11institutetext: INAF - Astrophysical Observatory of Torino, 10025 Pino
Torinese, Italy
11email<EMAIL_ADDRESS>22institutetext: Pulkovo Observatory,
Russian Academy of Sciences, 196140 Saint Petersburg, Russia

# Post-Newtonian gravity and Gaia-like astrometry

Effect of PPN $\gamma$ uncertainty on parallaxes

Alexey G. Butkevich 1122 Alberto Vecchiato 11 Beatrice Bucciarelli 11 Mario
Gai 11 Mariateresa Crosta 11 Mario G. Lattanzi 11

(Received …; accepted …)

###### Abstract

Context. Relativistic models of light propagation adopted for high-precision
astrometry are based on the parametrised post-Newtonian formalism, which
provides a framework for examining the effects of a hypothetical violation of
general relativity on astrometric data. Astrometric observations are strongly
affected by the post-Newtonian parameter $\gamma$ describing the strength of
gravitational light deflection.

Aims. We study both analytically and numerically how a deviation in the PPN
parameter $\gamma$ from unity, which is the value predicted by general
relativity, affects the parallax estimations in _Gaia_ -like astrometry.

Methods. Changes in the observable quantities produced by a small variation in
PPN $\gamma$ were calculated analytically. We then considered how such
variations of the observables are reflected in the parallax estimations, and
we performed numerical simulations to check the theoretical predictions.

Results. A variation in the PPN $\gamma$ results in a global shift of
parallaxes and we present a formula describing the parallax bias in terms of
the satellite barycentric distance, the angle between the spin axis and the
direction to the Sun, and the PPN $\gamma$ uncertainty. Numerical simulations
of the astrometric solutions confirm the theoretical result. The up-to-date
estimation of PPN $\gamma$ suggests that a corresponding contribution to the
_Gaia_ parallax zero point unlikely exceeds 0.2 $\mu$as. The numerical
simulations indicate that the parallax shift is strongly dependent on ecliptic
latitude. It is argued that this effect is due to an asymmetry in the _Gaia_
scanning law and this conclusion is fully validated by additional simulations
with a reversed direction of the precession of the spin axis around the
direction to the Sun.

###### Key Words.:

gravitation – methods: data analysis – methods: statistical – space vehicles:
instruments – astrometry

##  1 Introduction

Among the goals of the ESA _Gaia_ mission (Gaia Collaboration et al., 2016),
probably the most fundamental one is to produce an absolute astrometric
catalogue, which is a realisation of a quasi-inertial reference system on the
sky, down to magnitude $G=20.7$, and with accuracies ranging from 10 to 1000
microarcseconds ($\mu$as) for more than one billion stars. A real breakthrough
for astrometry, with a profound impact on many fields of astronomy and
astrophysics, is the availability of absolute parallaxes at $\sim 10\%$
accuracy on a Galaxy-wide scale. The estimation of these quantities is the
main goal of absolute astrometry, in contrast to relative astrometry, which is
usually able to determine parallaxes only with respect to background objects,
and thus it is affected by systematic errors that depend on these background
references.

Reaching this goal with _Gaia_ , however, is potentially hampered by the
coupling between the parallax and other parameters, which can result in a bias
between the computed and true parallaxes. This systematic error is generally
refered to as the parallax zero point.

Some of these correlations are intrinsically tied to the way _Gaia_ takes its
measurements; others have a more fundamental character and can affect
astrometric measurements in more general cases. The coupling of a global
parallax shift with variations in the basic angle between the two fields of
view (FOVs) (Butkevich et al., 2017) as well as the correlation between the
parallax and the rate of across-scan motion of sources in the FOVs (Lindegren
et al., 2021b) illustrate the correlations of the first type.

The second type of coupling is exemplified well by the correlation between
parallaxes and the so-called parameter $\gamma$, which is one of the
parameters utilised in the parametrised post-Newtonian (PPN) formalism (Will,
2014; Poisson & Will, 2014). This framework allows one to classify and
quantify deviations in a gravity theory from general relativity (GR) at its
first post-Newtonian order. The signature of such deviations is given by the
values of a set of ten parameters, each associated to a specific kind of
difference and related to one relativistic effect or more.

The $\gamma$ parameter of the PPN formalism is related to the amount of space
curvature generated by a unit of mass-energy, and for this reason it directly
enters in the formulae that describe the gravitational light bending. For this
reason, astrometric measurements are well suited to test GR through the
estimation of this PPN parameter.

This task was pursued in the past (Soffel, 1989), but its effectiveness could
only significantly improve with the advent of space astrometry, thanks to a
huge increase in the measurement accuracy, number of observed objects, and
observing time. Space-based astrometric missions have been proposed many times
in the past, but currently only two of them have actually been realised:
HIPPARCOS (ESA, 1997) and _Gaia_.

They are both survey missions based on the same sky-scanning principle, and
for which the most direct way to estimate the value of $\gamma$ is to include
it among the unknowns of the so-called global astrometric sphere
reconstruction (Lindegren et al., 2012; Vecchiato et al., 2018). In this way,
the parameter comes out from the reconstruction of the celestial global
reference system together with the standard astrometric parameters, namely
including the positions, parallaxes, and proper motions.

Therefore, the task of the sphere reconstruction turns into a huge test of
gravity theories at the same time, and this is exactly what is planned for
_Gaia_. On the other hand, its forerunner, HIPPARCOS, adopted a strategy based
on an a posteriori estimation of $\gamma$ from the residuals of the sphere
reconstruction (Froeschlé et al., 1997).

From what was mentioned above, however, it is clear that the existing
correlation between parallaxes and the PPN parameter $\gamma$ can jeopardise
the scientific results of an astrometric mission in many ways, and of a _Gaia_
-like one in particular. For example, a systematic bias on the parallaxes
could undermine the accuracy of the PPN $\gamma$ parameter coming from the
sphere reconstruction. Or, alternatively, an error on the latter might result
in biased parallaxes.

Such entanglement, from the point of view of a global astrometric mission, is
extremely worrisome because it can put the determination of absolute
parallaxes at stake, that is the most important scientific goal of these kinds
of missions. In this sense, it is worth mentioning that the estimated global
parallax zero point amounts to about $-17\leavevmode\nobreak\ \mu\mathrm{as}$
in the recently published _Gaia_ Early Data Release 3 (EDR3) (Lindegren et
al., 2021a).

General relativity is one of the cornerstones of modern physics and its
predictions have been confirmed in numerous experiments; alternative theories
of gravity are, nevertheless, thought to be viable (Will, 2014). The effects
of a deviation from GR on the outcome of a _Gaia_ -like astrometric experiment
have not been systematically studied so far. This paper aims to investigate
such issues, with particular attention being paid to the impact of a biased
PPN $\gamma$ on the determination of parallaxes in a _Gaia_ -like mission.

The outline of the paper is as follows. Section 2 gives an analytical
treatment for the problem. In Sect. 3 we present the results of the numerical
simulations that confirm the theoretical expectation. Section 4 explores the
dependence of the parallax bias on the star position. The results are
discussed in Sect. 5.

##  2 Theory

The correlation between parallaxes and the PPN-$\gamma$ parameter arises from
the similarity in the way the light deflection and parallax affect observed
directions. An increasing light deflection (i.e. a small increase in PPN
$\gamma$) results in a shift in the direction outwards from the centre of the
deflecting body, whereas rising parallaxes lead to a direction shift towards
the Solar System barycentre (SSB). The Sun is responsible for the most
significant part of the light deflection. For a scanning mission, utilising
the _Gaia_ -like design, large angular separations are kept between the Sun
and objects of observations; for example, the _Gaia_ observations are carried
out 45° to 135° away from the Sun. In such case, the great circles from the
observed source to the solar centre and to the SSB are almost coincident and
the effects of the light deflection and parallax are, therefore, nearly
parallel and opposite in direction (Fig. 2).

This intuitive explanation is more thoroughly detailed in the following
subsections, where we provide a theoretical background and derive some useful
formulae. An analysis of the effect of the trigonometric parallax and PPN
$\gamma$ on observables shows a strong correlation between these two
quantities, in agreement with earlier results. To further test the influence
of such correlation, we simulated Gaia-like observations using a perturbed
$\gamma$ value, and then we performed a least-squares solution of the sole
astrometric parameters by assuming the GR value of $\gamma$. Finally, we used
this approach to determine how a systematic parallax error biases the
estimation of the parameter $\gamma$ and vice versa. For brevity, all
distances are expressed in the astronomical unit $A$ hereafter.

###  2.1 Basic equations

We first calculated variations in the direction to a star due to small changes
in the trigonometric parallax $\varpi$ and PPN parameter $\gamma$. The
relationship between the direction $\@vec{u}$ to a star and its astrometric
parameters is described by the astrometric model. This model, given by Eq. (4)
of Lindegren et al. (2012), suggests that if the parallax receives a small
perturbation $\delta\varpi$, the direction is changed by

$\delta\@vec{u}_{\varpi}=\@vec{u}\times\left(\@vec{u}\times\@vec{R}\right)\delta\varpi\,,$
(1)

where $\@vec{R}$ is the position of the observer with respect to the SSB.

In the framework of the PPN formalism, the light deflection by the solar
gravitational field is given by (see, for example, Lindegren et al. (1992),
Klioner (2003) and Poisson & Will (2014), Sect. 13.3.2)

$\Delta\@vec{u}=-\@vec{u}\times\left(\@vec{u}\times\@vec{R}_{\mathrm{h}}\right)\frac{1+\gamma}{2}\frac{r_{\mathrm{s}}}{R_{\mathrm{h}}\left(R_{\mathrm{h}}+\@vec{u}^{\prime}\@vec{R}_{\mathrm{h}}\right)}\,,$
(2)

where $r_{\mathrm{s}}\equiv 2GM_{\sun}/Ac^{2}=1.974\times
10^{-8}\,\mathrm{au}$ is the Schwarzschild radius of the Sun,
$\@vec{R}_{\mathrm{h}}$ is the heliocentric position of the observer, and the
prime signifies vector scalar multiplication. As the Sun orbits the
barycentre, the barycentric distance of the solar centre does not exceed 0.01
au (Perryman & Schulze-Hartung, 2011). Therefore, $\@vec{R}$ and
$\@vec{R}_{\mathrm{h}}$ differ by, at most, 0.6° in direction and 1% in length
for a satellite operating in the vicinity of Earth or near the L2 point. Such
a small difference suggests that we can neglect the distinction between the
barycentric and heliocentric position. For this reason, we use $\@vec{R}$
instead of $\@vec{R}_{h}$ in what follows.

Thus, we found that the variation in the observed direction due to a small
change in PPN $\gamma$ can be written as

$\delta\@vec{u}_{\gamma}=-\@vec{u}\times\left(\@vec{u}\times\@vec{R}\right)\frac{r_{\mathrm{s}}}{2R^{2}}\,\frac{\delta\gamma}{1-\cos\chi}\,,$
(3)

with $\chi$ being the angular distance between the Sun and the star as seen by
the observer, $\@vec{u}^{\prime}\@vec{R}=-R\cos\chi$. The minus sign is
present because $\@vec{R}$ is the vector from the barycentre to the observer,
while $\@vec{u}$ points from the observer towards the star.

The barycentric distance $R$ is not strictly constant but slightly time-
dependent. For instance, the _Gaia_ barycentric distance ranges from 0.99 to
1.03 au. We regard $R$ as a constant value assuming $R=1.01\,\mathrm{au}$.

###  2.2 Gaia-like observations

_Gaia_ -like observations are conveniently described in terms of the so-called
scanning reference system (SRS) (Lindegren et al., 2012). The reference system
and relevant spherical coordinates are illustrated in Fig 1.

Figure 1: Scanning reference system (SRS). The axes $\@vec{x}$ and $\@vec{y}$
define the instantaneous scanning circle, with the axis $\@vec{z}$ being along
the spin axis. The heliotropic spin phase $\Omega$ and solar aspect angle
$\xi$ specify the direction towards the Sun. The observed direction $\@vec{u}$
is given by the along- and across-scan instrumental angles $\varphi$ and
$\zeta$, repectively.

The SRS is defined by the instrument axes $\@vec{x}$, $\@vec{y}$, and
$\@vec{z}$. The axis $\@vec{x}$ bisects the basic angle between the two
viewing directions, $\@vec{z}$ is the direction along the spin axis, and
$\@vec{y}=\@vec{z}\times\@vec{x}$ completes the right-handed triad. The
spherical coordinates $\varphi$ and $\zeta$, referred to as the instrument
angles, specify the direction towards the star:

$\@vec{u}=\@vec{x}\cos\varphi\cos\zeta+\@vec{y}\sin\varphi\cos\zeta+\@vec{z}\sin\zeta\,.$
(4)

Similarly, the two angles $\Omega$ and $\xi$ specify the direction to the Sun
in the SRS. The solar aspect angle $\xi$, that is the angle between the spin
axis and the direction to the Sun, is almost constant. For _Gaia_ , $\xi$ was
set to 45°; a slightly smaller value of 43° was chosen for Hipparcos. The spin
phase $\Omega$, being the angle between the direction to the Sun and the plane
$xz$, increases with time at a nearly constant rate. Thus, having considered
the SSB coincident with the centre of the Sun, the barycentric position of the
satellite in the SRS is

$\@vec{R}=R\left(-\@vec{x}\cos\Omega\sin\xi+\@vec{y}\sin\Omega\sin\xi-\@vec{z}\cos\xi\right)\,.$
(5)

This equation takes into account the fact that the vector $\@vec{R}$ is from
the barycentre to the satellite.

The dot product of $\@vec{u}$ and $\@vec{R}$ gives the angular separation
between the Sun and it observed the star in terms of the SRS:

$\cos\chi=-\@vec{u}^{\prime}\@vec{R}/R=\sin\zeta\cos\xi+\cos\zeta\sin\xi\cos\theta\,.$
(6)

Here, $\theta=\Omega+\varphi$ is the along-scan coordinate, that is the angle
between the Sun and $\@vec{u}$ measured around the $\@vec{z}$ axis in the
direction of the satellite spin. Disregarding the small across-scan extent of
the FOVs, it shows that a scanning satellite observes objects $90\degr-\xi$ to
$90\degr+\xi$ away from the Sun. For _Gaia_ , with $\xi=45\degr$, the angular
separation ranges from 45° to 135°.

Linearising Eq. (4) with respect to $\varphi$, we found that the change in the
direction due to a small variation $\delta\varphi$ is

$\delta\@vec{u}=\left(-\@vec{x}\sin\varphi\cos\zeta+\@vec{y}\cos\varphi\cos\zeta\right)\delta\varphi=\left(\@vec{z}\times\@vec{u}\right)\delta\varphi\,.$
(7)

We neglect the finite size of the FOV and simply assume that the observations
are only performed on the scanning circle $\left(\zeta=0\right)$ in what
follows. For _Gaia_ , with the across-scan FOV size being less than 1°, this
assumption is valid to within 1% (Butkevich et al., 2017). In this
approximation, the cross product $\@vec{z}\times\@vec{u}$ represents a unit
vector along the spin direction. Taking the dot product of Eq. (7) and
$\@vec{z}\times\@vec{u}$, we obtained the variation in the along-scan
coordinate due to a change in the direction

$\delta\varphi=\left(\@vec{z}\times\@vec{u}\right)^{\prime}\delta\@vec{u}\,.$
(8)

Substituting the variations of the observed direction from Eqs. (1) and (3)
into Eq. (8), we obtained the shifts of the along-scan coordinate due to
changes in the parallax and PPN $\gamma$

$\displaystyle\delta\varphi_{\varpi}$
$\displaystyle=-R\sin\xi\sin\theta\,\delta\varpi\,,$ (9)
$\displaystyle\delta\varphi_{\gamma}$
$\displaystyle=\frac{r_{\mathrm{s}}}{2R}\frac{\sin\xi\sin\theta}{1-\sin\xi\cos\theta}\,\delta\gamma\,.$
(10)

Deriving Eq. (10), we used $\cos\chi$ from Eq. (6) with $\zeta=0$.

Equations (9) and (10) describe the projections of displacement of the stellar
image on the scanning circle. Similar formulae were derived by Froeschlé et
al. (1997) and Hobbs et al. (2009) from geometrical considerations.

###  2.3 Size of the effects

Figure 2: Effect of changes in the light deflection and parallax on the along-
scan coordinate $\varphi$. The red line, calculated from Eq. (10) with
$\delta\gamma=2\times 10^{-5}$, shows the maximum shift expected under current
constraints on PPN $\gamma$. The blue line, computed using Eq. (9),
demonstrates how $\delta\varphi_{\varpi}$ varies with $\theta$ if the parallax
increases by 11.3 $\mu$as.

The most accurate estimate of PPN $\gamma$ as of yet, which was obtained from
Doppler tracking of the Cassini spececraft (Bertotti et al., 2003; Will,
2014), is $\gamma-1=\left(2.1\pm 2.3\right)\times 10^{-5}$. This places a
constraint on the expected variation in observables because the absolute value
of $\delta\gamma$ is not likely to be much larger than $2\times 10^{-5}$. The
shift $\delta\varphi_{\gamma}$ corresponding to this value of $\delta\gamma$
is shown in Fig. 2 as a function of the along-scan angle $\theta$, together
with the shift due to parallax.

The close similarity between the effects visible in Fig. 2 suggests a strong
correlation between them. Considering $\delta\varphi_{\varpi}$ and
$\delta\varphi_{\gamma}$ as functions $\theta$, Hobbs et al. (2009)
analytically demonstrated that the coefficient of correlation between the
along-scan components of the light deflection and parallactic effects is 0.91
for the _Gaia_ solar aspect angle.

We now estimate the size of the effects. The maximum absolute value of the
parallactic one is $R\sin\xi\,\delta\varpi$, while the maximum absolute value
of the along-scan light deflection, attained at $\theta=90\degr-\xi$ or
$\theta=270\degr+\xi$, is
$\left(r_{\mathrm{s}}/2R\right)\tan\xi\,\delta\gamma$. Assuming $R=1$ and
keeping in mind that the effect is of an opposite sign, we see that the
maximum effects are equal when

$-2\cos\xi\,\delta\varpi\simeq r_{\mathrm{s}}\,\delta\gamma\,.$ (11)

Expressing parallax in $\mu$as and $\gamma$ in units of $10^{-5}$, we find
that, for the _Gaia_ solar aspect angle of 45°, the following relation holds

$\frac{\delta\varpi\,\left[\mu\mathrm{as}\right]}{\delta\gamma\,\left[10^{-5}\right]}\simeq-\frac{1}{35}\,.$
(12)

This proportion illustrates the relative values of the effects. If, for
instance, a parallax increase of 10 $\mu$as occurs, this corresponds to a
variation in the observables, which can be attributed to a $\sim 350\times
10^{-5}$ drop in PPN $\gamma$. Conversely, a variation in the observables
resulting from a $10^{-5}$ rise in PPN $\gamma$ can be described by a parallax
descrease of $\sim 1/35\,\mu\mathrm{as}$, that is to say $\sim
30\,\mathrm{nas}$.

This treatment is applicable to the parallax of any star, and therefore
$\delta\varpi$ can be interpreted as a global shift of the parallaxes. A
systematic deviation in parallaxes from their true values is referred to as a
parallax zero point. Equation (12) thus relates the PPN $\gamma$ uncertainty
and the parallax zero point. Although this relation is approximate, it is
convenient for order-of-magnitude estimations. A more accurate formula for the
effect of PPN $\gamma$ on the parallax zero point is derived below in Sect.
2.4.

###  2.4 Impact of PPN $\gamma$ on the parallax zero point

The variation in the along-scan instrument angle $\varphi$ due to a change in
the parallax of a star is given by Eq. (9). Considering all observations of a
given star, this relation can be written in the matrix form as follows:

$\@vec{p}\,\delta\varpi=\delta\@vec{\varphi}\,,$ (13)

where $\@vec{p}$ and $\delta\@vec{\varphi}$ are the vectors of length $n$, the
number of observations of the star in question. The elements of vector
$\@vec{p}$ correspond to individual observations, labelled by the index $l$:

$p_{l}=-R\sin\xi\sin\theta_{l}\,.$ (14)

Equation (13) represents a system of $N$ linear equations with one unknown
$\delta\varpi$. To find its solution in the sense of a least-squares
estimation, we first constructed the normal equation by computing its scalar
product with vector $\@vec{p}$:

$\@vec{p}^{\prime}\@vec{p}\,\delta\varpi=\@vec{p}^{\prime}\delta\@vec{\varphi}\,.$
(15)

The solution of the normal equation gives a least-squares estimate of the
parallax shift

$\delta\varpi=\frac{\@vec{p}^{\prime}\delta\@vec{\varphi}}{\@vec{p}^{\prime}\@vec{p}}\,.$
(16)

This formula is generally valid because it is derived without making any
assumptions concerning the variations $\delta\@vec{\varphi}$. We now apply it
to the case of variations due to an uncertainty in the light deflection
parameter $\gamma$.

According to Eq. (10), variations of the along-scan instrument angle due to a
change in PPN $\gamma$ are

$\delta\@vec{\varphi}=\@vec{g}\,\delta\gamma\,,$ (17)

where the entries of the vector $\@vec{g}$ are

$g_{l}=\frac{r_{\mathrm{s}}}{2R}\frac{\sin\xi\sin\theta_{l}}{1-\sin\xi\cos\theta_{l}}\,.$
(18)

Substituting Eq. (17) into Eq. (16), we find an estimation of the parallax
bias

$\delta\varpi_{\gamma}=\frac{\@vec{p}^{\prime}\@vec{g}}{\@vec{p}^{\prime}\@vec{p}}\,\delta\gamma\,.$
(19)

It is convenient to use relevant mean values instead of the scalar products
$\@vec{p}^{\prime}\@vec{p}$ and $\@vec{p}^{\prime}\@vec{g}$:

$\@vec{p}^{\prime}\@vec{p}=\sum_{l=1}^{n}p_{l}^{2}=n\left\langle
p^{2}\right\rangle\,,\quad\@vec{p}^{\prime}\@vec{g}=\sum_{l=1}^{n}p_{l}g_{l}=n\left\langle
pg\right\rangle\,.$ (20)

Replacing the averaging over observations with averaging over the along-scan
angle, $\left\langle
x\right\rangle=\left(1/2\pi\right)\int_{0}^{2\pi}x\left(\theta\right)\,\mathrm{d}\theta$,
we find

$\displaystyle\left\langle p^{2}\right\rangle$
$\displaystyle=R^{2}\frac{\sin^{2}\xi}{2}\,,$ (21) $\displaystyle\left\langle
pg\right\rangle$ $\displaystyle=-r_{\mathrm{s}}\sin^{2}\left(\xi/2\right)\,.$
(22)

Finally, we find the parallax shift

$\displaystyle\delta\varpi_{\gamma}$
$\displaystyle=-\frac{r_{\mathrm{s}}}{2R^{2}}\frac{\delta\gamma}{\cos^{2}\left(\xi/2\right)}=-\frac{GM}{Ac^{2}}\left(\frac{A}{R}\right)^{2}\frac{\delta\gamma}{\cos^{2}\left(\xi/2\right)}$
(23)
$\displaystyle=-0.023\,\mu\mathrm{as}\times\left(\frac{\delta\gamma}{10^{-5}}\right)\quad\mbox{for
}R=1.01\mbox{ and }\xi=45\degr\,.$

This formula gives the solution to the problem. It determines the parallax
zero point originating from a PPN $\gamma$ uncertainty for _Gaia_ -like
astrometry. According to today’s best estimate by Bertotti et al. (2003), a
3-$\sigma$ deviation of $\gamma-1$ from the GR value of zero amounts to
$9\times 10^{-5}$. The expected effect of the parallax zero point is 0.2
$\mu$as. This tiny effect is truly negligible for the _Gaia_ astrometry.

Equation (23) presents the sky-averaged parallax bias. The bias is furthermore
subject to zonal variations due to different observational configurations for
various star positions. This is well exemplified by areas close to the
ecliptic poles. Straightforward geometrical arguments suggest that such stars
are observed only when $\theta\simeq 90\degr$ or 270°. Accordingly, in this
case, $\@vec{p}^{\prime}\@vec{p}\simeq R^{2}\sin^{2}\xi$ and
$\@vec{p}^{\prime}\@vec{g}\simeq-\left(r_{\mathrm{s}}/2\right)\sin^{2}\xi$ and
the parallax shift becomes

$\delta\varpi_{\gamma}\simeq-\frac{r_{\mathrm{s}}}{2R^{2}}\,\delta\gamma=-0.020\,\mu\mathrm{as}\times\left(\frac{\delta\gamma}{10^{-5}}\right)\quad\mbox{for
}R=1.01\,.$ (24)

Thus the parallax bias is independent of the solar aspect angle near the
ecliptic poles. We further address the bias dependence of star position in
Sect. 4.

##  3 Monte Carlo experiments

To test the consistency of the theory, numerical tests were conducted. In the
following, we first describe the basic computational tasks, that is the
generation of simulated _Gaia_ -like observations and the astrometric
solution, which uses a numerical technique to estimate astrometric parameters
from the simulated observations. We then employ a statistical approach to
assess the main characteristics of the simulations outcome. Finally, we
present the results of the Monte Carlo experiments.

###  3.1 Simulated data and astrometric solution

The key ingredients of the simulation process are a true astrometric
catalogue, and a procedure for generating the observations and measurement
uncertainties. For the true catalogue, we simply generated a set of stars
uniformly distributed over the sky. In accordance with the theoretical
treatment in Sect. 2.2, we considered only along-scan measurements. For each
star, we calculated the along-scan angle $\varphi$ at the moment when it
crosses the middle of the relevant FOV. We, of course, do not include in our
simulations’ numerous complexities, which hamper real satellite observations
such as attitude noise and inadequacy of the calibration. These effects are
modelled as random noise with formal uncertainty $\sigma_{\varphi}$ (specified
below).

The observations were simulated with the nominal _Gaia_ scanning law (Gaia
Collaboration et al., 2016), which is a combination of three rotations (Fig.
1). First, the satellite rotates about the spin axis $\@vec{z}$ with a period
of 6 hours, that is the spin phase $\Omega$ linearly increased at the rate
$\dot{\Omega}=60\arcsec\,\mathrm{s}^{-1}$. At the same time, the spin axis is
in a slow precessional motion about the direction to the Sun completing 5.8
revolutions per year, with the solar aspect angle $\xi$ kept fixed. Finally,
the Sun’s motion along the ecliptic provides the coverage of the whole
celestial sphere. The Sun advances 62° during one precession period. This
‘revolving scanning’ produces a quasi-uniform temporal sampling and features a
significant dependence on the ecliptic latitude.

Analysis of the post-fit residuals of the along-scan measurements in the EDR3
astrometric solution indicated that they are smallest for bright stars with
$9\la G\la 13$ (Appendix A in Lindegren et al., 2021b). Whereas the residuals
range from 100 to 200 $\mu$as, the formal photon-statistical errors are
between 50 and 70 $\mu$as for these stars. Attributed to unmodelled errors,
this difference will decrease in future data releases due to improved
calibration. This is the best current estimation of the accuracy of the along-
scan measurements and we adopted $\sigma_{\varphi}=100\,\mu\mathrm{as}$ as a
representative value of the uncertainty. Furthermore, both the residuals and
formal errors in EDR3 are weakly dependent on magnitude for $9\la G\la 13$.
We, therefore, ignored the difference in stellar brightness and used the same
uncertainty in all simulated observations.

Figure 3: Number of FOV transits per source in the equatorial coordinates for
simulations covering 5 years of satellite operation. The distribution displays
the essential characteristics of the _Gaia_ scanning law, such as the rough
symmetry around the ecliptic plane and high concentration of observations at
ecliptic latitudes $\pm\left(90\degr-\xi\right)=\pm 45\degr$.

The simulation process is straightforward. We assumed a Gaia-like FOV
geometry: FOVs are rectangular with a nominal across- and along-scan size of
0.7° and 1°, respectively. The along-scan size slightly exceeded the _Gaia_
value of 0.7°; this was done only for convenience of computation and does not
affect the results. The scanning law provided positions and orientations of
the FOVs on the celestial sphere at any instance of time $t_{l}$. For each
star fitting inside a FOV, its true instantaneous position was calculated
using the _Gaia_ astrometric model (Lindegren et al., 2012). To account for
the deviation from GR, light deflection was calculated with
$\gamma-1=10^{-5}$. Finally, a transformation to a SRS provided a true value
of the along-scan angle that was then perturbed with a Gaussian error with
variance $\sigma_{\varphi}$ to simulate the observed value $\varphi_{l}$. This
procedure generated, for each star, a set of pairs
$\left(t_{l},\varphi_{l}\right)$, which were treated as measurements and
served as input data for the subsequent astrometric solution.

In general, such an astrometric solution is a computationally heavy task, and
it requires high-performance computing facilities, such as in the case of
_Gaia_ (Lindegren et al., 2012; Vecchiato et al., 2018). Such a computational
complexity is mainly due to the need of solving for astrometric, attitude, and
calibration unknowns at the same time. For our purposes, however, we need to
solve the astrometric sphere for just the five conventional astrometric
parameters (positions, parallax, and proper motions) for each star. Such a
task corresponds to the solution of only the astrometric block of the _Gaia_
Astrometric Global Iterative Solution (Lindegren et al., 2012), namely a
linear system of equations whose normal matrix is block-diagonal, with each
block corresponding to the $5\times 5$ normal matrix of the astrometric
parameters of a single star. The system is therefore embarrassingly parallel,
which makes the solution much easier.

###  3.2 Small-scale solutions

As first test, we considered 20 000 astrometric solutions, each containing 100
000 stars uniformly distributed over the sky. The observations covered 5 years
with no dead time. Figure 3 shows the sky map of the observation density. For
each solution, we calculated the global parallax shift, defined as the average
difference between the computed and true parallaxes. Thus, this Monte Carlo
experiment provided 20 000 estimates of the parallax zero point, whose
distribution is shown in Fig. 4. The mean zero point is $-0.023$ $\mu$as and
the standard deviation is 0.072 $\mu$as. The mean agrees well with the
theoretical prediction given by Eq. (23) and we now demonstrate that the
scatter is also in agreement with the expected outcome of this kind of
experiment.

Figure 4: Histogram of the parallax zero point found in 20 000 astrometric
solutions for $10^{5}$ stars using simulated observations with
$\gamma-1=10^{-5}$. The distribution has a mean of $-0.023$ $\mu$as, shown
with the vertical solid line, while its standard deviation is 0.072 $\mu$as.
The red curve is a Gaussian distribution with the same parameters.

Equation (16) is valid for an arbitrary variation in the along-scan instrument
angle, including both systematic and random ones. Above, we considered the
systematic variations due to a change in PPN $\gamma$. With random errors
included, the instrument angle variation is modelled as the sum of two
components,

$\delta\varphi_{l}=g_{l}\,\delta\gamma+\varepsilon_{l}.$ (25)

The noise term $\varepsilon_{l}$ is assumed to be a centred, uncorrelated
random variable with standard deviation $\sigma_{\varphi}$. Substituting this
formula in the general Eq. (16), we find a change in the parallax of a given
star, designated by the index $i$,

$\delta\varpi_{i}=\delta\varpi_{\gamma}-\frac{2}{R\sin\xi}\frac{1}{n_{i}}\sum_{l=1}^{n_{i}}\varepsilon_{l}\sin\theta_{l}\,,$
(26)

where $\delta\varpi_{\gamma}$ is the parallax shift induced by a variation in
PPN $\gamma$, given by Eq. (23), and $n_{i}$ is the number of observations of
star $i$.

Averaging individual parallax errors gives the global parallax offset for a
specific solution

$\Delta\varpi=\frac{1}{N}\sum_{i=1}^{N}\delta\varpi_{i}=\delta\varpi_{\gamma}-\frac{2}{R\sin\xi}\frac{1}{N}\sum_{i=1}^{N}\frac{1}{n_{i}}\sum_{l=1}^{n_{i}}\varepsilon_{l}\sin\theta_{l}\,,$
(27)

where $N$ is the total number of stars in the solution. In our simulations,
the observations are statistically independent. Noticing that
$\left\langle\varepsilon^{2}\right\rangle=\sigma_{\varphi}^{2}$ and
$\left\langle\sin^{2}\theta\right\rangle=1/2$, we therefore found the
estimation of the standard deviation in the parallax zero point

$\sigma_{\Delta\varpi}=\frac{\sigma_{\varphi}}{R\sin\xi}\sqrt{\frac{2\left\langle
1/n\right\rangle}{N}}\,,$ (28)

where $\left\langle
1/n\right\rangle=\left(1/N\right)\sum_{i=1}^{N}\left(1/n_{i}\right)$ is the
harmonic mean of the number of observations per star; $\left\langle
1/n\right\rangle=0.012$ for the 5 yr mission.

Figure 5: Mean parallax shift, in two-degree bins, versus ecliptic longitude
in the solution with 14 million stars for a mission length of 10 years. The
blue curve is the running mean smoothed with a linear kernel of width 13. The
theoretical value of $-0.023$ $\mu$as is shown with the red line.

For the _Gaia_ parameters, Eq. (28) gives $\sigma_{\Delta\varpi}=0.069\
\mu\mathrm{as}$, which is in agreement with the experimental value of 0.072
$\mu$as. Figure 4 shows that the distribution of the parallax shift mainly
follows a normal one. The slight excess on the left to the peak is
attributable to various small effects, such as the finite FOV size and weak
dependence of $R$ and $\xi$ on time, as well as to the scanning law asymmetry
discussed in Sect. 4. Thus, the results of this Monte Carlo experiment confirm
the theoretical expectations for the basic statistical properties of the
parallax zero point.

###  3.3 Large-scale solution

The small-scale simulation, considered in Sect. 3.2, allows us to examine the
parallax zero point, only in a statistical sense, using a large number of
independent solutions. We now aim to study the impact of a PPN $\gamma$
variation on the zero point by conducting one large-scale astrometric solution
with better accuracy for a vastly increased number of stars. For a fixed
measurement uncertainty, the overall accuracy of a solution can be improved by
extending the time coverage. To this end, we considered a mission that was
twice as long, lasting 10 yr. The formal uncertainties of the parallaxes scale
with the observation duration as $T^{-1/2}$ (Lindegren et al., 2021b). Thus, a
twofold increase in the mission length results in an improvement in the
accuracy by a factor 0.71 for the parallaxes. The 10-yr duration corresponds
to the foreseen _Gaia_ extended mission lifetime (Gaia Collaboration et al.,
2021). Accordingly, we conducted a solution with 14 million stars, which is
roughly equal to the number of primary stars in the Gaia EDR3.

A vastly increased number of observations brought substantial improvement for
stability of the parallax bias estimation against noise. The parallax zero
point of $-0.020$ $\mu$as was detected in the large-scale astrometric
solution. The median parallax uncertainty was found to be 17 $\mu$as, and
therefore the expected precision of the zero point determination is $\left(17\
\mu\mathrm{as}\right)\times[14\times 10^{6}]^{-1/2}\simeq 4.5\times 10^{-3}\
\mu\mathrm{as}$. Thus, the detected value of the zero point agrees with the
theoretical prediction of $-0.023$ $\mu$as to within one standard deviation.
There is, however, a considerable scatter in the data, as illustrated in Fig.
5. The points show a mean parallax shift, computed in bins of two degrees, as
a function of ecliptic latitude. The bins contain 78 000 stars on average. The
values in the bins range from $-0.17$ to $0.12$ $\mu$as.

We carried out an additional test to examine the consistency of the parallax
zero point estimation: all of the data were split into odd- and even-numbered
stars, resulting in two subsets of 7 million stars each. Independent
astrometric solutions were calculated for the two data sets and the mean
parallax shifts were found to be $-0.025$ and $-0.016$ $\mu$as. The formal
uncertainty is $6.4\times 10^{-3}\ \mu\mathrm{as}$ for $7\times 10^{6}$ stars,
and therefore these results also confirm the theoretical prediction at a
1-$\sigma$ level. The good agreement between the theoretical expectation and
the experimental results demonstrates the validity of the mathematical
treatment and correct functioning of the simulation procedure.

Figure 6: Map of the parallax error in ecliptic coordinates obtained in the
astrometric solution using noise-free observations of five million stars for
an extended 10 yr mission with the nominal _Gaia_ scanning law.

##  4 Dependence of parallax bias on star position

We now turn to a study of how the parallax bias varies with the distance of a
star from the ecliptic. This kind of systematics presumably exists by virtue
of the strong dependence of the scanning law on ecliptic latitude (Fig. 3).
Providing clear evidence for the parallax zero point due to a variation in the
PPN, the large-scale solution does not enable us to study the finer details of
the bias because of a considerable scatter among individual stars. Even the
largely smoothed data, drawn in Fig. 5 with the blue curve, are dominated by
random variations around the mean.

In our numerical tests, with the GR violation parametrised as
$\gamma-1=10^{-5}$ and the current assessment of the Gaia observation
uncertainty of 100 $\mu$as, the mean value of the parallax bias, $-0.023$
$\mu$as, is three orders of magnitude less than the typical accuracy of
individual parallaxes, being 17 $\mu$as for a 10 yr mission. In the next
simulation run, we considered five million stars randomly distributed on the
sky. With the average star density of 121 deg-2, these sources constitute a
quasi-uniform grid dense enough to accurately map the parallax bias over the
entire celestial sphere. The generation of measurement noise was not activated
in this simulation to yield noiseless observations.

The all-sky map of the resulting parallax error is drawn in Fig. 6. The mean
of this distribution is $-0.024$ $\mu$as, with a dispersion of $0.001$
$\mu$as; this bias confirms the theoretical expectations. Although the
appearance of the parallax error generally bears the pattern of the revolving
scanning, it has a noteworthy feature: the north-south asymmetry in ecliptic
coordinates. The absolute value of the bias in the northern ecliptic
hemisphere is systematically larger compared to the southern one. This
phenomenon is further illustrated in Fig. 7, where the dependence of the
parallax error on the ecliptic latitude is shown. The asymmetry is clearly
seen in this plot.

The scanning law provides the sky coverage, which is geometrically symmetric
with respect to the ecliptic equator. As exemplified in Fig. 3, the scanning
density does not exhibit a systematic dependence on the ecliptic latitude,
apart from the slight non-uniformity in the ecliptic belt, that is for
$\left|\beta\right|\la 45\degr$, due to an interaction of the solar orbital
motion and the spin axis precession. It is, therefore, natural to assume that
the parallax bias possesses the same symmetry. The numerical experiments,
however, make us call this expectation into question. We argue below that such
an intricate behaviour of the bias results from an asymmetry inherent in the
revolving scanning.

Figure 7: Parallax error plotted against the ecliptic latitude in the
astrometric solution with five million stars with the nominal _Gaia_ scanning
law. The cyan curve is the running median.

The critical factor in both the parallactic effect and gravitational light
deflection is the angle between directions to the star and the Sun. We
examined the distribution of this angle and found that for stars in the
northern ecliptical polar cap $\left(\beta\ga 46\degr\right),$ its mean value
is 87$\aas@@fstack{\circ}$3; whereas, in the southern cap
$\left(\beta\la-46\degr\right),$ the mean is 92$\aas@@fstack{\circ}$7\. Thus,
the stars in the northern cap are mainly observed closer to the Sun, compared
to those in the southern cap.

The reason for that can be the following. In the nominal _Gaia_ scanning law,
the spin axis describes the loops is the ecliptic belt. The spin axis rotates
counterclockwise, as seen from the satellite (Gaia Collaboration et al.,
2016). As a result, the direction of the precession motion coincides with the
direction of the Sun for $\beta<0$, whereas for positive $\beta$ these
directions are the opposite. This means that the axis covers a longer path in
the southern ecliptic hemisphere. Corresponding scanning circles cover a
larger area and consequently contain more stars.

For this reason, stars in the northern hemisphere are observed closer than 90°
to the Sun on average, while the stars in the southern hemisphere are observed
further than 90° away from the Sun on average. The parallactic effect is
symmetric with respect to 90°. On the contrary, the light deflection rises as
the angle to the Sun decreases. In the north, this angle is systematically
smaller than 90°, while in the south it is systematically larger than 90°.
Consequently, the light deflection is stronger for the stars in the northern
polar cap for a Gaia observation. As a result, the detected parallax shift is
also larger in the north.

To verify this hypothesis, an additional simulation was performed with a
reversed motion of the spin axis. In this case, the spin axis spends more time
in the northern ecliptic hemisphere. The dependence of the parallax bias on
the ecliptic latitude found in the experiment with the reversed precession is
displayed in Fig. 8. Comparison of Figs. 7 and 8 shows that the changes of the
precession direction have an inverse effect on the parallax bias. This
provides evidence for the adequacy of the proposed explanation for the
asymmetry in the parallax bias.

Figure 8: As Fig. 7, but for the _Gaia_ scanning law with a reversed direction
for the precession, with the spin axis revolving around the Sun clockwise, as
seen from the satellite.

##  5 Discussion and conclusions

In this paper we investigated the consequences of the well-known correlation
between parallaxes and the PPN $\gamma$ parameter in astrometric observables,
with an emphasis on the impact of a bias of the PPN $\gamma$ parameter on the
resulting parallax uncertainties in _Gaia_ -like missions. This systematic
error translates into a corresponding bias for parallaxes, known as the zero-
point parallax error, and the existence of such an issue in the current _Gaia_
solutions was one of the motivations that inspired our investigation.

A straightforward statistical analysis provided a relation, never before
mentioned in the literature, between the zero-point parallax error and the PPN
$\gamma$ bias for _Gaia_ -like missions, namely for observations based on a
scanning strategy. Such a relation was then numerically verified with a series
of Monte Carlo runs of a simulated _Gaia_ mission. This proved that the size
of the zero-point parallax error observed in the actual _Gaia_ solution cannot
be attributed to the effect of a possible deviation in the PPN $\gamma$
parameter from the value assumed in GR, which has been adopted by all of the
current astrometric models in the reduction pipelines. Moreover, it also
implies that any bias of PPN $\gamma$ within the known experimental
uncertainty cannot generate a systematic effect on the parallaxes beyond the
target accuracy of this satellite, and therefore it is negligible for this
mission. Nonetheless, this remains potentially relevant for future similar
missions with the precision goal being improved by one order of magnitude.

In this respect, when we analysed the parallax bias effect on different
regions of the sky, we discovered an interesting, but qualitatively expected,
dependence on the ecliptic latitude that we dubbed ‘diavoletto’ (‘little
devil’ in Italian). Such a dependence is naturally introduced by the scanning
law, and thus has to be observed in every mission sharing this type of
observing strategy, and that depends on the precession direction direction of
precession of the spin axis of the satellite in a significant manner.

As mentioned in Lindegren et al. (2021b), the same precession direction of the
spin axis that produces the ‘diavoletto’ is also the very origin of a strong
correlation between the AL parallax factor and the AC rate. The cited paper
stresses that this correlation can bias the parallaxes of stars whose image
parameter determination is significantly impacted by the resulting AC
smearing. This effect was deemed important enough to reverse the precession
direction of the satellite for about one year in order to break this
correlation.

This calls for further investigation in this field, as it is thus quite
possible that these are just two examples of a whole set of previously
correlations, whose effects on the _Gaia_ and _Gaia_ -like missions are still
not known.

Following this line of reasoning, we may venture into a more general
extrapolation: even a pointed mission, not using any kind of scanning law, is
still susceptible to such an effect whenever the sampling scheme includes any
asymmetry among relevant regions of the sky. Even so, the systematic error can
be estimated for relevant cases, and taken care of in data reduction, if there
is sufficient confidence on the $\gamma$ parameter. A more reliable approach
would of course consist in mitigating the effect at its origin by enforcing
suitable ‘sampling rules’. By comparison of Figs. 7 and 8, a reversed scanning
law applied for $50\%$ of the mission lifetime would effectively compensate
for the systematic error by equalising the exposure time excess between
hemispheres and still provide adequate observations for the purposes of most
astrometric and astrophysical measurements. Optimal strategies may be devised
by means of dedicated studies in order to improve on efficiency with respect
to satellite operation and/or specific science cases.

###### Acknowledgements.

This work was supported by the Italian Space Agency ASI under contract
2018-24-HH.0 in support of the Italian participation to the Gaia mission, and
by the grant Astrometric Science and Technology Roadmap for Astrophysics
(ASTRA) from the Italian Ministry of Foreign Affairs and International
Cooperation (MAECI). The numerical simulations were executed at the CINECA
supercomputing centre. A.G.B. is thankful to the Astrophysical Observatory of
Torino for their warm hospitality during his short-term visits. A.G.B. is also
grateful to Lennart Lindegren (Lund Observatory) for kind permission to use
his drawing software. This research made use of NASA’s Astrophysics Data
System. Diagrams were produced using the astronomy data visualisation software
TOPCAT (Taylor, 2005).

## References

  * Bertotti et al. (2003) Bertotti, B., Iess, L., & Tortora, P. 2003, Nature, 425, 374
  * Butkevich et al. (2017) Butkevich, A. G., Klioner, S. A., Lindegren, L., Hobbs, D., & van Leeuwen, F. 2017, A&A, 603, A45
  * ESA (1997) ESA, ed. 1997, ESA Special Publication, Vol. 1200, The HIPPARCOS and TYCHO catalogues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission
  * Froeschlé et al. (1997) Froeschlé, M., Mignard, F., & Arenou, F. 1997, in Hipparcos-Venice ‘97 Proceedings, ESA SP-402, 49
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1
  * Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
  * Hobbs et al. (2009) Hobbs, D., Holl, B., Lindegren, L., et al. 2009, in Relativity in Fundamental Astronomy, IAU Symposium 261, ed. S. A. Klioner, P. K. Seidelman, & M. H. Soffel, 315
  * Klioner (2003) Klioner, S. 2003, ApJ, 125, 1580
  * Lindegren et al. (2021a) Lindegren, L., Bastian, U., Biermann, M., et al. 2021a, A&A, 649, A4
  * Lindegren et al. (1992) Lindegren, L., Høg, E., van Leeuwen, F., et al. 1992, A&A, 258, 18
  * Lindegren et al. (2021b) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021b, A&A, 649, A2
  * Lindegren et al. (2012) Lindegren, L., Lammers, U., Hobbs, D., et al. 2012, A&A, 538, A78
  * Perryman & Schulze-Hartung (2011) Perryman, M. A. C. & Schulze-Hartung, T. 2011, A&A, 595, A65
  * Poisson & Will (2014) Poisson, E. & Will, C. M. 2014, Gravity: Newtonian, post-Newtonian and relativistic (Cambridge: Cambridge Univ. Press)
  * Soffel (1989) Soffel, M. H. 1989, Relativity in Astrometry, Celestial Mechanics and Geodesy (Springer-Verlag)
  * Taylor (2005) Taylor, M. B. 2005, in Astronomical Society of the Pacific Conference Series, Vol. 347, Astronomical Data Analysis Software and Systems XIV, ed. P. Shopbell, M. Britton, & R. Ebert, 29
  * Vecchiato et al. (2018) Vecchiato, A., Bucciarelli, B., Lattanzi, M. G., et al. 2018, A&A, 620, A40
  * Will (2014) Will, C. M. 2014, Living Reviews in Relativity, 17, 4