458749.58439 2019.725 15.874 -18.513 24.387 33.521413 12.930810 15.792 16.175 15.243

Oberon 2459119.48147 2020.738 -10.222 -10.257 14.481 37.460952 14.266451 12.495 12.938 11.885

2458385.43593 2018.728 1.983 1.986 2.807 29.501437 11.456351 14.949 15.342 14.383

2458407.38451 2018.788 3.019 13.016 13.362 28.705900 11.149210 16.188 16.702 15.517

2458745.47045 2019.714 27.506 -3.827 27.771 33.644621 12.949972 13.244 13.631 12.685

Triton 2458421.20889 2018.826 -65.885 6.412 66.196 345.314881 -7.354610 14.920 15.445 14.157

2458727.43767 2019.664 -30.398 36.954 47.851 348.734026 -5.998857 16.124 16.898 15.266