# The history of V. A. Rokhlin’s ergodic seminar (1960–1970)

A. M. Vershik111St.Petersburg Department of Steklov Institute of Mathematics,
St.Petersburg, Russia, email<EMAIL_ADDRESS>
###### Abstract

The paper tells about the main features and events of the ergodic seminar
organized and headed by V. A. Rokhlin at the Leningrad State University. The
seminar was active in 1960–1970.

## 1\. V. A. Rokhlin’s moving to Leningrad

Vladimir Abramovich Rokhlin, with his wife Anna Alexandrovna Gurevich, his son
Volodya (born in 1952), and his daughter Lisa (born in 1955), moved to
Leningrad in the late summer of 1960. Before, they lived in Kolomna near
Moscow, where V.A. worked at a pedagogical institute. In 1959, the rector of
the Leningrad State University A. D. Alexandrov offered V.A. a position of
professor at the Chair of Geometry, of which he was the head. The
corresponding advice Alexandrov received from a friend and colleague N. V.
Efimov, a well-known Moscow geometer, whose family were old friends of the
Rokhlins: Efimov’s wife was a friend of A.A. from the 1930s, when they both
were L. S. Pontryagin’s graduate students in Voronezh.

Immediately upon arrival in Leningrad, in September 1960, V.A. announced his
research seminar on the theory of dynamical systems (“ergodic seminar”).
Besides, after a while he started to teach an elective course in combinatorial
topology. Neither the first area of mathematics (the theory of dynamical
systems), nor the second one (modern topology) was represented at the
university before. Another seminar, on topology, was announced by V.A.
somewhat later, since it was necessary first to train at least the first group
of participants. This seminar was started in 1961, became well known, and
existed almost until V.A.’s death in 1984. It deserves a separate memoir.

## 2\. The prehistory

Here I want to present the history of the ergodic seminar. Its rapid creation
and immediate activity are explained, first, by the fact that it coincided
with an impressing progress in the theory of dynamical systems caused by the
appearance of Kolmogorov’s entropy and other events, succeeded by an enormous
interest to the subject. But there was also another circumstance: the ground
for the seminar was well prepared. V.A.’s arrival in Leningrad was preceded by
one year by the arrival of his first graduate student in dynamical systems
Leonid Mikhaylovich Abramov, whose wife was from Leningrad. In 1955, Abramov
graduated from the Dnepropetrovsk University with a degree in approximation
theory and was sent to work at the Arkhangelsk Pedagogical Institute, where
V.A. held a position from 1952, when he was forced to leave Moscow after he
had been effectively expelled from the Steklov Institute of Mathematics by its
director I. M. Vinogradov.

L. M. Abramov, on V.A.’s advice, began to study ergodic theory, and in the
mid-1950s enrolled in the postgraduate program of the Moscow State University
under V.A.’s supervision (by agreement with A. N. Kolmogorov). L.M.’s thesis
became the first PhD thesis on entropy theory (defended at the Moscow State
University in 1960); he himself was the first V.A.’s graduate student. Many
knew about V.A.’s plans of moving to Leningrad, some (me included) knew from
Moscow friends about L.M.’s arrival. Several people, including O. A.
Ladyzhenskaya, who was friends with V.A. from the time of her university
studies in Moscow, helped L.M. to solve the problem, extremely difficult at
the time, of getting a job at the Leningrad State University. Coincidentally,
at this very time L. V. Kantorovich organized a new department of mathematical
economics at the Faculty of Economics, and this made it possible to get L.M. a
job at the Chair of Mathematical Economics, where he worked till the end of
his life. The times were like that.

In Leningrad, at G. P. Akilov’s (1921–1986) seminar, there was interest in
measure theory on locally compact groups and especially in measure theory on
linear topological spaces (the influence of I. M. Gelfand and D. A. Raikov).
The core members of the seminar were B. M. Makarov, V. P. Khavin, V. N.
Sudakov, and myself. Some of us had already heard about Kolmogorov’s entropy.
In the fall of 1956, the Second All-Union Conference on Functional Analysis
was held in Odessa. Like the first one (which took place in Moscow in January
1956), it attracted many well-known specialists. In particular, S. V. Fomin
gave a talk about the famous Kolmogorov’s paper on entropy (Dokl. Akad. Nauk
SSSR, 1958), which had just appeared. At the next, third, conference on
functional analysis, which was held in Baku in the spring of 1960, V.A. gave a
talk about the progress in the theory of dynamical systems related to the
notion of entropy. Together with other graduate students from Leningrad (B. M.
Makarov, V. N. Sudakov), I participated in both conferences. We were very
impressed by these talks, and they aroused a great interest in the subject.

An interesting episode occurred at the Odessa conference. After S. V. Fomin’s
talk about Kolmogorov’s entropy, an unknown graduate student of the Odessa
University D. Z. Arov unexpectedly rose from his seat to claim that A. N.
Kolmogorov’s results, presented in the talk, on applications of Shannon’s
entropy to Bernoulli schemes were contained in his master thesis, of which he
informed A.N. in a letter. A.N.’s second paper on entropy (Dokl. Akad. Nauk
SSSR, 1959) does indeed mention D. Z. Arov’s letter. But what exactly had been
done by D. Z. Arov and how this is related to the Kolmogorov–Sinai entropy
became clear only quite recently. The part of his master thesis related to
information theory was recently published in our series of Zapiski Nauchnykh
Seminarov POMI (Vol. 436, 2015) with commentaries written, at my request, by
B. M. Gurevich. Thus, D. Z. Arov may be safely called one of the pioneers of
entropy ergodic theory.

Later, in 1962, D. Z. Arov came to Leningrad in order to work with V.A. and
took an active part in the ergodic seminar, but then he switched to a field
close to his teacher M. G. Krein.

In the 1959–1960 academic year, we organized a home student seminar on measure
theory and functional analysis, where we studied books by A. Weil, P. Halmos,
and some papers on integration theory and measure theory.

It is appropriate to recall V.A.’s visit to Leningrad in 1957 to give a talk
at G. M. Fikhtengolts’s and L. V. Kantorovich’s seminar. I was present at this
meeting and remember the talk: it was one of the last meetings of the seminar,
soon it ceased to exist because of L.V.’s moving to Novosibirsk. The talk was
devoted to the recent V.A.’s paper in Uspekhi Mat. Nauk on metric invariants
of measurable functions based on his theory of measurable partitions. I
remembered this talk, and many years later used and generalized it in several
directions. During the same visit (or another one close in time), V.A. gave a
lecture at the Leningrad Department of Steklov Institute of Mathematics on
recent progress in algebraic and combinatorial topology. A rotaprint edition
of the text of the lecture was immediately issued, and for a long time it
attracted the attention of young mathematicians.

## 3\. The early days of the seminar and its participants

Upon arrival, V.A. asked L. M. Abramov, who by September 1960 already had good
contacts with Leningrad mathematicians, to put him in touch with those wishing
to take part in an ergodic seminar. The first organizational meeting, as far
as I remember, took place not later than October 1960 at the Faculty of
Mathematics and Mechanics. L.M. became the secretary of the seminar.

The fact that V.A. was now a professor at the Faculty of Mathematics and
Mechanics was known to many; the interest to him and to the new seminar was
great. That is why, the first meeting attracted many undergraduates, graduate
students, and professors. V.A. proposed to begin with the study of measure
theory in his understanding, the theory of Kolmogorov’s entropy, as well as a
series of classical works on dynamics: spectral theory, automorphisms of
compact groups, geodesic flows, relation to stationary random processes, etc.
All these subjects were a complete novelty for most of the audience. Now I
remember only those who took part in the seminar for a sufficiently long time
in its early years. Among them, I name only a few: L. M. Abramov, I. A.
Ibragimov, V. N. Sudakov, B. M. Makarov, A. M. Kagan, R. A. Zaidman. Later,
during the 1960s, some young mathematicians, who had graduated from the
University or the postgraduate program, joined in: R. M. Belinskaya, S. A.
Yuzvinsky, M. I. Gordin, A. A. Lodkin, A. N. Livshits, and others. Of course,
there were those who had just flickered and disappeared, there were,
especially at first, curiousity seekers; the corresponding list is long and
difficult to recover.

The first meeting of the seminar was opened by V.A., who told about his plans
for the seminar and suggested the topics of the first talks (see below). These
topics were rather quickly taken up, and soon regular meetings began. The
first invited speakers included two leading Leningrad mathematicians who
worked, in particular, on problems close to dynamics and with whom V.A.
maintained close contact: D. K. Faddeev, who spoke about the classification of
automorphisms of lattices, and Yu. V. Linnik, who gave a general outline of
his recent paper on the equidistribution of integer points on manifolds. I
have kept a copy of the first paper (later transformed into the book Ergodic
Method and $L$-Functions), which was presented by Yu.V. to V.A. and passed to
me by V.A.’s widow, bearing the inscription: “This is my best paper.”

An important feature of the seminar was the abundance of guests, mainly from
Moscow, where Ya. G. Sinai’s and V. M. Alekseev’s seminar on dynamical systems
was very active at the time. This seminar followed A. N. Kolmogorov’s
initiative of the late 1950s to revive the theory of dynamical systems in
Moscow and, in particular, to assert entropy theory, which was emerging after
his pioneering works, as one of its topics. At A.N.’s probability seminar,
this topic received some (little) attention, and V.A., which at the time
worked near Moscow, took part in it. He immediately appreciated A.N.’s idea
and became one of the main developers of this theory. Recall that in his
second note on entropy, A.N. emphasizes the role of the theory of measurable
partitions created by V.A. and in a footnote cites the counterexample
suggested by V.A. to one of the statements of the first note. V.A. repeatedly
mentioned the very great impression which Kolmogorov’s work on entropy made on
him. To this work he attributed his own renewed interest to the theory of
dynamical systems, the subject of his famous papers of the late 1940s, which
made him (along with the subsequent works on entropy theory) a leading expert
in this field.

All of these things largely predetermined also the topics of the Leningrad
ergodic seminar. And they explain the fact that its guests and speakers
included all active Moscow “dynamists” of the 1960s: V. I. Arnold, Ya. G.
Sinai, D. V. Anosov, V. M. Alekseev, and somewhat later their pupils B. M.
Gurevich, V. I. Oseledets, A. B. Katok, A. M. Stepin, G. A. Margulis, and
others. There were also guests from other cities: Odessa, Tashkent, Tbilisi,
Novosibirsk, Gorky, etc.

The seminar existed up to the late 1960s. Occasional meetings (say, on the
occasion of visits or special events) took place even later. I remember that I
spoke about my Doctor of Sciences dissertation in 1971. In 1965, I started my
own seminar on smooth dynamics, whose participants included some of those I
have already mentioned, and also M. L. Gromov, V. L. Eidlin, S. M. Belinsky,
and others. The seminar existed for one and a half year. In 1968, I organized
a wider seminar on measure theory in linear spaces, operator algebras,
representation theory, and dynamics, which exists to this day.

## 4\. Topics of the seminar

The main topic of the ergodic seminar was, of course, ergodic theory rather
than the entire dynamics. V.A. emphasized that he advocated distinguishing
between mathematical structures and preferred not to mix, say, smooth dynamics
with metric or topological dynamics. At the same time, he knew classical
dynamics (both smooth and number-theoretic) well, but for him they existed
separately, and he gave precedence to metric (measure-theoretic) dynamics; I
dare say, for aesthetic reasons. Moreover, he believed, for example, that it
is the methods of the theory of measure-preserving transformations that would
eventually allow one to solve many problems of both analytic number theory and
smooth dynamics.

He particularly emphasized the role of dynamical systems of algebraic origin
as natural examples. Topological dynamics, as compared with metric dynamics,
was for him a less natural area of mathematics. More exactly, he put emphasis
on the existence of an invariant measure with respect to a group of
transformations. Of course, such a measure does not always exist, but then
this is rather a flaw in the problem formulation, and it is the invariance of
a measure with respect to some group of transformations that ensures the
meaningfulness of studying this measure. On the other hand, V.A. believed that
measure theory itself is not an especially important field, in particular,
thought that the theory of nonseparable measure spaces is a nonessential and
somewhat pathological object. He believed that Lebesgue spaces, introduced and
studied by him, cover the needs of analysis and dynamics in measure spaces.
The widespread cliché “measure space” without further details invited his
criticism. He preferred the more accurate term “the theory of transformations
with an invariant measure” to the vague term “ergodic theory.” I plan to
express my views on the role of V.A.’s seminal paper “On the fundamental ideas
of measure theory” and on the modern understanding of what is the category of
measure spaces elsewhere.

Concluding the general description of the seminar, I want to list several
principles which, as I think, guided V.A. in mathematics in general and, in
particular, in his style of conducting a research mathematical seminar. First
of all, he insisted that all statements must be clear-cut and, if possible,
structurally (or categorically) pure. Further, he insisted on the invariance
(functoriality) of statements and, again if possible, proofs. He rejected
arguments that went beyond the framework of a given structure or category, and
this occasionally caused discussions, which usually ended in his favor. This
attitude could be regarded as an extreme kind of Bourbakism, but V. I. Arnold,
a well-known enemy of Bourbaki, eagerly supported V.A.’s recommendations and
followed them.

Besides, V.A. was able to appreciate the beauty of clear-cut statements and
the originality of arguments, and this feature is in a sense opposite to any
dogmatism. Anyway, the school of V.A.’s seminar had much to offer to those who
went through it and, possibly, did not subscribe to all his principles.

## 5\. On the style of conducting the seminar and the first meetings

Unfortunately, the agenda of all seminar meetings for almost ten years of its
existence has not survived. Now it seems astonishing: thanks to modern
electronic techniques, we easily save anything and everything. But at the
time, it was necessary to keep records, follow the sequence of talks, which
only few were capable of. As a result, we (and the history of science) have
lost very much. I wrote about this in relation to the famous I. M. Gelfand’s
seminar (see M. A. Shubin’s notes of talks at the seminar available in the
internet); but this applies as well to other seminars.

The above-mentioned numerous guests of the seminar spoke about their published
papers; it is hardly worthwhile (and anyway impossible) to list them or
reproduce the corresponding discussions. Of course, V.A.’s opinion on a result
or a talk, with which he usually concluded the meeting, as well as his
judgements and comments, were always of interest both to the audience and
speakers. He always expressed these opinions with authority and reserve no
matter whether the talk was received with enthusiasm or with doubt. It is
worth mentioning that V.A. disapproved of other styles of conducting a
seminar, when the leader, albeit a distinguished scholar, reacts to talks with
unrestrained emotions. The culture of mathematical seminars in Soviet time was
being worked out gradually, following, apparently, European and pre-
revolutionary traditions. V.A. followed the best of them.

It remains to extract from my memory what it has preserved. Unsurprisingly, I
remember mainly seminars somehow related to subjects of interest to myself,
and some of my own talks. To begin with, I present only a partial list of
talks of the first period about published results given mainly by the
participants of the seminar.

During the initial period, in 1960–1961, the talks were about famous papers on
ergodic theory and measure theory: about the Lebesgue space (V. N. Sudakov),
the definitions of entropy (I. A. Ibragimov), spectral theory (L. M. Abramov,
quasidiscrete spectrum; B. M. Makarov, the spectrum of Gaussian systems), the
martingale theorem and ergodic theorem (A. M. Kagan), etc.

I will write in slightly more detail about my own talks of the initial period.
First, V.A. asked me to tell about his little-known paper on unitary rings,
which he valued. It contained an attempt, in the spirit of Gelfand–Naimark and
Moscow functional analysis, to present measure theory and the notions of
ergodic theory in terms of the space $L^{2}$ of square-integrable functions.
For this, the Hilbert structure of the space of functions had to be equipped
with some additional structure, and V.A. choose a structure which he called a
unitary ring (later, it became known as a unitary algebra). In other words,
one considers an axiomatization of an unbounded (i.e., defined not for all
pairs of elements) multiplication in $L^{2}$ with obvious constraints. Indeed,
this defines a functor between the corresponding categories (the category of
Lebesgue spaces and the category of unitary rings; the terms did not existed
at the time). To an automorphism with an invariant measure there corresponds a
multiplicative unitary operator. This is a natural measure-theoretic
counterpart of Gelfand’s theory of normed rings. However, another category has
become more popular here, whose objects are $L^{2}$ spaces with a cone, the
cone of nonnegative functions. This functor proved to be more convenient, for
example, when dealing with operators: nonnegative automorphisms or (later)
polymorphisms and Markov operators. But my talk contained an answer to a
specific question indirectly posed by V.A.: how can one define a structure of
a unitary ring on an infinite-dimensional space with a Gaussian measure?

At the time, V. N. Sudakov and myself were interested mainly in geometric
aspects of Gaussian measures (quasiinvariant measures, extension of weak
distributions, etc.), partly under the influence of the Moscow school (I. M.
Gelfand, D. A. Raikov). In particular, I studied N. Wiener’s book Nonlinear
Problems in Random Theory (1958). But the interest to the dynamics of linear
automorphisms of Gaussian measures, or, as we said back then, “normal
dynamical systems,” was initiated by A. N. Kolmogorov and picked up by V.A.,
S. V. Fomin, and others. This applied to spectral theory, entropy, etc. A.N.’s
remarkable idea, implemented by his pupil I. V. Girsanov, led to constructing
an automorphism with simple singular spectrum (in the complement to the
constants). V.A. popularized this field, those who worked in it included L. M.
Abramov and later Ya. G. Sinai and many others, including myself. In my talk,
I described multiplication formulas for generalized Hermite polynomials which
defined the structure of a unitary ring. Then, and somewhat later, I
understood that these formulas are equivalent to formulas from the Itô–Wiener
stochastic analysis, or Wick’s regularization formulas (in physics), etc.

I remember that V.A. liked these results very much. Later, they were included
in my PhD thesis. Back then, I was a graduate student, formally under the
supervision of G. P. Akilov, a pupil and coauthor of L. V. Kantorovich, to
whom I owe much, but actually my thesis was inspired by V.A. Among other
theorems proved in the thesis, it announced a result on the isomorphism of
Gaussian automorphisms with absolutely continuous spectrum, which was proved
only for some special spectra. In full generality it was proved somewhat later
by D. Ornstein. Afterwards, I repeatedly spoke about this at the seminar.

I remember some of my other talks, initiated by V.A., about famous papers, for
example, my talk about the paper “Geodesic flows on manifolds of constant
negative curvature” by I. M. Gelfand and S. V. Fomin (Uspekhi Mat. Nauk, Vol.
7, No. 1(47), 1952) and the corresponding discussion, as well as my talk about
S. Smale’s remarkable paper on the orthogonal group as a retract of the group
of homeomorphisms.

## 6\. Talks of seminar participants about their own results

Almost all participants of the ergodic seminar worked, apart from ergodic
theory, also in other fields of mathematics, and even reported the obtained
results at the seminar. I remember V. N. Sudakov’s talks about measures in
functional spaces and my own talks about the axiomatics of measures in linear
spaces from the standpoint of Lebesgue spaces, etc. I had already begun to
work in representation theory, algebraic and combinatorial asymptotics, always
bearing in mind connections and analogies with dynamics. But, speaking about
not educational or “exterior” talks, but research talks on the topic of the
seminar, irrespective of the time when they were given, I must first of all
mention, at least very briefly, the most important results of the 1960s–1970s
obtained by the permanent participants of the seminar in ergodic theory
proper.

1\. V.A.: a proof of the existence of a countable generator for every
aperiodic endomorphism; later, he proved the existence of a generator with
finite entropy for endomorphisms with finite entropy. The ultimate result on
the existence of a finite generator for such endomorphisms with an estimate
was proved by W. Krieger and, independently, by my graduate student A. N.
Livshits. But V.A.’s result was the first in this direction.

2\. V.A. and L. M. Abramov: a formula for the entropy of skew products using
the new notion of the mixed entropy of fibers.

3\. V.A. and Ya. G. Sinai: a proof of the fundamental theorem on the
coincidence of the classes of Kolmogorov automorphisms and of automorphisms
with totally positive entropy; this is one of the fundamental facts of ergodic
theory. Later, it was extended by D. Ornstein and B. Weiss to actions of
amenable groups.

The next two permanent participants of the seminar from the mid-1960s deserve
special mention. During the years of V.A.’s teaching at the Leningrad State
University, he had many PhD and master students in topology, but, apart from
his very first PhD student L. M. Abramov, only two PhD students in ergodic
theory. The first of them was S. A. Yuzvinsky, who brilliantly completed his
evening studies at the Faculty of Mathematics and Mechanics of the Leningrad
State University (earlier, he had graduated from the Polytechnic Institute),
but nevertheless, in spite of V.A.’s recommendation, was not admitted to the
postgraduate program at the University. He enrolled in such a program at the
Hertzen Institute, where V.A. worked part-time for a while, and under V.A.’s
supervision prepared and defended his PhD thesis (in 1966).

4\. S. A. Yuzvinsky’s results on the entropy theory of automorphisms of
compact groups. This was one of V.A.’s favorite topics. He himself and many
his pupils and colleagues (Ya. G. Sinai and his pupils, D. Z. Arov, L. M.
Abramov) studied entropy formulas for group automorphisms. S. A.’s results and
subsequent work by K. Schmidt and others (on the Mahler measure) in a sense
settled this problem. Another specific result obtained by S. A., on the
genericity of automorphisms with simple continuous spectrum, also belongs to
the circle of problems traditional for approximation theory in ergodic theory.

5\. The second V.A.’s PhD student in ergodic theory was R. M. Belinskaya
(Ekhilevskaya, 1938–2011). Her main topic was time changes in automorphisms
and related problems of ergodic theory. The results of her thesis (1970) are
widely used in the literature. The main problem suggested to her by V.A. will
be described below. But first it is worth mentioning that V.A., at our
request, made several surveys of open problems in ergodic theory, first time
around 1962, and then in 1965 (mostly about entropy-related problems).

V.A. mentioned various problems, but all of them were related to his own work.
In the first talk, he promoted the study of endomorphisms and suggested to
classify finite monotone sequences of measurable partitions as their
invariants. It is appropriate to mention, in particular, O. V. Guseva, a
participant of the seminar during its early years, who worked mainly on the
theory of partial differential equations, but attended the ergodic seminar,
took up this subject, and, generalizing the classification of measurable
partitions given by V.A., obtained a complete classification in the case of
finite sequences.

Another problem was especially important: V.A. asked when two ergodic
automorphisms have isomorphic orbit partitions. He said that this question
appeared in his studies in the 1940s, but it was not known even whether this
is true for two nonisomorphic rotations of the circle. V.A. himself believed
that in this case the answer should be positive. And his bold conjecture was
that if two automorphisms have different entropies, then their orbit
partitions are nonisomorphic. It is this problem that R. M. Belinskaya
systematically tried to solve for a long time, discovering interesting facts
about time changes along the way. In particular, she observed that the orbit
partition of an automorphism is the intersection of a decreasing dyadic
sequence of partitions. But the main problem remained open. At the end of
1966, R.M. and S.A. informed me of this fact, and soon I proved that all
ergodic dyadic sequences are lacunarily isomorphic, and hence their
intersections are metrically isomorphic. Thus, V.A.’s conjecture turned out to
be false. V.A. was very glad that the answer was finally obtained. The results
were published in Functional Analysis and Its Applications in 1968. But while
the paper was in preparation, I discovered, selecting material for our survey,
joint with S.A., on dynamical systems and operator algebras for Itogi Nauki,
that as early as in 1963, the American mathematician H. Dye proved this fact
in terms of the theory of operator algebras etc. Dye mentioned that this
problem had been implicitly posed by J. von Neumann, in a paper from the
famous series of joint papers with F. Murray, as the isomorphism problem for
hyperfinite factors of type ${\rm II}_{1}$. Von Neumann believed that the
answer is most probably positive, but that it would require a detailed
analysis of ergodic automorphisms. It is this problem that Dye solved. It is
worth mentioning that V.A. knew about the isomorphism problem for factors, but
did not know about its relation to his question. Dye’s theorem is a classical
fact of ergodic theory and is much cited in the literature. My proof, relying
on the lacunary isomorphism theorem, is much simpler than the original one. Of
course, these results were reported both at the ergodic and Moscow seminars.

At first, it seemed to me that a strengthening of the method of proving the
lacunary isomorphism theorem would allow one to prove the isomorphism of any
two ergodic dyadic sequences. In my paper on lacunary isomorphism published in
Functional Analysis, I even made the corresponding remark. However, very soon
I became aware of the difficulty of the problem, and it took me a year before
I managed, on the one hand, to find a “standardness criterion,” i.e., a
necessary and sufficient approximative condition for a given dyadic sequence
to be isomorphic to a simplest (or standard) sequence, and, on the other hand,
to present a continuum of pairwise nonisomorphic ergodic dyadic sequences
(they are now called dyadic filtrations). The corresponding invariant was a
new metric invariant, the entropy of a filtration. Following the natural logic
of the subject, I defined the so-called “main invariant” of a filtration, and
somewhat later, the corresponding invariant of an action, the “scale” of an
automorphism. Soon after my results were reported at the Moscow ergodic
seminar, A. M. Stepin proved that the Kolmogorov entropy of the action of the
group $\sigma_{1}^{\infty}Z_{2}$ is also an invariant of a dyadic sequence.
The entropy of a filtration introduced by me coincides in this case with the
entropy of the action of the group. Curiously enough, V.A.’s intuition has
been eventually almost vindicated: the entropy of an action is an invariant
close to an orbit invariant, but still not exactly an orbit invariant. The
main results were included into my Doctor of Science dissertation
“Approximation in measure theory.” As far as I remember, it was reported at
probably the last meeting of the ergodic seminar (around 1971).

Later, this circle of problems (orbit theory, the theory of filtrations, etc.)
grew into a large research area and found applications in a vast variety of
fields: the theory of random processes, asymptotic combinatorics, and, of
course, ergodic theory. The theory of filtrations must provide the foundation
for a combinatorial (approximation) approach to dynamics. At the moment, it is
far from completed.

A few words about other seminar participants.

I have already mentioned the young mathematicians M. I. Gordin,A. A. Lodkin,
A. N. Livshits, and others, who took an interest in ergodic theory. Among my
graduate students of that time working in this field, A. N. Livshits
(1950–2008) stood out. He was a highly gifted and early matured mathematician.
He had caught only the very end of the ergodic seminar, and participated
mainly in my later seminar, to which he came as a first-year student. Very
quickly A. N. Livshits mastered the theory of Anosov systems and, while a
third-year undegraduate, proved his famous theorem on the cohomology of
hyperbolic automorphisms and flows. Then he worked much on encoding and
isomorphism, independently proved Krieger’s theorem (see above), and later we
jointly established a direct relation between adic transformations and
substitutions.

By the time of the new flourishing of ergodic theory connected with the name
of D. Ornstein, who solved the isomorphism problem for Bernoulli automorphisms
and discovered non-Kolmogorov Bernoulli systems, the seminar had already
slowed its activity. These papers were not discussed at the seminar. But it is
appropriate to mention that one of the seminar participants R. A. Zaidman,
prior to Ornstein’s works, claimed to have proved that the entropy is a
complete invariant of Bernoulli automorphisms. This claim was widely known.
However, repeated attempts of seminar participants to understand R. A.
Zaidman’s arguments failed. It is worth mentioning that he is a very talented
and interesting mathematician, and his ideas on encoding were later picked up
by A. N. Livshits, who greatly developed the theory of codes.

The works of other my graduate students in ergodic theory were done later.

In 1969, V.A. offered me to conduct, together with him, a small ergodic
seminar for beginners (first and second year students), and I gladly accepted.
V.A. only outlined the program of the seminar, but almost did not attend its
meetings. The seminar was educational and was active for one term. Its
participants and speakers included the students A. A. Suslin, V. V. Rokhlin
(Jr.), A. G. Reiman, V. M. Kharlamov, and many others, who later became well-
known mathematicians. By the way, this generation (graduated in 1972) was one
of the strongest at the Faculty of Mathematics and Mechanics in those years
(A. N. Livshits, B. S. Tsirelson, A. A. Suslin, V. M. Kharlamov, and others).

## 7\. Schools, conferences, the conclusion

Speaking about the ergodic seminar in the 1960s, one cannot fail to mention
the ergodic school (August 1965) at Khumsan (Uzbekistan). It took place during
the heyday of ergodic theory itself and relatively liberal times in the USSR.
The venue and conveniences were excellent. The organizers were mainly from
Moscow (Ya. G. Sinai and his pupils) and Uzbekistan (T. A. Sarymsakov and
others). The school attracted an exceptional group of participants and
speakers, including a very wide participation by experts in other fields of
mathematics. Here is an incomplete list: V. A. Rokhlin, O. A. Ladyzhenskaya,
T. A. Sarymsakov, N. P. Romanov, Yu. M. Smirnov, V. P. Maslov, A. Ya. Povzner,
I. I. Pyatetsky-Shapiro, Ya. G. Sinai, A. A. Kirillov, many of them with
families. Besides, a whole team of young talents arrived, mainly from Moscow:
I. N. Bernstein, M. L. Gromov, A. B. Katok, A. G. Kushnirenko, L. G. Makar-
Limanov, G. A. Margulis, A. M. Stepin, and others.

In conclusion, I should mention several more facts from the history of the
ergodic seminar. In August 1969, we organized a one-day informal conference
dedicated to V.A.’s 50th anniversary. Numerous participants listened with
interest to talks given by V. I. Arnold, S. P. Novikov, Ya. G. Sinai, D. V.
Fuks, M. L. Gromov, and A. M. Vershik. Ten years later, on December 10, 1979,
a similar one-day conference was organized in honor of V.A.’s 60th
anniversary, with the same speakers, except that instead of M. L. Gromov’s
talk there was V. M. Buchstaber’s talk. The day was chosen by chance, the
organizer forgot that in the free world calendar December 10 is the Human
Rights Day, and that in the USSR every mention of this day, let alone the
organization of any meetings on it, is punishable. That is why, as I was later
told by V. A. Zalgaller, the Party bosses of the faculty were terribly
frightened and ran to the District Party Committee (!) for consultations. But
eventually it was decided that it was safer for all involved to pretend that
there was no conference at all.

Soon after that, the faculty administration suggested that V.A. retire, under
a pretext that he declared that for health reasons he cannot teach at the new
building of the faculty in New Peterhof, but can give seminars in the city
(for example, at the Leningrad Department of Steklov Institute of
Mathematics). After much efforts and letters to the university and faculty
administrations, he was allowed to continue working at the Faculty of
Mathematics and Mechanics, but only for one year, as ordered by the then
rector of the Leningrad State University (V. B. Aleskovsky).

In V.A.’s notes, which I found after his death (December 3, 1984), the last
manuscript concerning ergodic theory was dated 1968 (classification of
measurable partitions); the text was unfinished. Later, V.A. worked only in
topology and real algebraic geometry, but followed the progress in ergodic
theory and attended the seminar up to the very last meetings.


Vol.0 (20xx) No.0, 000–000

11institutetext: Department of Computer Science, Jinan University, Guangzhou
510632, China<EMAIL_ADDRESS>
22institutetext: Sino-French Joint Laboratory for Astrometry, Dynamics and
Space Science, Jinan University, Guangzhou 510632, China
Received 20xx month day; accepted 20xx month day

# The Astrometric Performance Test of 80-cm Telescope at Yaoan Station and
Precise CCD Positions of Apophis

Bifeng Guo 1122 Qingyu Peng 1122 Ying Chen 1122 Zhongjie Zheng 1122 Yijia
Shang 1122 Dan Li 1122 Xiao Chen 1122

###### Abstract

The 80-cm azimuthal telescope is newly mounted at Yaoan Station, Purple
Mountain Observatory in 2018. The astrometric performance of the telescope is
tested in the following three aspects. (a) The geometric distortion of its CCD
attached. It is stable in both a single epoch and multi epochs. Eight
distortion solutions are derived over about one year. The maximum values range
from 0.75 to 0.79 pixel and the median values range from 0.14 to 0.16 pixel.
(b) The limit magnitude of stars. About 20.5 magnitude (Gaia-G) stars can be
detected with Johnson-V filter exposured in 300 seconds. The astrometric error
of about 20.5 magnitude stars is estimated at 0.14 arcsec using the fitted
sigmoidal function. (c) The astrometric accuracy and the precision of stacked
fast-moving faint object. 24 stacked frames of the potentially hazardous
asteroid (PHA) (99942) Apophis are derived on April 14 and 15, 2021 (fainter
than 18 mag) based on the ephemeris shifts. During data reduction, the newest
Gaia EDR3 Catalog and Jet Propulsion Laboratory Horizons ephemeris are
referenced as theoretical positions of stars and Apophis, respectively. Our
results show that the mean (O-C)s (observed minus computed) of Apophis are
-0.018 and 0.020 arcsec in right ascention and declination, and the
dispersions are estimated at 0.094 and 0.085 arcsec, respectively, which show
the consistency of the stacked results by Astrometrica.

###### keywords:

Methods: data analysis - Techniques: image processing - Minor Planet,
asteroid: general, Astrometry

##  1 Introduction

The 80-cm azimuthal-mounting telescope at Yaoan Station, Purple Mountain
Observatory has served for precise astrometry since it was set up in the year
of 2018. Although some precise positions by the telescope have been published
(Yuan et al. 2021, Li et al. 2021), the performance of the telescope has not
been specified. The potentiality of the telescope should be taped through the
performance test, which may serve for some survey or occultation observation.
In this paper, we aim to test its astrometric performance in the following
three aspects. (a) The geometric distortion of its CCD attached. (b) The limit
magnitude of stars. (c) The astrometric accuracy and the precision of stacked
fast-moving faint object.

The astrometric potentiality of the telescope can be tapped by the solutions
for the geometric distortions (GD). The Hubble Space Telescope (HST) is an
obvious example, whose WFPC2 chip has a maximum geometric distortion of about
5 pixels at the edge of its field (Anderson & King 2003). By increasing the
accuracy of the linear terms, a more accurate solution for GD is determined.
After the GD correction, the observations of Saturnian satellites have much
better precision than ever before (French et al. 2006). Later, the HST
observations of M92 are taken as a distortion-free reference frame to improve
the GD solution of Keck II 10-m telescope’s near-infrared camera (NIRC2) in
its narrow field mode, which is a major limitation for the proper motion
measurements of Galactic central stellar cluster (Yelda et al. 2010).

The limit magnitude of the telescope is also the embodiment of the astrometric
potentiality. Namely, it tells us how faint the telescope can observe the
object. Usually, it guides us to choose the objects of interest and develop
observational plans. For example, we can explore the physical properties of
the interested faint stars with high-precision astrometric positions. Also,
some interesting faint Kuiper-belt objects can be detected and explored.

We explore the potentiality of observing fast-moving faint object by analyzing
the astrometric accuracy and precision of stacked observations of Apophis.
Besides, the astrometric positions of Apophis are also precious. Apophis was
discovered by Bernardi, Tholen and Tucker on June 19, 2004 at Kitt Peak
observatory (Minor Planet Supplement 109613). In the same year, the impact
probability reached 2.7$\%$ in April 13, 2029 encountering with Earth, but
later it was ruled out (Giorgini et al. 2008, Farnocchia et al. 2013).
Nevertheless, some other close encounter events are forecast in 2068, 2085 and
2088 (Souchay et al. 2018). Successive observations remain needed to monitor
the positions of Apophis, which is of great importance to improve its orbital
parameters. Besides, the observations in 2021 are the key information of
analyzing the Yarkovsky effect for such a small-mass object (Vokrouhlický et
al. 2015, Brozović et al. 2018). According to Jet Propulsion Laboratory
Horizons ephemeris111http://ssd.jpl.nasa.gov/ (JPL), although the predicted
visual magnitude of Apophis is usually fainter than 20 mag in recent years, it
is brighter than 17 mag in February, 2013 when it is a valuable oppotunity to
observation. Some researchers (Wang et al. 2015, Thuillot et al. 2015) seized
the oppotunity and obtained the valuable precise positions of Apophis.

Because of its fast motion and faint brightness, high-quality observations of
Apophis are hard to obtain with a small-aperture ground-based telescope.
Observing in short-time exposures and stacking the frames are a good solution
to obtain high-quality images, by which the performance of the telescope can
be tested. This method was first proposed by Tyson et al. (1992) to detect
faint Kuiper-belt objects a few decades ago. Later, this technique is well
used to survey and find faint near-Earth asteroids (NEAs) (Shao et al. 2014,
Zhai et al. 2014). It is showed that the technique can discover asteroids 10
times fainter than conventional searches (Heinze et al. 2015). Wang et al.
2017 use the iterative stacking method to detect faint asteroids.
Subsequently, Li & Peng 2020 finds that shift-and-add method can also improve
the mesurement of astrometric positions for some faint satellites of Jupiter.
The developed open software also provides the function of stacking like
Astrometrica222http://www.astrometrica.at/ and MaxIm
DL333https://diffractionlimited.com/maxim-dl/. This paper explores an
alternative stacking way based on the ephemeris shifts of Apophis to obtain
the precise positions of it.

The contents of this paper are arranged as follows. In Section 2, we give the
specification and the operation performance of the telescope and its CCD
attached. In Section 3, we elaborate the performance test of the telescope in
geometric distortion, limit magnitude and the stack of fast-moving object
Apophis. The conclusions and outlooks are shown in the last section.

##  2 Introduction of the Telescope

###  2.1 Specifications of the Telescope and CCD

The 80-cm azimuthal-mounting telescope at Yaoan Station, Purple Mountain
Observatory is located at E101${\degr}$ 10${\arcmin}$ 51.0${\arcsec}$,
N25${\degr}$ 31${\arcmin}$ 43.0${\arcsec}$, whose IAU code is O49. The
specifications of telescope and the CCD attached are shown in Table 1.

###  2.2 Operation Performance

The azimuthal-mounting telescope is well-known to easily construct than the
equatorial-mounting telescope and is more stable for a large telescope. To
follow the rotation of the sky, both of the axes (azimuth axis and altitude
axis) must be turned around with a changing angle speed, which is compensated
for photography. However, if the pointing of the telescope goes close to the
zenith, its azimuth will change 180${\degr}$ in a short time and there is a
small region where the observations are impossible.

Table 1:  Specifications of the Telescope and its CCD Items | Parameters
---|---
Focal Length | 800cm
Diameter of Primary Mirror | 80cm
F Ratio | 10
CCD Field of View | 11.8′ $\times$ 11.8′
Size of Pixel | 13.5$\mu$m $\times$ 13.5$\mu$m
Size of CCD Array | 2048 $\times$ 2048
Angular Resolution | 0.346′′/px
Pointing Accuracy |  $<$8′′ RMS
Tracking Speed | 13${\degr}$ per second
Tracking Accuracy |  $<$0.25′′ in 5 minutes

##  3 Astrometric Test of the telescope

To explore the astrometric potentiality of the telescope, we test the
operation performance of the azimuthal-mounting telescope from the obtained
observations in geometric distortion (GD), limit magnitudes and the stack of
fast-moving object Apophis. Few stars may be observed in the sparse sky areas,
which leads to the worse accuracies and precisions due to the constructed
plate model (Lindegren 1980). Generally, the calibration field is observed
before observing the field of sparse sky areas so that the geometric
distortion can be solved. In this work, we focus on the crowded sky areas.

###  3.1 The Stability of Geometric Distortion

Two kinds of observations are carried out to test the stability of GD. One is
for single epoch observation data. In this case, multiple images are taken on
the same night with the stars always at the same pixel location. The highest
possible astrometric precision and the stability of GD during one night can be
tested. Another is for multi-epoch observation data. These observations allow
us to test the GD variation over time (nights, months, weeks and years).

For the stability of GD in the same epoch (the same night), we first solve the
GD of CCD by the observations of open cluster M35 with Johnson-I filter on Nov
28, 2019 using a convenient method (Peng et al. 2012). Then we retrieve the
observations taken at the same night with the stars always at almost the same
pixel location (some pixels’ shifts exist due to the tracking error of the
telescope). The pointing of the telescope is at (05h 41m 43.0s, -01${\degr}$
50${\arcmin}$ 30.0′′, J2000) with exposure time 60 seconds for each CCD frame
and totally 20 frames are taken. Three methods are used to perform data
reduction for the stars to investigate the stability of GD. (a) A simple
6-parameter model transformation. (b) A 6-parameter model transformation after
GD correction. (c) A 30-parameter model transformation. Fig. 1 and Fig. 2 show
the accuracies (mean (O-C)s) and precisions (STDs) of the reduced results,
respectively. From the results, the mean (O-C)s using 6-parameter model after
GD correction improve significantly, which is similar to the 30-parameter
model. And the three methods above have almost the same precisions. The
results show that the geometric distortion is stable during this epoch.

For the stability in different epochs (different nights), it’s an effective
way to test with the stars locating on the same pixel location. After GD
correction, other systematics effects of the stars (assuming zero proper
motion) from multi-epoch data sets can be estimated. For the time arrangement
of the telescope, such proper observations haven’t been taken. In our work, we
solve the distortion for eight nights (range from Dec 30, 2018 to Dec 26,
2019) including the time of adjacent nights, weeks, months and years to test
the stability of the distortion (see Fig. 3). In Fig. 3, the first eight
panels show the GD vector graphs of the CCD solved by the observations of the
open cluster M35 or NGC2324. They are all taken with Johnson-I filter. For the
details of the GD vector graphs, the large values of GD locate at the four
corners of the CCD, which point to the corresponding corner, while the
intermediate values mainly locate around the center of the CCD, which point to
the CCD center. There is an annular region of the CCD where the GD values are
small and the pointings are random. From the first eight panels, the shapes of
the vector graphs show the consistency and the maximum values of the
distortions range from 0.75 to 0.79 pixel over about one year. The median
values of the distortion range from 0.14 to 0.16 pixel. Our results show that
the distortion is stable over time. While the last panel of Fig.3 shows the
difference of the two geometric distortion solutions of Dec 26, 2019 and Dec
30, 2018. The difference of GD values of each region in the pixel coordinate
are stable and the median value is 0.04 pixel (about 0.014 arcsec). For the
stable GD solutions, it is easier to obtain precise positions of the objects
after GD correction even if there aren’t enough stars in the field of view to
construct high-order plate model.

We have also derived the distributions of the astrometric errors of the
calibration field for open cluster M35, which are also used to derive the
geometric distortion solutions. We perform data reduction with 30-parameter
model and compute the mean residual (O-C)s of each area in right ascention
(X-axis) and declination (Y-axis), which are shown in Fig.4. The resultant
values ($\sqrt{\sigma_{ra}^{2}+\sigma_{dec}^{2}}$) of the residuals are very
small and the obvious trend of the distribution hasn’t been found. The median
value of the residuals is 0.004 arcsec.

Figure 1: The left and right panel show the 3 different reduced method of the accuracies of the stars in right ascention and declination, respectively. Figure 2: The left and right panel show the 3 different reduced method of the precisions of the stars in right ascention and declination, respectively. Table 2: The specifications of the two sets of observations to explore the limit magnitude of the telescope. The first two columns list the observation dates and the obtained number of frames of stars. The last four columns shows the paramters of the fitted sigmoidal functions for the astrometric precisions. Date | Frames | Filter | Exptime(s) | Limit Mag | Para-A1 | Para-A2 | Para-X0 | Para-dx
---|---|---|---|---|---|---|---|---
Nov 5, 2018 | 8 | Johnson V | 300 |  20.5 | 0.00748 | 0.15577 | 19.30018 | 0.59252
Oct 9, 2019 | 4 | Johnson V | 300 |  19.0 | 0.01032 | 0.18495 | 18.56849 | 0.75234
Figure 3: The GD graphs are derived from the observations of open cluster M35
or NGC2324. A factor of 500 is used to exaggerate the magnitude of each
geometric distortion vector. The first eight panels show the GD graphs in
different epoch. The last panel shows the difference of the two geometric
distortion solutions of Dec 26, 2019 and Dec 30, 2018.  Figure 4: The left and
right panels show the residual graphs derived from the observations of open
cluster M35 on Dec 30, 2018 and Dec 26, 2019, respectively .A factor of 1,000
is used to exaggerate the magnitude of each vector.

###  3.2 The Limit Magnitude

The limit magnitude of the stars is also explored by reducing the observations
(reduction details in Section 3.3.6) with long exposure time (300 seconds).
The relationship between astrometric errors and exposure time can be consulted
from Lindergren’s work (Lindegren 1980). The specifications of the used
observations are shown in Table 2. Fig.5 and Fig.6 show the reduction results
of the observations and the pointings of the telescope are at (23h 01m 15.79s,
-07${\degr}$ 20${\arcmin}$ 17.56′′, J2000) and (19h 02m 24.72s, -22${\degr}$
51${\arcmin}$ 03.88′′, J2000), respectively. From the results over two nights,
the stars fainter than 20 magnitude (Gaia-G) can be detected using gaussian
centering algorithm where the detection threshold is set as 3 $\sigma$ of the
background in each frame. The signal-noise ratio (SNR) of the faintest stars
of each frame is about 5.

Figure 5: The left and right panel show the accuracies and the precisions of
the stars in right ascention and declination on November 5, 2018,
respectively. Figure 6: The left and right panel show the accuracies and the
precisions of the stars in right ascention and declination on October 9, 2019,
respectively. Figure 7: The left and the right panel show that the astrometric
errors fit with FWHM/SNR (SNR$>$10) using linear function and Gaia-G magnitude
using sigmoidal function, respectively. The fitted functions are shown in red
line.

The relationship between astrometric errors and FWHM/SNR is really an
effective way to estimate the astrometric precision, especially for high-SNR
stars. Considering both bright and faint stars across the field of view, it is
found that the sigmoidal function can fit well with Gaia-G Magnitude and the
astrometric precision (Lin et al. 2019). The expression of sigmoidal function
is shown in formula 1. We have compared the two methods above to estimate the
astrometric errors ($\sqrt{\sigma_{ra}^{2}+\sigma_{dec}^{2}}$). Using the
observations on Nov 5, 2018, the astrometric errors can be fitted in Fig. 7.
The left and the right panel show that the astrometric errors fit with
FWHM/SNR (SNR$>$10) using linear function and Gaia-G magnitude using sigmoidal
function, respectively. Both of them can be fitted well. Although the SNRs for
faint stars can’t be determined accurately, the precisions can be well
expressed with a sigmoidal function. Therefore, we use the sigmoidal function
to estimate the precision of the stars.

For the observations in 2018, about 20.5 magnitude stars can be detected and
the corresponding astrometric error is estimated at 0.14 arcsec using the
fitted sigmoidal function. While for the observations in 2019, about 19.0
magnitude stars can be detected, the corresponding astrometric error is
estimated at 0.12 arcsec. The difference of the limit magnitude over two sets
of observations are mainly caused by the atmosphere conditions. The
astrometric precisions of these observations are estimated with sigmoidal
functions, whose parameters are shown in Table 2. However, brighter stars
might have larger astrometric errors from the observations because they might
be saturated. The sigmoidal function only takes the unsaturated stars into
account.

$y=\dfrac{A1-A2}{1+e^{(X-X0)/dx}}+A2$ (1)

###  3.3 The Stack of Fast-Moving Object

In order to test the astrometric performance of the fast-moving object, we
explore the astrometric accuracy and precision of the stacked observations of
Apophis.

####  3.3.1 Observations

During the observation, the angular rate of Apophis is about 1 arcsec per
minute (details in Fig. 10). Given that the Full Width at Half Maximum (FWHM)
of the star images are usually 3-5 pixels and the astrometric precision, a CCD
frame of Apophis is required to be exposured within 20 s (about $\frac{1}{5}$
$\sim$ $\frac{1}{3}$ FWHM). Due to the object is so faint and invisible in a
frame, stacking method has to be performed to improve the signal-noise ratio.
The observations overview of Apophis are shown in Table 3. The observations
are taken on April 14 and 15, 2021 using Johnson I filter. The seeing over two
nights are similar and the predicted visual magnitude of Apophis is fainter
than 18 mag.

We evaluate the star images’ FWHMs and the shifts in pixel with respect to its
nearby CCD frame for each frame. We also derive the middle time of observation
for each frame from the headers of FITS files. Given the stacked SNR of the
object and the number of the total frames, we stack in group of 10 frames. To
ensure the quality of stacked frames, we select frames with similar FWHMs of
star images, near observation time and small pixel shifts into a group.
Finally, 13 and 11 groups of observations were selected on April 14 and 15,
2021, respectively.

Table 3: Observations Overview. The first two columns list the observation dates and the obtained number of frames of Apophis. The 5th column shows the stars’ FWHMs ranges evaluated from each frame and the last column shows the predicted visual magnitude of Apophis from JPL ephemeris. Date | Frames | Filter | Exptime(s) | FWHM(px) | Predicted Mag
---|---|---|---|---|---
April 14, 2021 | 141 | Johnson I | 20 | 2.7-4.3 | 18.16
April 15, 2021 | 135 | Johnson I | 20 | 2.8-4.5 | 18.20
Figure 8: The left and the right panel are the subframes after stacking. The
left shows a star-stacked subframe while the right shows an object-stacked
subframe.  Figure 9: The top two panels show the counts (ADU) across pixel
coordinate (in both X and Y axis) of Apophis in an individual frame. The
bottom two panels show the counts (ADU) across pixel coordinate (in both X and
Y axis) of Apophis in an individual frame. For each panel, X axis shows the
pixel location of Apophis, while Y axis shows the pixel counts, where we
define the center pixel location of Apophis is 0.

####  3.3.2 Alignment of Stars

In a group, we select a high-SNR frame as a master frame such that other
frames in this group should be aligned to. A stacked frame refers to the frame
after stacking, which is equivalent to a master frame taken by a larger-
aperture telescope with the same exposure time. According to what we align to
(the stars or the object), the stacked frames includes star-stacked frames and
object-stacked frames. Flat fielding and bias substracting are made to remove
the difference of pixel response to photoelectrons before stacking. The
procedures of aligning stars are described as follows. Firstly, we obtained
the position of each star image and performed photometry. Thus, the pixel
positions $(x,y)$ and the instrumental magnitude of each star image of the
master frame can be obtained. In the same way, we can derive the positions
$(x$$,y$$)$ and the instrumental magnitude of each individual frame. The same
star locates in both master frame and individual frame in a group can be
indentified conveniently through the relationships of the positions and the
instrumental magnitudes. As for each individual frame, we can calculate the
corresponding position of the master frame through a 6-parameter model
transformation (see equation 2) fitted by a least square method. We have tried
higher order polynomial models (including 12-parameter model and 20-parameter
model) to align the images for the observations of Apophis. However, no
obvious improvement is found for the accuracy and the precision in this set of
observations, which might reflect the stable atmosphere conditions and the
geometric distortion. We also think that high-order polynomial model alignment
will work better in the group of images with different pointings (dithered
frames), different epochs or in a rapidly changing atmosphere. According to
the transformed model, the pixel positions of each individual frame in a group
can be transformed to the master frame using Drizzle method (Fruchter & Hook
2002).

$\left\\{\begin{aligned} x&=ax^{\prime}+by^{\prime}+c\\\
y&=dx^{\prime}+ey^{\prime}+f\\\ \end{aligned}\right.$ (2)

####  3.3.3 Alignment of Object

After each individual frame has been aligned to the master frame according to
the positions of the stars, we are to calculate the shifts in pixel of each
individual frame with respect to the master frame to align the object such as
Apophis. Specifically, we first derive the middle of exposure time of both
master frame and individual frames. Then, we can calculate the object’s right
ascention and declination (apparent positions) according to its ephemeris. In
these frames (including the master frame and each individual frame), we can
obtain the pixel positions $(x,y)$ and the standard coordinate
$($$\xi$,$\eta$$)$ of each star (more details in Section 3.3.6). The standard
coordinate positions $($$\xi_{o}$,$\eta_{o}$$)$ of the object can also be
calculated through the central prejection after considering the atmosphere
refraction. A least squares scheme is used to solve the plate model with a
6-parameter model (see equation 3). Next, we can derive the positions of the
object in the pixel coordinate $(x_{o},y_{o})$ of both master frame and each
individual frame according to the plate model. Finally, the shifts in pixel of
each individual frame with respect to the master frame $\Delta$$x_{o}$ and
$\Delta$$y_{o}$ can be calculated and added to the parameter c and f in
equation 2, respectively after the alignment of stars.

$\left\\{\begin{aligned} \xi&=ax+by+c\\\ \eta&=dx+ey+f\\\
\end{aligned}\right.$ (3)

Figure 10: The angular rate (i.e. proper motions) and the position angle
(P.A.) of Apophis according to JPL ephemeris (5 minutes for interval) during
the observation time on April 14 and 15, 2021. The zero point of the time axis
is set as UTC 12:00 on April 14, 2021.

####  3.3.4 Stack of CCD Frames

After alignment, the frames (including a master frame and nine individual
frames in a group) are stacked with all the corresponding average pixel values
using Drizzle method (Fruchter & Hook 2002). Then, we perform both star-
stacking and object-stacking. In this way, we can derive the positions of
stars from star-stacked frames and the positions of the object from object-
stacked frames. The left and the right panel of Fig.8 show a star-stacked
frame and an object-stacked frame, respectively. We can see the trailing of
Apophis in the star-stacked frame and the trailing of stars in the object-
stacked frame. Fig.9 shows the counts (ADU) across the pixel coordinate (in
both X and Y axis) of Apophis in an individual frame and an object-stacked
frame. In each panel, X Axis shows the pixel location of Apophis, and Y Axis
shows the counts of the corresponding pixel locations. The sky backround is
easier to be distinguished in the object-stacked frames. Compared with the
panels in the individual frame, the distributions of the object-stacked frame
are closer to Gaussian profile.

####  3.3.5 Comparison with Astrometrica’s Stacking

The stacking function of Astrometrica is usually used to survey where we can
set the angular rate (i.e. proper mostions) and the position angle (P.A.) of
the moving object. However, the software assumes that the object goes along a
uniform straight linear motion. Namely, the two parameters (angular rate and
P.A.) have to be set as the constant in a group of frames. To obtain higher-
precision postions of the object with a fast motion, we try to explore the
effect of the changes of angular rate and P.A.. Fig.10 shows the angular rate
and the P.A. of Apophis according to JPL ephemeris changing over time during
the observation. From Fig.10, the angular rate and P.A. change obviously with
respect to the observation time. Compared with Astrometrica, we calculate the
relative shifts of each frame based on the object’s ephemeris shifts and we
consider the relative positions from ephemeris is accurate in a short time.

Figure 11: The left and the right panel show the common stars’ mean (O-C)s of
Astrometrica’s stacking and our stacking method, respectively based on the
observations of April 14, 2021. Figure 12: The left and the right panel show
the common stars’ mean (O-C)s of Astrometrica’s stacking and our stacking
method, respectively based on the observations of April 15, 2021. Figure 13:
The left and the right panel show the common stars’ SDs of Astrometrica’s
stacking and our stacking method based on the observations of April 14, 2021
and April 15, 2021, respectively. Figure 14: The left and the right panel show
the object’s (O-C)s of Astrometrica’s stacking and our stacking method in
right ascension and declination, respectively.

####  3.3.6 Data Reduction

We perform data reduction to the stacked observations from Astrometrica and
stacked observations based on the ephemeris shifts stacking, respectively.
Each frame is reduced by the following procedures. Firstly, we measure the
pixel positions $(x,y)$ of each stars in the star-stacked frames by two-
dimentional Gaussian centering algorithm. In the same way, we can obtain the
pixel positions $(x_{o},y_{o})$ of the object in the object-stacked frames.
Secondly, we calculate the standard coordinate $($$\xi$,$\eta$$)$ of each star
through the central projection (Green 1985). The reference equatorial
coordinates $($$\alpha$,$\delta$$)$ here are taken from the newest Gaia EDR3
catalog (Gaia Collaboration et al. 2021) and calculated to the Astrometric
positions at the observational epoch. The reference positions of the object
(Apophis) are obtained from the JPL ephemeris. 101 and 82 Gaia stars are found
for observations on April 14 and 15, respectively. The least squares scheme is
used to solve the plate model with a weighted fourth-order polynomial (Lin et
al. 2019). In this way, the (O-C)s (observed minus computed) of stars and the
object can be calculated.

####  3.3.7 Astrometric Results

Fig.11-Fig.13 show the mean (O-C)s and standard deviations (SD) of the stars
of star-stacked frames in right ascension and declination stacked by
Astrometrica and stacked based on the ephemeris shifts, respectively. The two
stacking methods show the consistency of both the accuracy and the precision.
Fig.14 shows the (O-C)s of the object (Apophis) by the two stacking methods in
right ascension and declination, respectively and Table 4 gives the mean
(O-C)s and the SDs of the object. The observed topocentric astrometric
positions of the object are listed in Table 5. Also, they can be downloaded
from the website of Sino-French Joint Laboratory for Astrometry, Dynamics and
Space Science of Jinan
University444https://astrometry.jnu.edu.cn/download/list.htm.

Table 4: Statistics of mean (O-C)s and SDs for Apophis over Two Nights Date | Software | Mean (O-C) in RA (arcsec) | SD(arcsec) | Mean (O-C) in DEC (arcsec) | SD(arcsec)
---|---|---|---|---|---
April 14, 2021 | Astrometrica | 0.013 | 0.077 | 0.002 | 0.090
April 14, 2021 | This work | -0.004 | 0.086 | -0.003 | 0.084
April 15, 2021 | Astrometrica | -0.017 | 0.103 | 0.050 | 0.089
April 15, 2021 | This work | -0.036 | 0.104 | 0.046 | 0.082
Total | Astrometrica | -0.001 | 0.090 | 0.024 | 0.091
Total | This work | -0.018 | 0.094 | 0.020 | 0.085
Table 5: The observed topocentric astrometric positions of Apophis. The first column is the Julian Date and it is corresponding to the mid-time of each stacked frame. The observed topocentric astrometric positions in right ascention and declination of Apophis are listed in the second and the third column, respectively. JD | RA(h m s) | DEC(∘ ′ ′′)
---|---|---
2459319.0557558 | 08 02 27.177 | +17 15 57.418
2459319.0640116 | 08 02 26.805 | +17 16 8.856
2459319.0684479 | 08 02 26.604 | +17 16 14.892
2459319.0726389 | 08 02 26.420 | +17 16 20.718
2459319.0768032 | 08 02 26.230 | +17 16 26.452
2459319.0811944 | 08 02 26.039 | +17 16 32.593
2459319.0853657 | 08 02 25.871 | +17 16 38.275
2459319.0894873 | 08 02 25.674 | +17 16 44.046
2459319.0938843 | 08 02 25.490 | +17 16 50.064
2459319.0980764 | 08 02 25.320 | +17 16 55.891
2459319.1022569 | 08 02 25.125 | +17 17 1.474
2459319.1066470 | 08 02 24.956 | +17 17 7.679
2459319.1108032 | 08 02 24.779 | +17 17 13.420
2459320.0561435 | 08 02 3.376 | +17 39 39.363
2459320.0624560 | 08 02 3.115 | +17 39 47.871
2459320.0666019 | 08 02 2.937 | +17 39 53.486
2459320.0713600 | 08 02 2.736 | +17 39 59.924
2459320.0764097 | 08 02 2.524 | +17 40 6.715
2459320.0805058 | 08 02 2.343 | +17 40 12.368
2459320.0897905 | 08 02 1.969 | +17 40 24.804
2459320.0940972 | 08 02 1.811 | +17 40 30.810
2459320.0982535 | 08 02 1.632 | +17 40 36.332
2459320.1042315 | 08 02 1.393 | +17 40 44.267
2459320.1083866 | 08 02 1.222 | +17 40 49.819

From the results above, the dispersions of the two stacking methods show the
consistency. Astrometrica is an open well-integrated astrometric software and
the difference of object’s (O-C)s by the two methods might mainly derive from
the considered velocity model. However, the difference of object’s (O-C)s by
two methods in such precision makes little sense to orbital determination. The
results above also show that the object fainter than 18 mag can be detected
through stacking. With high-credibility accuracy, the positions’ precision of
the object is better than 0.1 arcsec in both right ascension and declination.

####  3.3.8 Discussion

To detect unknown objects or those whose ephemeris is inaccurate, iterative
stacking method (Zhai et al. 2014, Heinze et al. 2015) according to the
stacked SNR of the object would be a better solution, which is promising to
survey. However, this iterative method costs more time and computing resources
of the computer. Besides, the astrometric precisions of the objects are to be
explored because this method concentrates on detecting unknown object after
all. Astrometrica also provides the stacking function, assuming the object
going along a uniform linear motion. The model of iterative stacking survey
has not been supported so far. The iterative stacking method usually adopts
the way of uniform straight linear motion for the objects. However, if the
motion of the object changes significantly during the observation time such as
NEAs, sometimes the effect of the changes of angular rate and its position
angle should be considered. If the relative positions of the object from the
ephemeris are accurate in a short time, we can take the effect of velocity
change into account, which might help us obtain higher-precision astrometric
positions of the object. However, this work only tests the performance of the
stacking object with angular rate of about 1 arcsec per minute. The
performance of moving object with different angular rates hasn’t been tested.
The potential of this telescope at observing fast-moving objects hasn’t been
fully addressed.

##  4 Conclusion and Outlook

In this paper, we have a test to the performance of the 80-cm azimuthal-
mounting telescope at Yaoan Station, Purple Mountain Observatory in stability
of GD, the limit magnitude and the astrometric accuracy and precision of the
stack of fast-moving object Apophis. We find that the geometric distortion of
the CCD is stable in a single epoch and multi epochs. From 8 derived GD
solutions over about one year, the maximum values of each vector ranges from
0.75 to 0.79 pixel. The median value of each vector ranges from 0.14 to 0.16
pixel. About 20.5 magnitude (Gaia-G) stars can be detected with Johnson-V
filter exposured in 300 seconds. The astrometric error is estimated at 0.14
arcsec using the fitted sigmoidal function. 24 groups of stacked observations
are derived in total over two nights based on the ephemeris shifts of Apophis.
After data reduction, our results show that the mean (O-C)s of Apophis are
-0.018 and 0.020 arcsec in right ascention and declination, and the standard
deviation are 0.094 and 0.085 arcsec, respectively. The results based on the
ephemeris shifts have a little systematic deviation of object’s (O-C)s
compared with the results by uniform linear motion stacking of Astrometrica,
which is mainly probably caused by the velocity model of the object. The
astrometric results show that the fast moving object fainter than 18 mag can
be detected through stacking method with the precision better than 0.1 arcsec.
However, this work only tests the stacking of object Apophis, and the
potential of the telescope at observing fast-moving object hasn’t been fully
addressed. We are to explore the limit magnitude of the telescope with other
filters (such as Johnson I filter) and objects with different angular rates
for stacking. Of course, the telescope also has the potentiality to perform
photometry, which will be explored with more proper observations.

###### Acknowledgements.

We appreciate the reviewers (Valéry Lainey and another anonymous reviewer) who
provide the constructive comments. We are grateful for Drs. Li F., Yuan Y.,
and others for the group of 80-cm telescope providing the help of obtaining
observations. We also thank Prof. Fu Y.N. from Purple Mountain Observatory and
you give some valuable advice to our work. This research is supported by the
National Natural Science Foundation of China (Grant No.11873026, 11273014), by
the Joint Research Fund in Astronomy (Grant No.U1431227), by the science
research grants from the China Manned Space Project with NO. CMS-CSST-2021-B08
and Excellent Postgraduate Recommendation Scientific Research Innovative
Cultivation Program of Jinan University. This work has made use of data from
the European Space Agency (ESA) mission _Gaia_
(https://www.cosmos.esa.int/gaia), processed by the _Gaia_ Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the _Gaia_ Multilateral Agreement.

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