###### Abstract

To achieve mass, power, and cost reduction, there is a trend to reduce the
volume of many instruments aboard spacecraft, especially for small spacecraft
(cubesats or nanosats) with very limited mass, volume and power budgets. With
the current trend of miniaturizing spacecraft instruments one could naturally
ask if is there a physical limit to this process for star sensors. This paper
shows that there is a fundamental limit on star sensor accuracy, which depends
on stellar distribution, star sensor dimensions and exposure time. An estimate
of such limit is given for our location in the galaxy.

###### keywords:

star sensors; star trackers; attitude sensors; stellar distribution;
photometry; astrometry; star catalogs; fundamental limits

xx 1 5 Received: date; Accepted: date; Published: date Theoretical Limits of
Star Sensor Accuracy$\dagger$ Marcio A. A. Fialho 1,†,* and Daniele Mortari 2
Marcio A. A. Fialho and Daniele Mortari Correspondence<EMAIL_ADDRESS>Tel.: +55-12-3208-6145 This paper is a revised version of the work previously
published in Fialho2017  as a doctorate thesis chapter.

##  1 Introduction

Many progresses in a variety of fields in science and technology could be
accomplished thanks to the miniaturization obtained in microelectronics in the
recent decades. One of the most remarkable examples is the prediction by
Gordon Moore that the computational power would increase exponentially, an
empirical observation that became known as Moore’s law Moore ; Ball ; Charles
; Beigel . Yet, this rate of improvement is not expected to last forever.
Eventually a fundamental limit will be reached when the size of transistors
reaches atomic scales. Likewise, in other fields of science and technology,
fundamental limits to miniaturization and performance improvements are often
found. For instance, in the field of telecommunications, there is a
theoretical minimum amount of energy that must be spent to transmit a bit in a
digital message from one point to another within a given time interval, being
this quantity closely related to Planck’s constant Bekenstein . Hence, it is
natural to ponder whether there is a fundamental limit to the accuracy
attainable by star sensors, given constraints such as the _volume_ in space it
occupies, the length of _time_ available for observations, and the
distribution and brightness of stars around it.

An attitude sensor is any instrument used aboard spacecraft to provide data to
estimate its orientation in space (attitude). Many spacecraft need to have an
accurate knowledge of their attitude in order to accomplish their mission
goals (e.g., point cameras/telescopes, communication antennas, thrusters,
etc.). In order to do so, they use a variety of attitude sensors, such as sun
and horizon sensors, magnetometers, and star sensors (specifically, _star
scanners_ for spinning spacecraft and _star trackers_ for three-axis
stabilized spacecraft) Wertz1978 ; Eterno1999 . Star trackers (STRs) are among
the most accurate attitude sensors available for spacecraft use, by providing
absolute triaxial attitude measurements with errors typically in the order of
few arc-seconds or less Eisenman1997 ; Wang2012 . These sensors are in essence
computerized optical cameras with the appropriate software for star
extraction, star identification, and attitude determination. Reference
Liebe2002  provides a good explanation on how star trackers work.

The purpose of this study is to present an estimate for the ultimate limits
for attitude determination from stars, imposed by fundamental laws of Physics,
i.e., limits that cannot be overcome by technology improvements, for our
location in our galaxy. These estimates are useful as a basis for assessing
real star sensors as to their potential for improvements through technology
advancements. We will not discuss in this work practical limitations faced by
existing, real world star sensors, such as readout noise, non-ideal point
spread function (PSF) in centroiding, and distortions introduced by the
optics, since these limitations have already been well covered by existing
literature Enright2012 ; Sun2013 ; Zakharov2013 . Even though our discussion
will focus on an idealized star tracker, the limits derived here are also
applicable to star scanners and any other possible future type of star sensor,
regardless of the technology employed.

This work is organized as follows: Section 2 describes the methodology used,
Section 3 presents and discusses the results, Section 4 describes how the
model adopted here could be improved, and Section 5 concludes this paper.

##  2 Methodology and Model Description

The star sensor model analyzed in this work is an ideal _spherical star
tracker_ , capable of measuring the direction and energy of every photon
incident on its surface. This ideal STR is able to observe stars from any
direction, that is, it has a field of view of $4\pi\ \rm sr$. The knowledge on
the incoming direction of photons in this model is limited only by diffraction
at the star tracker aperture, assumed to be circular with the same radius of
the STR itself. In other words, it is assumed that the STR aperture is given
by the projection of the STR body on a plane perpendicular to the direction of
incoming photons. Fig. 1 provides a sketch of the STR model adopted in this
work. In this model, the accuracy of the centroids of each star is limited
only by diffraction and shot noise. These effects depend only on the STR
aperture, stellar spectra, and integration time (exposure time). This ideal
STR is completely black, as it absorbs every photon impinging on it. Sections
2.1 and 2.2 provide more details on the assumptions adopted in this model.

Figure 1: Ideal star tracker (STR) model with stars on the background. To aid
visualization, unrealistically represented by a gray sphere here.

###  2.1 Basic assumptions

The following basic assumptions were made:

  1. 1.

the star tracker has a spherical shape with a diameter $D$;

  2. 2.

it is able to detect _every_ photon of stellar origin impinging on its
surface;

  3. 3.

it is capable of registering the incoming direction and energy of every
detected photon with an accuracy limited only by Heisenberg’s Uncertainty
Principle;

  4. 4.

only photons detected during a period of length $t$ – the exposure or
integration time – are considered for attitude determination;

  5. 5.

it is at absolute zero temperature;

  6. 6.

the coordinates of the stars in an inertial reference frame with origin in the
star tracker are known with absolute precision;

  7. 7.

an unbiased optimal estimator is used to determine the star sensor attitude,
and computations are performed with infinite precision;

  8. 8.

measurements obtained with this ideal STR are not merged with external
measurements;

Assumption 2 implies that the STR field of view is $4\pi\ \rm sr$, in other
words, it is capable of observing the whole celestial sphere simultaneously, a
fact that coupled to its spherical shape, implies that the accuracy of this
ideal STR _does not depend on its attitude_.

Assumption 3 and the fact that _every_ photon is detected implies that the
optics are ideal: $100\%$ transmission, with no _defocusing_ and _blurring_ ,
except for the blurring dictated by diffraction.

Assumption 5 means there is no noise of thermal origin within the star
tracker.

Assumptions 4 and 8 limit the number of photons that will be observed by the
ideal STR. If exposure time were not constrained, it would be possible to get
attitude measurement uncertainty as low as desired, just by increasing the
exposure time. In addition, this model does not consider the possibility of
combining current observations with previous observations to improve accuracy.
Assumption 4 also implies that the STR is able to measure just photons and no
other particles111The only other particle known to science that could,
perhaps, convey better the positions of stars are neutrinos emitted at their
core. However, these particles interact so weakly with ordinary matter that
their detection in star sensors is currently impossible and may never become a
realityJayawardhana ..

Assumption 6 implies that the star catalog is perfect and that all corrections
needed to bring the coordinates, brightness and colors from the star catalog
reference frame origin to the STR location (corrections for stellar
aberration, parallax, and redshift/blueshift) are performed with no errors.

Assumption 8 expresses the goal of obtaining a lower bound on attitude error
for a single star tracker used in isolation. If measurements from multiple
sensors were permitted to be merged, a significant improvement in attitude
measurement accuracy would become possible. For example, by
interferometrically combining measurements from a small number of STRs mounted
in a rigid structure and separated by a distance much greater than their
diameters, it would be possible to improve attitude determination by many
orders of magnitude in comparison to the theoretical estimate presented in
this work, with attitude uncertainty being roughly inversely proportional to
the distance between them Baldwin ; Molinder .

###  2.2 Simplifying assumptions

In addition to the previous assumptions, to make this study feasible, the
following additional assumptions were also made:

  1. 1.

the whole Universe is assumed to be composed only by stars;

  2. 2.

stars are considered as polychromatic point sources of light;

  3. 3.

stellar spectra are approximated by the spectra of black bodies that best
match the cataloged star intensity given by star catalogs adopted
here222Ideally, the actual spectra of stars should be used, at least for the
brightest stars, something to be attempted in future works. Section 3.2
discusses the adequacy of this approximation.;

  4. 4.

all Solar System bodies (including the Sun) are disregarded;

  5. 5.

stellar proper motion is disregarded;

  6. 6.

the star tracker is not rotating;

  7. 7.

it is assumed that each detected photon can be univocally associated with the
star from where it originated;

  8. 8.

stars are considered to be at an infinite distance.

Simplifying assumption 1 means that we are not considering as additional
sources of attitude information extended bodies, such as interstellar clouds,
given that these sources are difficult to precisely model and would hardly
significantly increase our attitude knowledge. However, it does not
necessarily mean that all non-stellar pointlike sources will be excluded from
analysis. This means simply that any non-stellar pointlike source present in
star catalogs, such as some quasars and some distant galaxies, will be treated
as if they were stars.

Regarding assumption 6, had the star sensor be rotating, but with knowledge of
the precise instant each photon were detected and a very accurate knowledge of
its own angular velocity vector, it would be possible to compute the incoming
direction of every photon in a non-rotating reference frame attached to the
star sensor, thus reducing the problem of attitude determination of a spinning
star sensor to the problem of attitude determination of a non-rotating star
sensor. This makes the limits derived in this work also applicable to star
scanners and other possible future types of star sensors.

Our computations disregard the Sun and other Solar System objects as
additional references for attitude determination, since these sources are
difficult to model accurately. Also, being the Sun many orders of magnitude
closer and brighter than the other stars, from our vantage point in the
Universe, it is too bright to be directly observed by most, if not all, star
sensors. However, an attitude sensor in a distant future which is able to use
and model very accurately the Sun and a neighboring planetary body as
additional attitude references, could, perhaps, overcome the estimates on the
lower bound of attitude uncertainty computed in this work. This is a topic to
be better investigated in the future.

###  2.3 Model description

Figure 2 presents a flowchart for the model used in this work. Basically, for
each star in the selected star catalog, an estimate for the lower bound on
centroiding uncertainty is computed, and these estimates are used together
with the unit vectors that represent the stars in the star catalog reference
frame to determine the lower bound on attitude determination uncertainty (box
at the lower right corner).

Figure 2: Model for estimating the theoretical lower bound on attitude
uncertainty for star sensors.

Unfortunately, no star catalog is complete. Therefore, any estimate obtained
from an existing star catalog will be incomplete, since the missing stars in
that star catalog still can contribute to attitude knowledge if they are
observed by the star tracker, no matter how far or dim they are. To work
around this limitation, we plot the relation of attitude knowledge upper bound
with star catalog size for a number of publicly available star catalogs and
extrapolate that to the estimated number of stars in our galaxy, plus some
margin, to account for extragalactic sources, as described in Section 3.3. In
the following sections, a more detailed description of the model used is
given.

###  2.4 Black body model for stars

In the model adopted in this work, the spectrum of each star is considered to
be the spectrum of an equivalent spherical black body, diluted by a non-
dimensional geometric factor $C$ arising from its distance to the STR. Given
that the spectral exitance of a black body is uniquely determined by its
temperature, only two parameters are needed in this model to determine the
spectral distribution and intensity of the electromagnetic radiation received
by the STR from each star, the temperature $T$ and the dilution factor $C$.
Mathematically:

$E_{e,\lambda,i}(\lambda)=C_{i}\cdot M_{e,\lambda}(T_{i},\lambda)$ (1)

where:

  * •

$E_{e,\lambda,i}(\lambda)$ is the spectral irradiance received from star $i$
by a surface located at the same place of the star tracker and perpendicular
to incoming rays, evaluated at wavelength $\lambda$;

  * •

$C_{i}$ is the geometric dilution factor for star $i$;

  * •

$T_{i}$ is the temperature of the black body that represents star $i$;

  * •

$M_{e,\lambda}(T_{i},\lambda)$ is the spectral exitance of the surface of the
equivalent black body, at wavelength $\lambda$.

In this equation, both $E_{e,\lambda,i}(\lambda)$ and
$M_{e,\lambda}(T_{i},\lambda)$ are given in unit of power per unit of area and
per unit of wavelength (e.g.: W/m2/nm).

To uniquely determine these two parameters ($T$ and $C$) for each star, at
least two samples of their flux taken at different wavelengths or at different
spectral bands are needed. The following sections describe how $T$ and $C$ are
derived for each star from Hipparcos catalog
data333https://www.cosmos.esa.int/web/hipparcos/catalogues ESA , using the
cataloged $m_{V}$ magnitudes and $B-V$ color indexes. A similar procedure is
performed with data from Hipparcos using the $V-I$ color indexes and data from
other star catalogs.

###  2.5 Black body temperatures from B – V color indexes

Taking as an example data from the Hipparcos catalog, the spectra of stars is
taken as the spectra of black bodies with intensities adjusted so that the
integrated spectra over the Johnson’s $B$ and $V$ bands Bessell2005 ;
Bessell2012  match simultaneously the flux at these bands derived from catalog
data. To determine equivalent black bodies temperatures for stars in the
Hipparcos catalog, an empirical relation is established in this section,
linking the $B-V$ color indexes given in the Hipparcos catalog with black body
temperatures.

The spectral exitance at wavelength $\lambda$ of a black body at a temperature
$T$ can be computed as follows Budding2007 :

$M_{e,\lambda}(T,\lambda)=\frac{2\pi
hc^{2}}{\lambda^{5}}\frac{1}{e^{\frac{hc}{\lambda kT}}-1}$ (2)

where $h$ is the Planck’s constant, $c$ is the speed of light in vacuum and
$k$ is the Boltzmann constant. The spectral exitance will have units of power
per unit area per unit wavelength ($\rm[W\cdot m^{-2}\cdot m^{-1}]$ in SI
units). Numerical values of $h$, $c$ and $k$ used in computations were those
adopted in the 2019 redefinition of the SI base unitsBIPM2019 .

By integrating the product of the spectral exitance of a black body with the
Johnson’s $B$ and $V$ bands energy responses it is possible to obtain the
black-body fluxes in the $B$ and $V$ bands at its surface. This procedure is
described in detail by Bessell in Bessell2005 , section 1.6 – Synthetic
Photometry:

$\varphi_{BB,B}(T)=\int_{\lambda=0}^{\infty}M_{e,\lambda}(T,\lambda)R_{B}(\lambda){\mathrm{d}}\lambda$
(3)
$\varphi_{BB,V}(T)=\int_{\lambda=0}^{\infty}M_{e,\lambda}(T,\lambda)R_{V}(\lambda){\mathrm{d}}\lambda$
(4)

where $\varphi_{BB,B}(T)$ = flux at the surface of a black body at temperature
$T$ in the Johnson’s $B$ band and $R_{B}(\lambda)$ = spectral energy response
function of the Johnson’s $B$ band. Analogously, $\varphi_{BB,V}(T)$ and
$R_{V}(\lambda)$ are quantities related to the Johnson’s $V$ band.

The $R_{B}(\lambda)$ and $R_{V}(\lambda)$ response functions were obtained by
converting the tabulated values recommended by Bessell (Table 1 on page 146 of
Bessell2012 ) from normalized photonic responses to normalized energy
responses and interpolating the resulting values. The energy response
functions adopted in this work are shown in Fig. 3.

Figure 3: Spectral energy response of the $B$ (blue) and $V$ (visual) bands.
The $B$ band is the blue dashed curve to the left.

The conversion from normalized photonic response to normalized energy response
was done by multiplying the photonic response by the wavelength and
renormalizing the results (Eq. A9 in Bessell2012 ). The explanation for this
procedure is given in section A2 in the appendix of Bessell2012 , on page 153.
The method of interpolation used was a “shape-preserving piecewise cubic
interpolation,” provided by the MATLAB function `interp1` with method “pchip.”
Computations were performed in MATLAB R2015b with the script
`plot_BV_BB_script.m` from the .zip archive which supplements this work444See
“Supplementary Materials” on page 5..

From the fluxes in the $B$ and $V$ bands, the magnitudes in these bands can be
computed:

$m_{BB,B}(T)=-2.5\ \log_{10}(\varphi_{BB,B}(T)/\varphi_{REF,B})$ (5)
$m_{BB,V}(T)=-2.5\ \log_{10}(\varphi_{BB,V}(T)/\varphi_{REF,V})$ (6)

These equations give the apparent magnitudes in the $B$ and $V$ spectral bands
of a spherical black body for an observer situated just above its surface
looking down towards its center. $\varphi_{REF,B}$ and $\varphi_{REF,V}$ are
the reference fluxes that define the zero points of the magnitude scales in
these bands, having being obtained by numerically integrating the spectrum of
Vega ($\alpha$-Lyr) multiplied by the band responses, and adjusting their
values such that the computed $B$ and $V$ magnitudes of Vega matches those in
the star catalog ($m_{\rm Vega,B}=0.029$ and $m_{\rm Vega,V}=0.030$ in
Hipparcos). Mathematically:

$\varphi_{REF,B}=10^{0.4m_{Vega,B}}\int_{\lambda=0}^{\infty}E_{Vega}(\lambda)R_{B}(\lambda){\mathrm{d}}\lambda$
(7)
$\varphi_{REF,V}=10^{0.4m_{Vega,V}}\int_{\lambda=0}^{\infty}E_{Vega}(\lambda)R_{V}(\lambda){\mathrm{d}}\lambda$
(8)

where $E_{Vega}(\lambda)$ is the spectral irradiance from Vega measured at the
top of Earth’s atmosphere. The spectrum of Vega used in equations 7 and 8 was
obtained from file `alpha_lyr_stis_008.fits` from the CALSPEC
database555http://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/calspec. Bohlin2014 .

Figure 4 shows the apparent magnitudes of black bodies versus temperature in
the Johnson’s $B$ and $V$ bands for an observer located at their surface. In
this plot, brighter sources (more negative magnitudes) are at the top. Note
that the magnitude scale used in astronomy is reversed, with smaller
magnitudes meaning brighter sources. The magnitudes are said to be apparent
because they depend on the observer location, contrasting to stellar absolute
magnitudes which are magnitudes of a star as seen from a standardized distance
Zeilik1998 .

Figure 4: Apparent magnitudes of a black body for an observer lying on its
surface and looking down towards its center versus black body temperature, in
the Johnson-Morgan $B$ and $V$ bands. Note that the vertical axis of this plot
is reversed, with more negative magnitudes (brighter black-bodies) at the top.

Figure 5: Relation between temperature (or its reciprocal) with $B-V$ color
index for black bodies.

The difference between the $B$ and $V$ magnitudes of a celestial body is its
$B-V$ color index. Figure 5 presents the relation between the $B-V$ color
index and temperature for black-bodies. The plot to the right relates the
$B-V$ color index with the multiplicative inverse of its temperature. Note
that this curve is much more linear than the direct relation between
temperature and $B-V$ color index. Therefore, to get equivalent black-body
temperatures for stars in the catalog, we use the 1/$T$ versus $B-V$ curve for
interpolation. To avoid temperature estimates with large errors from
appearing, the $B-V$ color indexes in the Hipparcos catalog are clamped into
the interval $[-0.2357,+2.7028]$ before conversion. These limits correspond to
black-body temperatures of $30,\\!000$ K and $2,\\!000$ K, respectively. Most
stars have effective temperatures in that range.

It should be noted that for many stars, the temperature $T$ used in our model
will not be equal to the effective temperature666Effective temperature is the
temperature that a black body with the same physical size of a star should
have to irradiate the same amount of electromagnetic radiation as the star. of
the star but will usually be smaller. This is caused by interstellar reddening
– selective absorption by dust in the intervening light path from that star to
the star tracker. Likewise, the constant $C$ will also have a different value.

###  2.6 Determination of the geometric dilution factor $C$ from Hipparcos
data

From temperature $T$, the equivalent black body’s visual magnitude at its
surface ($m_{BB,V,surface}$) is determined by interpolating the solid black
curve in Fig. 4. The dilution factor $C$ is then obtained by comparing this
magnitude with the cataloged visual magnitude ($m_{V}$) in the Hipparcos
catalog, using the following equation:

$C_{i}=10^{0.4\cdot(m_{BB,V,surface,i}-m_{V,i})}$ (9)

The geometric dilution factor $C$ will typically be between $10^{-20}$ and
$10^{-14}$ for stars in the Hipparcos catalog. In this equation, the subscript
$i$ indicates that the values refer to star $i$.

###  2.7 Number of photons detected per unit wavelength

This section derives equations for the number of photons that will be
detected, per wavelength, by the idealized star tracker used in this model,
for a given exposure time $t$ and a given STR diameter $D$, also assumed to be
equal to its aperture diameter. The energy of each photon is related to its
frequency $\nu$ by the following equation:

$E_{ph}=h\nu=\frac{hc}{\lambda}$ (10)

Dividing the spectral irradiance at the location of the star tracker due to
the black-body equivalent of star $i$ (equations (1) and (2)) by the energy of
a photon of wavelength $\lambda$, the following expression for the spectral
photon flux density received by the STR from the equivalent of star $i$ is
obtained:

$\varphi_{ph,\lambda,i}=C_{i}\frac{2\pi
c}{\lambda^{4}}\frac{1}{e^{\frac{hc}{\lambda kT_{i}}}-1}$ (11)

This flux density has units of photons per unit of time per unit of area per
unit of wavelength. Multiplying this by the star tracker’s cross section area
$A=\pi D^{2}/4$ and by the integration time $t$ we obtain:

$n_{ph,\lambda,i}=C_{i}\cdot
t\cdot\frac{\pi^{2}D^{2}c}{2\cdot\lambda^{4}}\frac{1}{e^{\frac{hc}{\lambda
kT_{i}}}-1}$ (12)

which is the number of photons from star $i$ equivalent being collected by the
STR, per unit wavelength.

###  2.8 Diffraction and shot noise

Diffraction and optics blurring set the format of the point spread function
(PSF) of stellar image. In an ideal star tracker, there’s no optical blurring,
except for that set by diffraction. Therefore, for the STR model adopted in
this work, the PSF function will be the diffraction pattern given by a
circular aperture of diameter $D$ contained in a plane perpendicular to the
incoming direction of photons. This diffraction pattern consists of a disk
(Airy disk) with a series of concentric rings, being first derived by Airy in
1835 Airy1835 .

If the description of Nature given by Classical Mechanics were correct, it
would be possible, at least in theory, to measure the intensity of the
electromagnetic fields at the detector plane with no error, from where the
true, error free, direction of the incoming light rays would be obtained.
However, the fact that light is discretized in photons leads to the situation
where the number of detected photons will be finite, even with an ideal
detector. Therefore, instead of precisely defining the intensity of the
electromagnetic fields at each point in the detector (as thought by 19th
century physicists), the PSF will define the probability density function that
a photon coming from a point source at infinity will be detected on a
particular location at the detector. Since the number of photons detected will
be finite, even for the case of an ideal star tracker, and these photons are
detected at random positions, with probabilities given by the PSF, the
centroid estimate for each observed star will have a noise. The lower bound
for this noise was determined by Lindegren Lindegren2013 , being discussed in
the next section.

###  2.9 Lower bound on centroiding error for single stars

According to Lindegren Lindegren2013 , Heisenberg’s uncertainty principle sets
a fundamental limit for centroiding, and this limit assumes the following form
for monochromatic light of wavelength $\lambda$:

$\sigma_{xc}\geqslant\frac{\lambda}{4\pi\Delta x\sqrt{N}}$ (13)

where:

  * •

$\sigma_{xc}$ = angular centroiding uncertainty along an axis $x$
perpendicular to the direction of incoming photons, in radians;

  * •

$\Delta x=\sqrt{\int(x-\overline{x})^{2}\,{\mathrm{d}}S\,/\int{\mathrm{d}}S}$
is the root mean square extension of the star tracker aperture (entrance
pupil) along the $x$ axis, being $\overline{x}=\int
x\,{\mathrm{d}}S\,/\int{\mathrm{d}}S$ the position in $x$ of the aperture
geometric center; and

  * •

$N$ = number of photons detected.

For circular apertures of diameter $D$, $\Delta x=D/4$. Substituting this into
Eq. (13) the following expression for the reciprocal of the lower bound of
variance of centroiding error (the Fisher information $F$) over a circular
aperture of diameter $D$, for monochromatic sources of light, is obtained:

$\frac{1}{\sigma_{xc}^{2}}\leqslant\frac{1}{\sigma_{\min}^{2}}\triangleq
F_{N,\textrm{mono}}=\frac{\pi^{2}D^{2}N}{\lambda^{2}}$ (14)

Since stars are incoherent sources of light, the detection of a given photon
is not correlated with the detection of another photon from the same star.
This means that the number of detected photons from a given star will follow a
Poisson distribution with parameter $\iota$, being $\iota$ the expected number
of detected photons777We are using the Greek letter $\iota$ instead of the
more common $\lambda$ for the Poisson distribution parameter to avoid
confusion with $\lambda$ used for wavelength.. This parameter can be obtained
by integrating Eq. (12). For large values of $\iota$, the Poisson distribution
narrows down in comparison to the value of $\iota$. This means that when the
expected number of detected photons is significantly large, the true value of
the lower bound of centroiding accuracy will be very close to the value
predicted by Eq. (14) if we substitute $N$ by $\iota$. Numerical tests have
shown, assuming that the centroiding error for exactly $N$ detected photons
follows a Gaussian distribution with a standard deviation given by Eq. (13),
that the error between the actual centroiding error and the value estimated by
Eq. (14) using $\iota$ in place of $N$ will be smaller than $23\%$ for $N\geq
1$, $6.4\%$ for $N\geq 10$ and $0.51\%$ for $N\geq 100$. It is true that the
actual probability density function for centroiding error along one axis will
not be exactly Gaussian, specially for a low number of detected photons, but a
Gaussian distribution provides a good approximation, even when only one photon
is detected.

Another consequence of the fact that the detection of a given photon is not
correlated with the detection of another photon from the same star is that the
centroiding error of a centroid computed using photons in the wavelength
interval $[\lambda_{1},\lambda_{2}]$ is independent on the centroiding error
using photons in the wavelength interval $[\lambda_{3},\lambda_{4}]$ when
these intervals do not overlap ($\lambda_{2}<\lambda_{3}$ or
$\lambda_{4}<\lambda_{1}$). Therefore, we can consider each wavelength
interval individually and then merge the centroid estimates for each
wavelength.

For the discrete case of having $n$ independent unbiased estimates of the same
physical variable (e.g., the $x$ coordinate of a star centroid), each having a
variance $\sigma_{i}^{2}$, the best estimate for that variable is obtained by
summing these estimates using the reciprocal of their variances as weights (a
procedure sometimes known as inverse variance weighting)Hartung ; JamesYen .
In that case, the variance of this optimal estimate will be given by:

$\sigma_{T}^{2}=\left(\sum_{i=1}^{n}\sigma_{i}^{-2}\right)^{-1}$ (15)

where $\sigma_{T}^{2}$ = total variance in the estimate of a scalar physical
variable obtained by merging $n$ independent measurements and $\sigma_{i}^{2}$
= variance of each individual measurement $i$. Since the spectra of black
bodies is continuous, the following adaptation of Eq. (15) is used to compute
centroid estimates for black bodies:

$\frac{1}{\sigma_{xc}^{2}}=\int_{\lambda=0}^{\infty}\frac{{\mathrm{d}}(\sigma^{-2})}{{\mathrm{d}}\lambda}{\mathrm{d}}\lambda$
(16)

The contribution from each wavelength to the knowledge of the centroid
position can be obtained from Eq. (14) by replacing $N$ with
$n_{ph,\lambda,i}{\mathrm{d}}\lambda$, where
$n_{ph,\lambda,i}={\mathrm{d}}N_{ph,i}/{\mathrm{d}}\lambda$ is the derivative
with wavelength of the number of photons from star $i$ entering the star
tracker aperture within an integration time of $t$, as given by Eq. (12) from
section 2.7. Hence, for each star $i$, the wavelength derivative of the
maximum knowledge physically attainable of its centroid position (derivative
of its centroiding Fisher information) is given by:

$\frac{{\mathrm{d}}F}{{\mathrm{d}}\lambda}=\frac{{\mathrm{d}}\left(\sigma^{-2}_{\min}\right)}{{\mathrm{d}}\lambda}=\frac{\pi^{2}D^{2}}{\lambda^{2}}\,n_{ph,\lambda}$
(17)

Here we have dropped the subscript $i$ to simplify notation. Plugging Eq. (12)
into Eq. (17) yields:

$\frac{{\mathrm{d}}F}{{\mathrm{d}}\lambda}=C\cdot
t\cdot\frac{\pi^{4}D^{4}c}{2\cdot\lambda^{6}}\frac{1}{e^{\frac{hc}{\lambda
kT}}-1}$ (18)

Integrating this equation for $\lambda=0$ to $\infty$ gives $F_{i}$, the
Fisher information for stellar centroid $i$, and its reciprocal
$\sigma_{\min}^{2}$, the minimum variance for the centroid position error in
$x$ direction, being $x$ an axis perpendicular to the incoming light rays:

$\frac{1}{\sigma_{\min}^{2}}=F_{i}=\int_{\lambda=0}^{\infty}\frac{{\mathrm{d}}F_{i}}{{\mathrm{d}}\lambda}{\mathrm{d}}\lambda=12\zeta(5)\pi^{4}\cdot\frac{k^{5}}{h^{5}c^{4}}\cdot
D^{4}t\cdot C_{i}T_{i}^{5}$ (19)

where $\zeta(5)=1.0369277551...$ is the Riemann zeta function evaluated at 5.
Since the aperture is symmetrical, Eq. (19) gives the minimum centroiding
variance for star $i$ ($\sigma_{\min,i}^{2}$) along any axis perpendicular to
the direction of incoming light rays. From this equation, it can be noted that
the lower bound of the standard deviation on centroiding error along any axis
perpendicular to the true direction of the star is proportional to $D^{-2}$
and $t^{-1/2}$, when the number of detected photons is sufficiently large.
This means that the star tracker diameter has a much larger effect in the
ultimate centroid accuracy and precision than the exposure time.

###  2.10 Estimating the lower bound of attitude error from many stars

This section follows the formulation given by Markley and Crassidis in
Markley2014 , Section 5.5. This formulation is valid when measurement errors
are small, uncorrelated and axially symmetric around the true direction of
stars, conditions fulfilled by our model, except for ideal star trackers with
very small diameters, much less than 1 mm.

According to equations 5.113 and 5.114 in Markley2014 , the covariance matrix
($\mathbf{P}_{\vartheta\vartheta}$) of the rotation vector error
($\boldsymbol{\delta\vartheta}$) for an optimal attitude estimator is the
inverse of the Fisher information matrix $\mathbf{F}$:

$\mathbf{P}_{\vartheta\vartheta}=\mathbf{F}^{-1}$ (20)

with:

$\mathbf{F}=\sum_{i=1}^{N}\frac{1}{\sigma_{i}^{2}}\left[\mathbf{I}_{3\times
3}-\mathbf{r}_{i}^{\mathrm{true}}(\mathbf{r}_{i}^{\mathrm{true}})^{T}\right]$
(21)

where $\sigma_{i}^{2}$ is the measurement variance associated with star $i$,
as defined in Markley2014 , $\mathbf{I}_{3\times 3}$ is a $3\times 3$ identity
matrix and $\mathbf{r}_{i}^{\mathrm{true}}$ is the true direction of star $i$,
represented by a unit vector expressed as a $3\times 1$ column matrix.
$\mathbf{r}_{i}^{\mathrm{true}}$ is given in some reference frame R and $N$ is
the number of identified stars used in attitude computation888Since the
attitude error does not depend on the reference frame chosen, any one can be
chosen for analysis. For simplicity, Markley and Crassidis have chosen the
body frame B as the reference frame R in their analysis, making the true
direction of stars in the body frame $\mathbf{b}_{i}^{\mathrm{true}}$ equal to
$\mathbf{r}_{i}^{\mathrm{true}}$, that’s the reason they use
$\mathbf{b}_{i}^{\mathrm{true}}$ instead of $\mathbf{r}_{i}^{\mathrm{true}}$
in their formulation..

In the ideal STR model adopted in this work, the measurement variance
$\sigma_{i}^{2}$ is identical to the lower bound of centroiding error variance
$\sigma_{\min,i}^{2}$ given by Eq. (19). This statement is proved in the
Appendix.

###  2.11 A compact metric for the attitude error

Even though the covariance matrix $\mathbf{P}_{\vartheta\vartheta}$ provides
detailed information about the attitude uncertainty, as it has six independent
parameters it has the disadvantage of being hard to visualize. Therefore, to
perform comparisons, we use a more compact metric derived from it:

$(\bar{\vartheta}_{rms})^{2}=E\\{\vartheta^{2}\\}=\operatorname{tr}(\mathbf{P}_{\vartheta\vartheta})$
(22)

The trace of the covariance matrix $\mathbf{P}_{\vartheta\vartheta}$ gives the
variance of the overall attitude error, that is, the sum of the variances of
the attitude error around the three defining axes of the reference frame. It
is also equal to the square of the limiting value of the root mean square
(rms) of the angle theta ($\vartheta$) of the Euler axis/angle
parameterization of the attitude error when the number of attitude
determinations tends to infinity.

When the STR diameter and exposure time are large enough so that most stars
contributing to the Fisher information matrix $F$ have many detected photons,
the lower bound of the expected rms value of theta
($\bar{\vartheta}_{rms,min}$) can be computed by the Eqs. (19), (20), (21) and
(22). These equations can also be rearranged in the following manner, which
makes more explicit the dependence of $\bar{\vartheta}_{rms,min}$ with $D$ and
$t$:

$\bar{\vartheta}_{rms,min}=G\cdot D^{-2}\cdot t^{-1/2}$ (23)

with

$G=\sqrt{\frac{h^{5}c^{4}}{12\zeta(5)\pi^{4}k^{5}}\cdot\operatorname{tr}{\left[\left(\sum_{i=1}^{N}C_{i}T_{i}^{5}\left[\mathbf{I}_{3\times
3}-\mathbf{r}_{i}^{\mathrm{true}}(\mathbf{r}_{i}^{\mathrm{true}})^{T}\right]\right)^{-1}\right]}}$
(24)

$G$ is a constant that depends only on stellar distribution around the star
tracker, stellar brightness and on attenuation of stellar light by the
intervening medium.

##  3 Discussion and Results

###  3.1 Star catalogs used

The Hipparcos star catalog was initially selected because it was until very
recently one of the most accurate star catalogs available for precise attitude
work. Therefore we had already all the tools needed to process it.
Unfortunately, the Hipparcos star catalog having less than 120,000 stars is
too short to give an adequate basis for extrapolation. Therefore it was
decided to include data from two larger catalogs, the Tycho-2 Hog2000 ;
Turon2009  with around 2.5 million stars and 2MASS Skrutskie2006  with about
470 million objects.

The Hipparcos and Tycho-2 star catalogs give magnitudes in the optical regime
(near ultraviolet, visible and near infrared), whereas the 2MASS star catalog
gives magnitudes in the near/shortwave infrared bands $J$(1.25 $\mu$m),
$H$(1.65 $\mu$m) and $K_{s}$(2.16 $\mu$m).

###  3.2 Adequacy of the black-body approximation

In order to check the adequacy of the black-body approximation used in Section
2.4, we have performed a numerical integration of Eq. (17) for some selected
stars, using their actual spectra. It was observed that, given the color index
used, the black-body approximation provides a good fit for some stars, but the
fitting is not so good for all of them. Figure 6 compares the actual spectra
of two stars with the spectra of their black-body equivalents, derived from
their $B-V$ color indexes and $V$ magnitudes ($m_{V}$) given in Table 1 using
the methods described in sections 2.5 and 2.6. Spectral fluxes in Figure 6 are
given in power per unit area per frequency (or wavelength) decade.

Figure 6: Comparison between the actual spectra for the Sun and Vega ($\alpha$-Lyr) with the spectra of their black-body equivalents derived from their $B-V$ color indexes and $V$ magnitudes with the methodology explained in sections 2.5 and 2.6. Actual spectra represented by continuous line. Dashed lines represent the spectra of equivalent black-bodies. Table 1: Comparison for some selected stars when $D=1\,\rm m$ and $t=1\,\rm s$. Parameter | Star
---|---
name/symbol | unit | Vega | 1757132 | Sun* | KF06T2 | VB8
spectral type | - | A0V | A3V | G2V | K1.5III | M7V
$m_{V}$ | mag | 0.030 | 11.81 | $-26.75$ | 13.97 | 16.80
$B-V$ | mag | $-0.001$ | 0.26 | 0.65 | 1.18 | 2.01
$T$ | K | 13,231 | 8,580 | 5,711 | 3,951 | 2,613
$C$ | 1 | 2.96E-17 | 1.87E-21 | 2.42E-5 | 9.59E-21 | 2.01E-20
$\sigma_{\min,BB}$ | rad | 3.76E-13 | 1.40E-10 | 3.40E-18 | 4.29E-10 | 8.32E-10
$m_{H}$ | mag | $-0.004$ | 11.23 | $-28.24$ | 11.26 | 9.17
$H-K_{s}$ | mag | $-0.005$ | 0.02 | 0.04 | 0.10 | 0.34
$T$ | K | 10,417 | 8,961 | 8,059 | 6,050 | 3,282
$C$ | 1 | 4.80E-17 | 1.94E-21 | 1.41E-5 | 3.72E-21 | 1.05E-19
$\sigma_{\min,BB}$ | rad | 5.37E-13 | 1.23E-10 | 1.88E-18 | 2.37E-10 | 2.06E-10
$\sigma_{\min,num}$ | rad | 5.50E-13 | 1.75E-10 | 3.61E-18 | 4.67E-10 | 2.92E-10
* Not used in the results presented in this work, due to its extreme proximity
(see explanation at the end of section 2.2).

Table 1 also presents a comparison between the lower bound of centroiding
error obtained by numerical integration ($\sigma_{\min,num}$, in the last row
of the table) and the lower bound of centroiding error $\sigma_{\min,BB}$
obtained from the black-body approximation. To show how $\sigma_{\min,BB}$ can
vary depending on the spectral bands used for estimating the equivalent black-
bodies, results are presented for two photometric systems: Johnson’s UBV and
2MASS JHKs, with the derived black-body parameters ($T$ and $C$) also shown.
As can be seen, the error in $\sigma_{\min,BB}$ is typically less than a
factor of 2, but sometimes it can be much larger (see for example star VB8).

The magnitudes and color indexes listed in Table 1 were computed from spectra
downloaded from the CALSPEC
library999http://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/calspec. For the UBV system, the
zero points that define the origin of the magnitude scales were computed using
the method described in section 2.5. For the 2MASS JHKs system, the zero
points were considered to be the zero-magnitude in-band fluxes listed on the
third column of Table 2 from Cohen et al. Cohen2003 .

####  3.2.1 Color index limiting values

As explained in Section 2.5, the color indexes were limited to the interval
that corresponds to a temperature range of $2,\\!000$ K to $30,\\!000$ K. It
was observed that, when the upper temperature limit was raised to more than
$100,\\!000$ K, the Fisher information matrix would be dominated by a few very
blue, hot stars where the interpolation from the color index curve versus
temperature would give a very high temperature, much higher than their actual
temperatures, leading to a significant underestimate of
$\bar{\vartheta}_{rms,min}$. In fact, even the $30,\\!000$ kelvins upper limit
adopted in this work might be too high, resulting that the
$\bar{\vartheta}_{rms,min}$ estimated here is probably lower than the actual
lower bound of attitude error attainable by star trackers.

The lower limiting temperature of $2,\\!000$ K could perhaps be set to a lower
value (e.g.: 500 K), in order to better accommodate interstellar absorption
and the existence of brown dwarfs. However, it was noted that this lower
temperature limit has very little effect in the estimated value of
$\bar{\vartheta}_{rms,min}$.

The optimal selection of temperature limits to be adopted for the black-body
model will be a subject of a future work, if this model is not abandoned in
favor of a more accurate stellar spectra model.

###  3.3 Results from catalogs and extrapolation

Some scripts101010See section “Supplementary Materials” on page 5. were
written to numerically evaluate the lower bound on star tracker attitude error
for different star catalogs, different spectral bands and limiting the number
of stars used in the computations to the $N$ brightest cataloged stars, with
$N$ varying from two stars to the whole star catalog. Figure 7 shows results
obtained with the catalogs described in Section 3.1 for $D=1$ m and $t=1$ s.
The letter codes B-V, V-I, $\rm B_{T}$-$\rm V_{T}$, J-H, J-$\rm K_{s}$ and
H-$\rm K_{s}$ indicate the spectral bands and catalog used for each curve.
These curves form a basis for extrapolation (dashed lines) from where we can
obtain an interval that will very likely contain the true value of the
parameter $G$ of Equation 24 for our location in the galaxy.

Figure 7: Estimates of $\operatorname{tr}(\mathbf{P}_{\vartheta\vartheta})$
for an ideal star sensor with ${D=1}$ m and ${t=1}$ s, obtained from the
following star catalogs: Hipparcos, Tycho-2 and 2MASS and their subsets of
brightest stars. Extrapolation curves shown as dashed lines.
$\operatorname{tr}(\mathbf{P}_{\vartheta\vartheta})$ is a measure of the
attitude uncertainty (see equations 20-22).

Performing a rough extrapolation, we obtain for N = 300 billion
stars111111Many sources give a number between $10^{11}$ and $4\cdot 10^{11}$
stars in our galaxy, with Guo2009  giving about $3\cdot 10^{11}$ stars.
,121212The word billion can be ambiguous, meaning either $10^{9}$ or $10^{12}$
depending on the language, date and culture Merr2019 ; Cambridge2019 ; RAE2019
; GrandDic2019 ; Lex2019 . In this work we use it as a synonym to $10^{9}$. A
similar ambiguity also exists for the words “trillion”, “quadrillion” and
others that follow the same pattern Lex2019 .  (the estimated number of stars
in our galaxy)
$\operatorname{tr}(\mathbf{P}_{\vartheta\vartheta}),_{\min}\approx$ $1.4\cdot
10^{-28}$ rad2 for the lower extrapolation curve and $\approx 3.1\cdot
10^{-27}$ rad2 for the upper extrapolation curve. However, there were many
approximations made in the model, mainly the assumption of black-body spectra
for stars. Therefore, the
$\operatorname{tr}(\mathbf{P}_{\vartheta\vartheta}),_{\min}$ upper and lower
estimates for $D=1$ m and $t=1$ s could still be wrong by a factor of 2 or 3.
Hence, additional safety factors represented by the red vertical arrows along
the $3\cdot 10^{11}$ stars dashed dotted line in Figure 7 were included. With
these safety factors, and considering that the contribution of extragalactic
sources is negligible (subsection 3.3.2), it should be safe to assume that the
true value of $\operatorname{tr}(\mathbf{P}_{\vartheta\vartheta}),_{\min}$
lies between $5\cdot 10^{-29}$ rad2 and $10^{-26}$ rad2 for $D=1$ m and $t=1$
s. From this, we conclude that the bounds for the $G$ constant of Eq. (23)
are: $7\cdot 10^{-15}$ rad m2 s${}^{1/2}<$ $G$ $<10^{-13}$ rad m2 s1/2, with
the upper bound of $10^{-13}$ rad m2 s1/2 not being surpassed when $D>0.1$ m
and $t>0.01$ s.

####  3.3.1 Explanation for the upper extrapolation curve in Fig. 7

For $D=0.1$ m and $t=0.01$ s (which gives a value of 0.001 m2 s1/2 for the
combined $D^{2}t^{1/2}$ metric used in Fig. 8), it was found that the expected
number of detected photons ($\iota$), using values derived from $J-H$ color
index in the 2MASS catalog, is positively larger than 1 only for the $8\cdot
10^{8}$ brightest stars. For the remaining stars, $\iota$ may be less than 1,
meaning that these stars may have a large probability of not being detected.
Therefore, conservatively, we ignored these dim stars when building the upper
extrapolation curve in Fig. 7, making this curve flat starting at $8\cdot
10^{8}$ stars. This makes the attitude uncertainty lower bound
($\bar{\vartheta}_{rms,min}$) obtained from this curve being slightly
overestimated.

####  3.3.2 Contribution from extragalactic sources

The contribution of all existing extragalactic sources in the known Universe
for the attitude accuracy is probably very small (probably less than 10% of
the overall Fisher information). The reason for that is the vast distances
between galaxies in comparison to their dimensions. For example, the nearest
galaxy about the same size or larger than our galaxy is the Andromeda Galaxy.
It’s center lies about at a distance of 780 kpc from us Ribas2005 , which is
about 10-20 times the diameter of their disks.

Our galaxy, the Milky Way Galaxy, is orbited by many dwarf galaxies, such as
the Small and Large Magellanic Clouds, but the total number of stars in these
dwarf galaxies is less than 10% of the number of stars in our galaxy,
therefore their contribution is also negligible.

Considering that the light intensity (and number of detected photons per unit
time) falls off with the square of the distance from the source and that the
Fisher information contributed by a star is proportional to the number of
detected photons from that source, it is easy to see that the contribution
from extragalactic sources will be small.

####  3.3.3 Need to consider some stars as extended sources

The lower bound on attitude uncertainty is so low that future star trackers
would probably need to consider some stars as extended bodies and correct the
effects of stellar spots (akin to sun spots, but in other stars) in their
atmospheres to be able to come close to this theoretical lower bound,
something that is unthinkable for current generation star sensors. For
example, the star R Doradus, the star with largest apparent diameter after the
Sun, has an apparent diameter of 57 $\pm$ 5 mas [(2.76 $\pm$ 0.25)$\cdot
10^{-7}$ rad] ESO2017 .

###  3.4 Comparison with commercial star trackers

To give a feeling on how much room for improvement there is for future
technology developments, Fig. 8 compares the reported accuracy of some
commercially available star trackers Astro15 ; Terma ; SED26 ; Leonardo ;
VST41M ; VST68M ; Sinclair ; BallCT633 ; Spacemicro  with the theoretical
lower bounds of an equivalent spherical star tracker having approximately the
same volume of a sphere that circumscribes the optical head of the star
tracker131313the optical head is the box that houses the optics and image
sensor. In some models, it includes the whole star tracker with the exception
of its baffle. In other models, the processing electronics is in a separate
box., excluding its baffle141414A baffle is a protective light shade used in
star trackers to prevent blinding by the Sun or other bright sources..

The comparison is performed in terms of the combined metric $D^{2}t^{1/2}$,
according to Eq. (23), which makes it possible to compare many different star
trackers in a single plot. The solid line at the bottom left of this plot
denotes the lower estimate of the lower bound of the attitude error
$\bar{\vartheta}_{rms,min}$, derived from the lower curve in Fig. 7. The
dashed line immediately above this solid line is the upper estimate of the the
lower bound of the attitude error $\bar{\vartheta}_{rms,min}$, derived from
the upper curve in Fig. 7. For $D>0.1$ m and $t>0.01$ s the true lower bound
of attitude error ($\bar{\vartheta}_{rms,min,true}$) that can be obtained
using solely electromagnetic radiation emitted by bodies outside the Solar
System should lie between these two curves. No star sensor that satisfies the
constraints 4 and 8 stated in Section 2.1 should be able to surpass
$\bar{\vartheta}_{rms,min,true}$ without making use of additional attitude
reference sources, such as the Sun and other Solar System objects (excluded
from analysis by the simplifying hypothesis 4 in Section 2.2) or artificial
attitude references.

Figure 8: Comparison between some commercial star trackers with the upper and
lower limits of the theoretical lower bound on attitude uncertainty.

We have opted to exclude the baffle in the estimate of $D$, because the sole
reason for including a baffle in star trackers is to protect them from being
temporarily blinded by stray light coming from the Sun and other bright
sources (Earth, Moon and other spacecraft parts), something that would not
happen in the absence of Solar System objects. Also, including the baffle
would greatly inflate the diameter $D$ of the circumscribing sphere.
Unfortunately, star tracker product briefs and other publicly available
information only provide their total dimension, including the baffle.
Therefore, we had to estimate what would be their sizes without the baffle, by
measuring lengths in photos included in the product briefs, from where $D$
could be estimated with an uncertainty probably around $25\%$. It would be
interesting as well to compute the $D^{2}t^{1/2}$ metric considering only the
optical head aperture, typically much smaller than its dimensions.
Unfortunately this information is almost never provided in product briefs. The
accuracy used in this plot was derived from the reported (1-$\sigma$) noise
equivalent angle or attitude accuracy, using the equation
$\vartheta_{rms}^{2}=\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{z}^{2}$, being
$\sigma_{x}=\sigma_{y}$ the uncertainties around the cross-boresight axes
(pitch and yaw angles) and $\sigma_{z}$ the uncertainty around the boresight
axis (roll angle). For single head star trackers, the uncertainty in the roll
angle is usually much larger than those in the yaw and pitch angles.

It should be noted that the positions of the commercial star trackers listed
in this plot have large errors, being probably wrong by a factor of 2 or 3 in
the combined $D^{2}t^{1/2}$ metric (horizontal axis) and also by a factor of 2
or 3 in the overall attitude error (vertical axis). The reason for that is the
scarcity of information that is available in product briefs and websites. In
many cases, to obtain accurate information about star tracker dimensions,
exposure time and accuracy it would be necessary to perform lengthy
negotiations with star tracker manufacturers.

As can be seen, there’s still a lot of room for improvement in star sensors.
The theoretical lower bound is around 6 or 7 orders of magnitude lower than
what is currently attained by most star trackers.

##  4 Future work

Here are some ideas for future works that could improve the accuracy of the
lower bound of attitude error estimates given in this work:

  * •

Use more complete star catalogs. A more accurate estimate on the Cramér-Rao
bound on attitude accuracy obtainable from stars could be computed using a
more complete star catalog, such as the USNO-B star
catalog151515http://www.usno.navy.mil/USNO/astrometry/optical-IR-
prod/usno-b1.0 with over 1,000,000,000 objectsMonet , or the forthcoming final
star catalog from the Gaia astrometric missionGaia
161616http://sci.esa.int/gaia/58277-towards-the-final-gaia-catalogue/.

  * •

Better model for stellar spectra. Using a more accurate model for the stellar
spectra, that takes into account the peculiarities of stellar spectral types
and interstellar absorption is something that could be done in the future to
improve the accuracy of the results obtained in this work.

  * •

Use magnitudes and color index uncertainties provided in star catalogs. Due to
operational limitations we have ignored the uncertainties in magnitude and
color indexes provided in star catalogs. An overhaul of the software used to
determine the estimates of the lower bound of attitude error is planned for
the future. With it, the propagation of uncertainties provided in star
catalogs will be included.

  * •

More rigorous statistical treatment. We plan in the future to reevaluate these
estimates, using a more rigorous statistical treatment, taking into
consideration what happens when the expected number of detected photons from a
source is small, less than one. In this scenario, there exists a high
probability that no photon from that source will be detected.

  * •

Include the Sun and other Solar System bodies in the computation. It would be
interesting to know how much the lower bound on attitude determination
uncertainty would be improved if the detected photons from the Sun and other
bodies in the Solar System were fully exploited to improve attitude
determination, in a system without any technical limitation.

  * •

Compute estimates for other parts of the galaxy. The estimates derived in this
work are valid only in the Solar System neighborhood, since they are based on
star coordinates, color indexes and magnitudes as seen from the Solar System.
It would be interesting in the future to expand this work for other regions of
our galaxy. For example, in star dense regions, the ultimate accuracy
attainable by star sensors should be better than the ultimate accuracy
attainable in our part of the galaxy.

##  5 Conclusion

To our best knowledge, Chapter 7 in the doctorate thesis of the first
authorFialho2017  was the first work which provided estimates on the lower
bound of attitude uncertainty attainable by star sensors in the Solar System’s
stellar neighborhood. These derived limits are valid in our stellar
neighborhood regardless of the star sensor technology, when no Solar System
body can be used to significantly augment attitude determination171717For
example, for a spacecraft a few light years from home, in a distant future
probably hundreds of years from now, when interstellar travels should become
as common as is interplanetary travel today.. Being the first work to attempt
to derive the numerical values of this limit, it did not aim for much
accuracy. This explains the large factor (of about one order of magnitude)
between the upper and lower estimates of this lower bound on attitude
uncertainty presented in that work181818This article is a thoroughly revised
version of that chapter. and revised here. Nevertheless, these results
suffices for the purpose of obtaining an order of magnitude evaluation on how
much room for improvement there exists for current state of the art star
trackers. It is shown that the accuracy of current star trackers can still
improve by about 6 or 7 orders of magnitude before reaching the ultimate
limits imposed by laws of Physics and stellar distribution in our stellar
neighborhood.

To facilitate verification of the results presented in this work and to foster
a culture of open collaboration in the scientific community, the authors make
the source code of the routines used to generate results presented in this
work available to anyone interested as a free open source software (see
section “Supplementary Materials” below).

The routines used to generate most of the figures and results presented in
this work are made available inside STR_Limits_r02.zip under an OSI approved
BSD 3 clause copyright license (https://opensource.org/licenses/BSD-3-Clause).
STR_Limits_r02.zip is 512,102 bytes long and its MD5SUM is
E8BAE12D0F594440A5F40B86F78A206C. To assure long term preservation,
STR_Limits_r02.zip has been published in a public data
repositorySTRLimits_code  and also in an institutional repository
(http://urlib.net/rep/8JMKD3MGP3W34R/3U448R8).

M.A.A.F developed the software and wrote the bulk of this paper as a part of
his doctorate thesis. M.A.A.F and D.M. extensively revised the manuscript.

This research was funded by CNPq scholarship grant 207308/2015-2.

###### Acknowledgements.

The authors would like to thank for the patient review and recommendations
given by friends and anonymous reviewers. Special thanks for Dr. Leonel
Perondi, who has suggested many improvements to the manuscript. The first
author would like also to thank CNPq for the research scholarship
207308/2015-2. Without their financial support, it would be much more
difficult to conclude this work. We also want to express our gratitude to the
many people and institutions that in a way or the other contributed and gave
support for this research to become a reality, in special, for many people
from INPE, CNPq, CAPES, AEB, MCTIC and Texas A&M University. yes

##  Appendix A Proof that $\sigma_{i}^{2}=\sigma_{\min,i}^{2}$

Being $\sigma_{\min,i}^{2}$ the lower bound of centroiding error variance for
star $i$ along any axis perpendicular to the true direction of that star,
given by Eq. (19), and being $\sigma_{i}^{2}$ the overall measurement variance
associated with that star, used in Eq. (5.114) in Markley2014 , the goal of
this appendix is to prove they are the same for the STR model discussed in
this work.

###### Proof.

_Part 1:_ Let $\mathbf{b}_{i}^{\mathrm{true}}$ be a unit vector representing
the true position of star $i$ in the star tracker reference frame (body frame)
B and $\mathbf{s}_{i}^{\mathrm{true}}=[0\ 0\ 1]^{T}$ the same unit vector in a
reference frame Si where the line joining the star tracker to star $i$ is the
$z$-axis of that reference frame191919For each star $i$, there are many
different reference frames Si with this property, but any one of them will
suffice for our proof.. The attitude matrix $\mathbf{A}_{i}$ linking those two
frames will satisfy:

$\mathbf{s}_{i}^{\mathrm{true}}=\mathbf{A}_{i}\,\mathbf{b}_{i}^{\mathrm{true}}$
(25)

Due to measurement errors, the measured direction of star $i$
($\mathbf{s}_{i}$) will differ from its true position
$\mathbf{s}_{i}^{\mathrm{true}}=[0\ 0\ 1]^{T}$ by
$\varDelta\mathbf{s}_{i}\equiv$
$\mathbf{s}_{i}-\mathbf{s}_{i}^{\mathrm{true}}\equiv$ $[\varDelta s_{i,x}\ $
$\varDelta s_{i,y}\ $ $\varDelta s_{i,z}]^{T}$. Under the assumptions of
Section 2 and considering that $\sigma_{\min,i}^{2}$ gives the lower bound on
centroid error per axis, the expected values of the variances of the $x$ and
$y$ components (components in the Si reference frame) of
$\varDelta\mathbf{s}_{i}$ will be equal to $\sigma_{\min,i}^{2}$. In
mathematical terms:

$E\left\\{(\varDelta s_{i,x})^{2}\right\\}=E\left\\{(\varDelta
s_{i,y})^{2}\right\\}=\sigma_{\min,i}^{2}$ (26)

with $E\\{x\\}$ denoting the expected value of a random variable $x$. Since
$\mathbf{s}_{i}$ is a unit vector very close to
$\mathbf{s}_{i}^{\mathrm{true}}$ and $\mathbf{s}_{i}^{\mathrm{true}}=[0\ 0\
1]^{T}$, the $z$ component of $\varDelta\mathbf{s}_{i}$, $\varDelta s_{i,z}$,
will be given by:

$\varDelta s_{i,z}\ =\ \sqrt{1-\Big{(}(\varDelta s_{i,x})^{2}+(\varDelta
s_{i,y})^{2}\Big{)}}\,\ -1\ \ \approx\ \ -\frac{1}{2}\left((\varDelta
s_{i,x})^{2}+(\varDelta s_{i,y})^{2}\right)$ (27)

As we are retaining only first order terms, like Markley and Crassidis did
Markley2014 , $\varDelta s_{i,z}\approx 0\;\Rightarrow\;E\\{\varDelta
s_{i,x}\,\varDelta s_{i,z}\\}=E\\{\varDelta s_{i,y}\,\varDelta
s_{i,z}\\}=E\\{(\varDelta s_{i,z})^{2}\\}=0$, in a first order approximation.
Given that the star tracker aperture is circular and contained in a plane
perpendicular to the direction of incoming rays from star $i$, from symmetry
considerations we also have $E\\{\varDelta s_{i,x}\,\varDelta s_{i,y}\\}=0$,
that is, the errors are axially symmetric about the true vectors
$\mathbf{s}_{i}^{\mathrm{true}}$. Hence, the measurement covariance matrix for
star $i$

$\mathbf{S}_{i}\equiv E\\{\varDelta\mathbf{s}_{i}\
\varDelta\mathbf{s}_{i}^{T}\\}$ (28)

will be in a first order approximation:

$\mathbf{S}_{i}\approx\begin{bmatrix}\sigma_{\min,i}^{2}&0&0\\\
0&\sigma_{\min,i}^{2}&0\\\ 0&0&0\end{bmatrix}$ (29)

Its trace will be:

$\operatorname{tr}\left(\mathbf{S}_{i}\right)\approx 2\sigma_{\min,i}^{2}$
(30)

_Part two:_

Markley and Crassidis define the following measurement covariance matrix for
the errors in the measured star direction vectors $\mathbf{b}_{i}$ (Eq.
(5.104a) in Markley2014 ):

$\mathbf{R}_{b_{i}}\equiv
E\left\\{\varDelta\mathbf{b}_{i}\varDelta\mathbf{b}_{i}^{T}\right\\}$ (31)

Given the assumption that the vector errors are axially symmetric about the
true vectors (in our model, this arises from the consideration that the star
tracker has a spherical shape) and ignoring the components along the true star
directions (components along vectors $\mathbf{b}_{i}^{\mathrm{true}}$ =
components $\varDelta s_{i,z}$), since these are of higher order than the
terms that we retain, this measurement covariance matrix can be expressed as
(Eq. (5.107b) in Markley2014 ):

$\mathbf{R}_{b_{i}}=\sigma_{b_{i}}^{2}\left[\mathbf{I}_{3\times
3}-\mathbf{b}_{i}^{\mathrm{true}}(\mathbf{b}_{i}^{\mathrm{true}})^{T}\right]$
(32)

being $\sigma^{2}_{b_{i}}$ the variance in the measured vector position for
star $i$ and $\mathbf{I}_{3\times 3}$ the $3\times 3$ identity matrix.

Considering that the inverse of an attitude matrix is its transpose, from Eq.
(25) we have that
$\mathbf{b}_{i}^{\mathrm{true}}=\mathbf{A}_{i}^{T}\mathbf{s}_{i}^{\mathrm{true}}$,
hence:

$\mathbf{R}_{b_{i}}=\sigma_{b_{i}}^{2}\left[\mathbf{I}_{3\times
3}-\mathbf{A}_{i}^{T}\mathbf{s}_{i}^{\mathrm{true}}\left(\mathbf{s}_{i}^{\mathrm{true}}\right)^{T}\mathbf{A}_{i}\right]$
(33)

Considering that the trace of a matrix is a linear operator:

$\operatorname{tr}(\mathbf{R}_{b_{i}})=\sigma_{b_{i}}^{2}\left[\operatorname{tr}(\mathbf{I}_{3\times
3})-\operatorname{tr}\\!\left(\mathbf{A}_{i}^{T}\mathbf{s}_{i}^{\mathrm{true}}\left(\mathbf{s}_{i}^{\mathrm{true}}\right)^{T}\mathbf{A}_{i}\right)\right]$
(34)

Using the matrix trace property,
$\operatorname{tr}(\mathbf{D}\mathbf{C})=\operatorname{tr}(\mathbf{C}\mathbf{D})$,
with $\mathbf{D}=\mathbf{A}_{i}^{T}\mathbf{s}_{i}^{\mathrm{true}}$ and
$\mathbf{C}=\left(\mathbf{s}_{i}^{\mathrm{true}}\right)^{T}\mathbf{A}_{i}$ and
considering that the trace of a $3\times 3$ identity matrix is 3:

$\displaystyle\operatorname{tr}(\mathbf{R}_{b_{i}})$
$\displaystyle=\sigma_{b_{i}}^{2}\left[3-\operatorname{tr}\\!\Big{(}\big{(}(\mathbf{s}_{i}^{\mathrm{true}})^{T}\mathbf{A}_{i}\big{)}\big{(}\mathbf{A}_{i}^{T}\mathbf{s}_{i}^{\mathrm{true}}\big{)}\Big{)}\right]$
$\displaystyle=\sigma_{b_{i}}^{2}\left[3-\operatorname{tr}\\!\Big{(}(\mathbf{s}_{i}^{\mathrm{true}})^{T}\big{(}\mathbf{A}_{i}\mathbf{A}_{i}^{T}\big{)}\mathbf{s}_{i}^{\mathrm{true}}\Big{)}\right]$
(35)

Since $\mathbf{A}_{i}\mathbf{A}_{i}^{T}=\mathbf{I}_{3\times 3}$ and
$\mathbf{s}_{i}^{\mathrm{true}}=[0\ 0\ 1]^{T}$, the trace of
$\mathbf{R}_{b_{i}}$ reduces to:

$\displaystyle\operatorname{tr}(\mathbf{R}_{b_{i}})$
$\displaystyle=\sigma_{b_{i}}^{2}\left[3-\operatorname{tr}\\!\Big{(}(\mathbf{s}_{i}^{\mathrm{true}})^{T}\mathbf{s}_{i}^{\mathrm{true}}\Big{)}\right]$
$\displaystyle=\sigma_{b_{i}}^{2}\left[3-\operatorname{tr}([1])\right]=2\,\sigma_{b_{i}}^{2}$
(36)

Considering that both the expectation $E\\{\cdot\\}$ and matrix trace are
linear operators and using the identity
$\operatorname{tr}(\mathbf{D}\mathbf{C})=\operatorname{tr}(\mathbf{C}\mathbf{D})$,
the following result is obtained from Eq. (31):

$\displaystyle\operatorname{tr}(\mathbf{R}_{b_{i}})$
$\displaystyle=\operatorname{tr}\\!\big{(}E\\{\varDelta\mathbf{b}_{i}\varDelta\mathbf{b}_{i}^{T}\\}\big{)}=E\big{\\{}\\!\operatorname{tr}(\varDelta\mathbf{b}_{i}\varDelta\mathbf{b}_{i}^{T})\big{\\}}$
$\displaystyle=E\big{\\{}\\!\operatorname{tr}(\varDelta\mathbf{b}_{i}^{T}\varDelta\mathbf{b}_{i})\big{\\}}$
(37)

In the same manner that
$\mathbf{b}_{i}^{\mathrm{true}}=\mathbf{A}_{i}^{T}\mathbf{s}_{i}^{\mathrm{true}}$,
we have $\varDelta\mathbf{b}_{i}=\mathbf{A}_{i}^{T}\varDelta\mathbf{s}_{i}$.
Using the matrix property,
$(\mathbf{B}\mathbf{C})^{T}=\mathbf{C}^{T}\mathbf{B}^{T}$, we also have
$\varDelta\mathbf{b}_{i}^{T}=\varDelta\mathbf{s}_{i}^{T}\mathbf{A}_{i}$.
Substituting these into the last equation:

$\displaystyle\operatorname{tr}(\mathbf{R}_{b_{i}})$
$\displaystyle=E\big{\\{}\\!\operatorname{tr}(\varDelta\mathbf{s}_{i}^{T}\mathbf{A}_{i}\mathbf{A}_{i}^{T}\varDelta\mathbf{s}_{i})\big{\\}}=E\big{\\{}\\!\operatorname{tr}(\varDelta\mathbf{s}_{i}^{T}\varDelta\mathbf{s}_{i})\big{\\}}$
$\displaystyle=E\big{\\{}\\!\operatorname{tr}(\varDelta\mathbf{s}_{i}\varDelta\mathbf{s}_{i}^{T})\big{\\}}=\operatorname{tr}\\!\big{(}\\!E\\{\varDelta\mathbf{s}_{i}\varDelta\mathbf{s}_{i}^{T}\\}\big{)}$
(38)

But, from Eq. (28),
$E\\{\varDelta\mathbf{s}_{i}\varDelta\mathbf{s}_{i}^{T}\\}$ is the measurement
covariance matrix $\mathbf{S}_{i}$ for star $i$, whose first order
approximation is given by Eq. (29) and trace by Eq. (30). Therefore:

$\operatorname{tr}(\mathbf{R}_{b_{i}})=\operatorname{tr}(\mathbf{S}_{i})=2\sigma_{\min,i}^{2}$
(39)

From Eqs. (A) and (39) the following is obtained:

$\sigma_{\min,i}^{2}=\sigma_{b_{i}}^{2}$ (40)

Equation (5.109) in Markley2014  gives $\sigma_{i}^{2}$ as:

$\sigma_{i}^{2}=\sigma_{b_{i}}^{2}+\sigma_{r_{i}}^{2}$ (41)

where $\sigma_{r_{i}}^{2}$ is the variance in the reference vector for star
$i$. In other words, $\sigma_{r_{i}}^{2}$ gives the uncertainty in the
cataloged position of that star. Since it is being assumed that these
cataloged position are known with no errors (Item 6 in Section 2.1),
$\sigma_{r_{i}}^{2}=0$, implying that $\sigma_{i}^{2}=\sigma_{b_{i}}^{2}$.
Substituting this into Eq. (40) the following is obtained:

$\sigma_{i}^{2}=\sigma_{\min,i}^{2}$ (42)

completing the proof. ∎

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