Optical tracking of deep-space spacecraft in Halo L2 orbits and beyond: the Gaia mission as a pilot case

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by  Alberto Buzzoni

Optical tracking of deep-space spacecraft in Halo L2 orbits and beyond:
✩
the Gaia mission as a pilot case
Alberto Buzzoni∗, Giuseppe Altavilla, Silvia Galleti
INAF - Osservatorio Astronomico di Bologna, ViaRanzani 1 40127 Bologna Italy
6Abstract
1
We tackle the problem of accurate optical tracking of distant man-made probes, on Halo orbit around the Earth-Sun
0
2librationpointL2andbeyond,alonginterplanetarytransfers. Theimprovedperformanceofon-targettracking,especially
whenobservingwithsmall-classtelescopesisassessedprovidingageneralestimateoftheexpectedS/Nratioinspacecraft
n
detection. The on-going Gaia missionis taken as a pilot case for our analysis,reporting on fresh literature and original
a
Joptical photometry and astrometric results.

8 The probe has been located, along its projected nominal path, with quite high precision, within 0.13±0.09 arcsec, or
10.9±0.6 km. Spacecraft color appears to be red, with (V Rc) = 1.1±0.2 and a bolometric correction to the Rc band
of (Bol Rc) = 1.1±0.2. The apparent magnitude, Rc−= 20.8±0.2, is much fainter than originally expected. These
] − −
Mfeatures lead to suggest a lower limit for the Bond albedo α=0.11±0.05 and confirm that incident Sun light is strongly
reddened by Gaia through its on-board MLI blankets covering the solar shield.
I
. RelyingontheGaiafigures,wefoundthatVLT-classtelescopescouldyetbeabletoprobedistantspacecraftheading
h
Mars, up to 30 million km away, while a broader optical coverage of the forthcoming missions to Venus and Mars could
p
be envisaged, providing to deal with space vehicles of minimum effective area 106 cm2. In addition to L2 surveys,
-
A≥
o2m-class telescopes could also effectively flank standard radar-ranging techniques in deep-space probe tracking along
r
tEarth’s gravity-assistmaneuvers for interplanetary missions.
s
aKeywords: Space vehicles; Techniques: high angular resolution; Astrometry and celestial mechanics; Techniques:
[
photometric

1
v
91. Introduction to be locatedin a so-calledHalo L2orbitin the forthcom-
1
ingyears. TheseincludetheJames-WebbSpaceTelescope
7
4 The exploitation of the Sun-Earth Lagrangian points, (Jwst; Gardner et al., 2006), the Euclid cosmological
0especially L1 and L2, along the Sun-Earth direction (Far- probe (Laureijs et al., 2010), and the Athena X-ray ob-
.quhar&Kamel,1973;Nariai,1975;Rawal,1991;Farquhar servatory (Barret et al., 2013).
1
0etal.,2002)hasbeenanextraordinarychallengeforspace Optical ground tracking is yet of recognized impor-
exploration in the recent years. For their particular posi-
6 tance for any L2 mission. For the co-rotating orbit to be
1tion,some1.5106kmaway,onoppositesidesofEarthand maintained within its nominal figures, in fact, we need to
:therefore well beyond the Moon, both locations are ideal
v carefully check spacecraft during its course along a com-
ilookouts for astrophysical observatories aimed at study- plex Lissajous trajectory, as seen from Earth (e.g. Bray
X
ing the Sun (L1) and the deep Universe (L2), far from
& Gouclas, 1967; Zagouras & Markellos, 1985; Liu et al.,
ranyanthropiccontamination. TheL2point,inparticular,
a 2007; Kolemen et al., 2012; Dutt & Sharma, 2011; Qiao
hasbeenhostinganumberofimportantastrophysicalmis-
et al., 2014). This is also of special interest for any space
sions, starting with the Wmap, Planck and Herschel observatory (like Gaia, or the next Jwst) as its abso-
probes, and currently continuing with the Gaia mission,
lute inertial position is required with exquisite precision
aimed at performing an exhaustive census of the Milky
toallow,forinstance,aconfidentmeasureofastronomical
Way stellar population (de Bruijne, 2012; Cacciari,2015).
parallaxesof distant stars with the on-boardinstruments.
Following Gaia, other major space facilities are planned
Inthisregard,radar-rangingtechniquesmayactuallypro-
vide a better measurement of spacecraft distance and ra-
dial velocity (Imbriale, 2003), but telescope observations,
✩ Based on observations collected at the Cassini Telescope of the
from their side, take advantage of a superior angular res-
LoianoObservatory,Italy
∗Correspondingauthor: olution, providing in principle more accurate astrometry
Email address: [email protected] (Alberto and finer proper motion estimates (Altmann et al., 2011).
Buzzoni) Compared to the observation of near-Earth satellites,
Preprint submittedto Advances inSpace Research January 20, 2016
Figure1: Theapparentbolometricmagnitudeforman-madespacecraftatincreasingdistancefromEarth,accordingtoeq.(2). Probescale-
sizeislabelledalongeachcurve. ThealtitudeofLEO(setto500km),MEO(5000km)andGEOterrestrialorbitsismarked,together with
a few small bodies in the Solar System and their reference interplanetary distances at Earth’s opposition. The limitingmagnitude reached
by a 2m mid-class telescope, the 8m ESO VLT and the forthcoming 40m E-ELT telescope, when observing distant Sun-type stars, is also
sketched ontheplot.
however, optical tracking of distant probes, in L2 and on If a probe offersa cross-sections2 to Sun’s light, being
route to even farther interplanetary distances, has to deal s its reference scale-size,and if we assume the illuminated
with much fainter target magnitudes, a drawback that body to reflect isotropically, then the apparent bolomet-
urgesasubstantialimprovementintermsoftelescopeskills ric magnitude of a spacecraft placed at a distance d from
and especially ofobservingtechniques to effectively assess Earth’ssurface(atSun’s opposition)1 simplyscalesasthe
our deep-space situational awareness (e.g. Mooney et al., ratio of the incident solar flux at Earth and at the space-
2006; Ruprecht et al., 2014; Woods et al., 2014). craft distance, so that
In this contribution we want therefore to briefly assess
sometechnicalissues(Sec.2)dealingwithaccurateground m mbol = 2.5log D⊙ 2 s2 ,(1)
trackingofdeep-spaceprobesatopticalwavelength,taking bol− ⊙ − "(cid:18)D⊙+d(cid:19) (cid:18)4πd2(cid:19)#
freshobservationsoftheGaiaspacecraft(Sec.3)asapilot
case for tuning up our theoretical analysis. The relevant where D⊙ = 1.491013 cm is the astronomical unit (AU)
and mbol = 26.85 is the apparent bolometric magnitude
photometric figures for Gaia will constrain the required ⊙ −
oftheSun,asseenfromEarth(e.g.Karttunenetal.,1996).
telescopeperformance,fortheopticalobservationstocon-
Withthe relevantsubstitutions,andexpressingthespace-
sistently complement standard radar-tracking techniques
as in the forthcoming space missions to Mars and other craft distance u=d/D⊙ in AU, eq. (1) takes the form:
planets of the solar system (Sec. 4). Our conclusions will m = 5logs+5log[u(1+u)]+k, (2)
bol
be briefly stressed in Sec. 5. −
where the numerical constant is
2. Apparent magnitude of distant spacecraft k=2.5log(4πD2)+mbol =41.77, (3)
⊙ ⊙
Depending on its physical properties, a satellite under providing the satellite scale-size s is set in cm.
solarillumination reflects a fractionα (the so-calledBond In Fig. 1 we report an illustrative summary of the ex-
albedo) of the incident flux. The remaining fraction of pectedbolometricmagnitudefordistantman-madeprobes
the input energy is retained and heats the body up to an of different characteristic size, compared with a few small
equilibrium temperature that leads to a balance between planetary bodies. Just as a guideline, the limiting magni-
the absorbed and re-emitted flux. At Earth’s heliocen- tude asforobservingSun-type stars,reachedbymid-class
tric distance, this temperature cannot exceed 120◦C (e.g.
Gilmore, 2002), so that thermal emission of spacecraft in
1Although,strictlyspeaking,disatopocentricdistance,toallex-
the terrestrialneighborhood(andbeyond)isonlyrelevant
tent,foradistantspacecraftinL2andbeyond,itbasicallycoincides
at mid/far-infrared wavelength. withthegeocentric distance,aswell.
2
Figure 2: The surface-brightness dimmingfortrailingsatellites, ac-
cordingtoeq.(5). Differentexposuretimesareassumed,aslabelled.
Figure 3: The magnitude limit reached by a 8m telescope, with a
Eachstriphasalowerandupperenvelopefora2”and0.5”FWHM
Dqe=0.8,ata5and100S/Ndetectionlevelin100(upperenvelope)
seeingfigure,respectively. Thereferenceangularspeedforsatellites
and 1000 sec (lower envelope of the curves) exposure time, as from
inLEO,MEO,andGEO (Veis,1963) isreportedtogether withthe
meanskymotionforotherrelevantsolarbodies. Inaddition,wealso eq. (6). We assume to observe in the Rc band under two extreme
seeing conditions, namelywithaFWHM of 0.5and 2.0arcsec, and
displaythemeanangularvelocityofspacecraftGaia,alongitsHalo withadarksky(µR =20.5magarcsec−2). Theindicativeproper
L2orbit. sky
motionofsomereferenceobjects isreported,asfromFig.2.
(2maperture)andnew-generationtelescopes(i.e. theESO
latterwouldactuallybehardertocatch,becauseofalower
8m VLT and the forthcoming 40mE-ELT) is also marked
meansurfacebrightnessandacorrespondinglypoorerS/N
on the plot.
ratio. The surface-brightness dimming can be quantified
As a fraction (1 α) of the incident solar flux is “di-
− in
verted” into the infrared, to convert the bolometric fig-
4t ζ
ures to other broad-band optical magnitudes, say for in- ∆µ =2.5log(a′/a)=2.5log 1+ exp . (5)
stance the Johnson-Cousins R band, we have to dim the sat πFwhm
c (cid:18) (cid:19)
re-processed optical flux such as Notethattheeffectdoesnotdependontargetmagnitude.
m =m BC′ 2.5 logα, (4) Rather, and quite interestingly, it becomes more severe
R bol− R− with increasing exposure time and with improving seeing
beingBC′ =(m m )thebolometriccorrectiontothe (that is for a better FWHM value), as shown in Fig. 2.
R bol− R
band,ifsatellitewereaperfectacromaticreflector(i.e. for According to telescope diameter and the exposure
D
α 1). This value has to be estimated on the basis of time texp, the S/Nratioofatrailingobjectcaneventually
→
satellite’s physical and geometrical properties. If we lack be computed as
tchoiusldinfboermtaaktieonn,atsheanfieristthearptphreosxoimlaartviaolnueor(,BiCfR′a=co+lo0r.3is) S = 10−0.4hmsat−(cid:16)µsky−2∆µsat(cid:17)i, (6)
N R
known for our target, the BC′ correction for a more ap- (cid:18) (cid:19)trail
R
propriate star could be chosen, as from standard calibra- with
tions in the literature (e.g. Johnson, 1966; Bessell, 1979;
= D (n Dqet )1/2. (7)
Buzzoni et al., 2010). R Fwhm o exp
In previous equations, µ is sky surface brightness, n
2.1. Motion dimming sky o
and Dqe are the reference photon number for the magni-
For fixed exposure time, any moving satellite tends to tude zero point and the detector quantum efficiency, re-
appear more elusive than fixed stars of the same magni- spectively, according to the photometric band of our ob-
tude on CCD images. In fact, while a star is supposed servations. By definition, n = (f λ ∆ )/(hc), being f
o o o λ o
to concentrate most of its photons across a spot of area the zero-magreferenceflux, λ and∆ the effective wave-
o λ
a = π(Fwhm/2)2, whose FWHM depends on the tele- length and width of the photometric band, respectively, h
scopepoint-spread-function,atargetmovingwithangular thePlanckconstantandcthespeedoflight,asusual. For
speed ζ, along a time texp, would spread its light across a the Rc band, no = 1.05 106 photons cm−2 s−1 (Buzzoni,
larger area a′ = ζtexpFwhm+π(Fwhm/2)2. If both the 2005). Noticethat,ifζ 0,thenthe∆µsat termvanishes
star and the moving target have the same magnitude, the and eq. (6) approaches→the S/N ratio for a fixed star.
3
Table1: TheOct17-18,2014andMarch26-27,2015Gaiaobserva-
tions
UTC(a) RA(b,c) (J2000) Dec(b,c) Mag(c,d) Exp.(e)
hh:mm:ss.s hh:mm:ss.sss dd:mm:ss.ss sec
Oct17-18,2014observingsession
20:08:55.5 01:49:13.459±009 +15:30:11.54±43 20.6±3 wR 120d
20:15:45.6 01:49:14.134±049 +15:30:20.50±62 20.7±3 wR 300d
20:23:56.1 01:49:14.951±015 +15:30:31.71±21 21.1±3 wR 300d
20:28:55.5 01:49:15.434±028 +15:30:38.12±29 20.9±3 wR 300d
20:34:08.3 01:49:15.907±016 +15:30:45.32±21 20.6±3 wR 300d
20:44:07.5 01:49:16.758±014 +15:30:57.88±13 20.9±3 wR 300d
20:55:32.7 01:49:17.695±015 +15:31:11.71±14 20.5±3 wR 300s
21:13:16.1 01:49:18.975±017 +15:31:34.35±21 21.0±2 Rc 300d
Figure4: ThespacecraftGaia,asdetectedinthenightofOct17-18, 21:29:09.9 01:49:20.007±016 +15:31:52.66±16 22.2±2 V 420d
2014 with the 1.52m telescope of the Loiano Observatory. The two 21:35:46.8 01:49:20.363±016 +15:32:00.45±54 20.6±4 zR 300d
21:44:25.3 01:49:20.896±015 +15:32:09.63±09 20.7±2 Rc 300d
panels areconsecutive 300secexposures in”white” light(i.e. CCD 00:50:46.7 01:49:29.027±005 +15:34:37.50±34 21.0±2 Rc 300d
with no photometric filter) of the same field with sideral (left) and 00:58:26.2 01:49:29.475±011 +15:34:40.94±25 20.8±3 wR 300d
differential(right)telescopetrackingon. Thedisplayedfieldis3 3 01:10:54.5 01:49:30.163±008 +15:34:45.55±27 21.9±2 V 600d
×
arcminacross. Northisup,Easttotheleft. Pixelsizeis0.58arcsec. March26-27,2015observingsession
Seetextforadiscussion. 00:00:14.5 12:28:27.063±013 +04:09:23.99±20 20.7±3 Rc 480d
00:13:46.8 12:28:27.499±013 +04:09:02.05±20 21.7±3 V 900d
As an illustrative case,in Fig. 3 we report the limiting (a) Luminosity-weightedtimebarycenter,accordingtoeq.(9)
(b) Assumedtopocentriccoordinatesofthetelescope:
magnitude in the Rc band, that can be achieved for a (λ,φ,h)=(11o20m02.7sE,44o15′33.2′′N,745.3ma.s.l.)
trailingtargetataS/N=5and100detectionlevelwitha (c) Errorfiguresapplytothelastdigitsofeachentry
8m telescope (Dqe=0.8) in a 100 and 1000 sec exposure (d) Photometricbands: Johnson-Cousinsbands(V,Rc),
R-convertedwhitelight(wR),andGunnzband(zR)
time. Weassumetoobservefromagoodastronomicalsite (e) Trackingmode: d=differential(“on-target”),s=sideral
(µR = 20.5 mag arcsec−2, e.g. Patat, 2003), under two
sky
extremeseeingconditions(namely0.5”and2.0”FWHM).
Animportantfeatureoftheplotisthatmaglimitforhigh- position and apparent luminosity in the Rc and V bands.
speed satellites does not depend on exposure time but it Bothnightshadphotometricconditionsalongtheobserva-
only improves with a better seeing or a bigger telescope. tionwindows,withanR-bandFWHMseeingfigureabout
Inthelattercase,eq. (6)(andthecurvesinFig.3)offsets 1.5′′-1.7′′. The relevant data of our observations are sum-
by ∆R =2.5log( /800), by expressing in cm. marized in Table 1. All frames were taken in “on-target”
lim
D D tracking mode, with the only exception of one image (as
markedinthe table), trackedinsideralmodeforthe illus-
3. The Gaia observations as a benchmark
trative purpose of Fig. 4.
Inthefigureweactuallyreporttwoconsecutive300sec
Previous arguments make clear that, when observing
exposuresofthesamefield,alongtheOct2014night,with
distant spacecraft, on-target telescope tracking, such as
sideral (left panel) and differential (right panel) telescope
to compensate target motion, is the mandatory require-
tracking on. Just at first glance, it is evident that Gaia
ment for letting any faint target literally “emerge” from
can barely be appreciated when trailing across the field,
the backgroundnoise. Comparedtothe outputofeq. (6),
while it clearly stands out when concentrating its light as
infact,theexpectedimprovementintheS/Nratioforthe
in the right panel.
latter case is of the order of
S = S 10+0.2∆µsat. (8) 3.1. Astrometry
N N
(cid:18) (cid:19)track (cid:18) (cid:19)trail Although limited, our observed database allowed us
As a striking example in this sense, we report here on to benchamark the astrometric and photometric reduc-
tworecentobservingsessionsontheGaiaprobe,alongits tionprocedure,inordertoassesstherealisticperformance
HaloL2orbit. Observationshavebeencarriedoutwiththe in spatially locating the spacecraft at such large distance
Cassini1.52mtelescopeoftheLoianoObservatory(Italy), fromEarthandcharacterizeitsapparentphotometricprop-
inasix-monthsintervalalongthenightofOct17-18,2014 erties. A first crucial piece of information deals with as-
and on March 26-27, 2015. The telescope was equipped trometry.
with the BFOSC camera, carrying a EEV 1300 1340 px For the astrometric solution of the CCD images, af-
coated and back-illuminated CCD, with a FOV×of 12.6 ter standard reduction procedures with IRAF2, we relied
13.0 arcmin, and a pixel scale of 0.58 arcsec px−1. × on the WCSTools package3 to pick up a reference grid of
In addition to a set of eight “white”-lightframes (that template stars across the field of view and finally set the
is by exposing CCD with no photometric filter), the 2014
frames also consistedof six V, R , andGunn z exposures.
c 2SeetheURLs: http://iraf.noao.edu.
Overall, the spacecraft was tracked along five hours. The
3SeetheURLs: http://tdc-www.harvard.edu/wcstools.
2015nightwasjust devotedto aone-shotcheckofGaia’s
4
WorldCoordinateSystem(WCS)ofthecorrespondingim-
age. The 50 brightest objects in each field, as detected
by the IRAF task STARFIND, have been matched by the
WCSTools IMWCS task with the corresponding HST Guide
star Catalogue II (GSC-II) (Russell et al., 1990)4 avail-
able by default to constrain the WCS across the frame.
The WCS is the relationship between the pixel coordi-
nates and the celestial coordinate, and it is written in
a standard way in the FITS header. The precise astro-
metric solution is then refined by means of the WCSTools
SCAT and the IRAF CCFIND and CCMAP tasks. In particu-
lar, SCATretrievesthe GSC-II catalogueofthe field,while
CCFIND uses the image WCS to convert the celestial coor-
dinates into image pixel coordinates and refine the latter
ones by using a centroid algorithm. The matched coordi-
nates are finally used by CCMAP to compute a new plate
solution in the RA and DEC domainby a low-orderpoly- Figure 5: The angular residuals of Gaia’ssky path along the night
of Oct 17-18, 2014, as seen from the Loiano Observatory (UAI ob-
nomial fit. The astrometric solution is directly inspected
servatory code “598”). The data of Table 1 are compared with the
by over-plottingthe referencecataloguesourcesonthe as-
corresponding JPL topocentric ephemeris. Residuals are in arcsec
trometrically calibrated image. units,bothforRA(upperpanel)andDec(lowerpanel),inthesense
Overall,our procedure secured an internal astrometric “Observed–Computed”,(O-C).Theonlysideraltrackingobservation
inoursampleissingledoutwitharombmarkerinbothpanels.
accuracyof0.38±0.18 arcsec(rms)inthe photometriccen-
troiddeterminationofGaia’sindividual observations(see
Table 1).5 Intheequation,θ isexpressedinmas(milliarcsec)unit
mas
Though slightly elongated, the star figures in the tele- and ζ in arcsec hr−1, as in Fig. 2. With the Gaia typical
scope images did not prove to severely affect the astro- figures, t has to be knownwithin a few seconds,at most.
o
metric solution across the field, provided the trailing in- According to the proper-motionconstraints of Fig. 2, this
tensity ofthe astrometriccalibrators(I)doesnotvary(or isalsotheaccuracylevelfortrackinginterplanetaryspace-
just vary in a predictable way) along the exposure time.6 craft, while a better tuned clock (σ(t ) 0.2 sec) is de-
o
This is, actually, a crucial requirement for the photomet- vised, on the contrary, when observing f≪ast-moving (ζ
ric barycenter of stellar tracks to consistently tie to the 103) Earth- and Moon-orbiting satellites. ≫
luminosity-weighted time barycenter (t ) of the exposure.
o
By definition, for the latter, we have 3.1.1. Locating L2 spacecraft
Figure 5 shows that Gaia position consistently com-
t τI(τ)dτ 1
o
= 1. (9) pares with its nominal trajectory along the Oct 17, 2014
t I(τ)dτ dτ ≤
exp R night, as predicted by the topocentric ephemeris from the
IncaseI(τ)=Rconst,thiRs relationshipdeliversthe obvious JPL Horizons Solar System Dynamics (SSD) Interface.7
solutiont =t /2,sothatthespacecraftpositionshould The mean coordinate offsets of our data (in the sense
o exp
be attributed to the mid-exposure time. “observed–computed”) and their corresponding σ uncer-
Clearly, the attainable astrometric resolution (θmas) tainty amount to (∆RA, ∆Dec) = (0.10±0.08, 0.09±0.10)
−
also constrains the required accuracy, σ(t ) in setting t , arcsec, which lead to a mean transverse offset ∆ =
o o GAIA
depending on the spacecraft proper motion: 0.13±0.09arcsec. Ourobservationsdemonstrate,therefore,
that the spacecraft can correctly be located along its ex-
3.6θ
σ(t ) mas [sec]. (10) pectedorbitalfigurewithanangularaccuracyθ ofthe
o mas
≃ ζ
order of 90 mas (see Fig. 6).8 At the reported distance of
Gaia along the night (namely 1.39106 km, according to
4SeetheURL:http://tdc-www.harward.edu/software/catalogues/ the JPLephemeris),ourmeasurespointtoa meanorbital
gsc2.html. offset of 0.9±0.6 km.
5For the illustrative scope of our analysis, we relied here on the These figures nicely compare also with other almost
GSC-II catalog, a default reference for WCSTools to set the astro-
parallel sets of observations, taken a few nights earlier
metric solution. Other catalogs could, however, be implemented
providing to set them in appropriate format. Of these, especially
the PPMXL (Roeser, Demleitner, & Schilbach, 2010) and CMC-15
(Niels Bohr Inst. et al., 2014) compilations may prove to be viable 7SeetheURL:http://ssd.jpl.nasa.gov/horizons.cgi.
alternatives toimproveaccuracyatleastinselectedskyregions. 8Quite consistently, notice that the derived 90 mas rms of the
6Among others, relevant change along trailing intensity of stars average coordinate residuals of Fig. 6 fully compares with the
couldeitherbeduetoanysmalldriftinthesideraland“on-target” (380/√Nobs) ∼ 100 mas theoretical figure, as for the independent
telescopetracking,ortoanyunperceivedtinycloudcrossingthefield observations ofTable1.
duringexposureetc.
5
Figure 8: The Gaia sky path along the March 10-26, 2015 period.
Figure 6: Arcsec coordinate residuals (in the sense “observed– TheTerskol(UAIcode“B18”)(Velichkoetal.,2015,cyanmarker),
computed”) of the Gaia positions along the night of Oct 17, 2014, PanSTARRS(UAIcode“F51”)(Gibsonetal.,2015,greendots)and
with respect to the JPL topocentric ephemeris. Mean RA and Dec Loiano(reddot)observationsaresuperposedtotheJPLtopocentric
offsets,togetherwiththeir1-σuncertainty, arereportedintheplot. ephemeris(nominallyfortheB18location,justasaguideline). Note
Whencombined,theseleadtoameanpathoffsetwithrespecttothe thedaily“ripples”oftheGaiaapparentorbit,duetotheparallaxef-
nominal figure of only ∆GAIA =0.13±0.09 arcsec (big circle in the fectofEarthrotation,thatsuperposestotheoverallLissajousfigure
plot). onlargerscales.
(namelyonOct14,2014)bythe1.5mMt.Lemmon(Kowal- ( 0.11±0.13, 0.11±0.10) arcsec,leading to a mean orbital
− −
ski et al., 2014) and the 1.8m LPL Spacewatch II (Tubbi- offset of ∆GAIA = 0.16±0.11 arcsec (or 1.1±0.7 km) with
olo,2014)telescopes. Asummaryofthesedataisshownin respect to the corresponding JPL ephemeris.
Fig.7. AfterrejectingoneclearoutlierintheLPL/Space- Closer to our 2015 observations, a supplementary set
watchII sample,they indicate,overall,a(∆RA, ∆Dec)= of10astrometricmeasurementswereprovidedbyVelichko
et al. (2015) in the night of March 10, 2015, at the 2m
telescope of the Terskol Observatory (Russia), together
with six estimates of Gibson et al. (2015) with the 1.8m
Pan-STARRStelescopeinHaleakala(Hawaii,USA)onthe
nights of March 23 and 24, 2015 (see Fig. 8).
ThiscoarsersetofobservationsismergedinFig.9with
ourdataandleads,overall,to(∆RA,∆Dec)=(0.21±0.06,
0.07±0.04)arcsec,thatisameanorbitaloffsetof∆GAIA =
0.22±0.05 arcsec (i.e. 1.6±0.4 km).
3.1.2. Locating deep-space spacecraft
The brief overview of the Gaia case convincingly sup-
portsthepreliminaryresultsoftheon-goingtrackingcam-
paigncarriedonintheframeworkoftheGaiamissionplan
(seeAltmannetal.,2012,fordetails),andprovesthatop-
ticalmeasurements,evencarriedoutwithsmall-classtele-
scopesandunderquitestandardobservingconditions,can
easilyachieveasuperioraccuracywhencomparedtoradar
tracking, to locate distant spacecraft across the sky. This
better score is the obvious consequence of the fact that,
Figure 7: Same as Fig. 6 but for a set of astrometric measure- for fixed angular resolving power, it must be
ments forthenightofOct14, 2014, taken withthe1.5m Mt. Lem-
mon (UAI code “G96”) (Kowalski et al., 2014, dots) and the 1.8m (νD) (νD) . (11)
LPL/Spacewatch II (UAI code “291”) telescopes (Tubbiolo, 2014, radio ≃ opt
rombs). Arcsec coordinate residuals are computed with respect to Therefore,ifaradardishissettooperateatsomeX-band
thecorrepsondingJPLtopocentric ephemerisleadingtoameanor-
frequency (say 8 GHz, as for the Deep Space Network an-
bitaloffsetof∆GAIA=0.16±0.11 arcsec(bigcircleintheplot). tennas; Imbriale,2003),comparedtothe 51014 Hzofa
∼
6
Figure10: Theexpectedspatialresolutionindetectingdistantspace-
craftwithintheSolarSystem. Thetransversecomponent,projected
Figure9: SameasFig.6butforacoarsersetofobservations taken
on the sky, is assessed in terms of absolute resolution in kilome-
duringMarch2015attheTerskol2mtelescope(triangles),the1.8m
ters at the different distances from Earth. Two representative val-
Pan-STARRStelescope(pentagons)andtheLoiano1.52mtelescope
ues for angular resolution of optical tracking are assumed, namely
(rombs,seeTable1). Arcseccoordinateresidualsarecomputedwith
θµas=300masand30mas,aslabelledontheplot.
respect to the appropriate JPL topocentric ephemeris for each ob-
servatory. Errorbarsareonlyavailableforourobservations,asfrom
Table 1. The merged set of data leads to a mean orbital offset
Gaiabenchmarkingshowedthataninternalaccuracybet-
of ∆GAIA = 0.22±0.05 arcsec (yellow circle in the plot) along the
spannedperiod. ter than 100 mas is a fully attainable goal for standard
observations carried out with 2m-class telescopes.
As far as even farther distances from Earth are con-
telescopeobservinginthe V band,thenafactorof60,000
sidered, our arguments lead us to conclude that absolute
largerantennaisrequiredtoachievethesameangularper-
position of man-made probes could be constrained within
formance of the optical instrument. This means that a
a nominal accuracy ∆ of the order of
sat
60 km-wide(!) antenna (or radio-interferometric baseline)
is neededto offsetthe (diffraction-limited)performanceof d6θmas
∆ = [km], (12)
sat
a 1m telescope. 206
Clearly, one could argue that any ground-based tele- or
scope is eventually seeing (and not diffraction) limited,
∆ =0.72uθ [km], (13)
thus restraining its resolving power to a fraction of arc- sat mas
sec, at best. Another important issue also deals with the
assuming to express the geocentric distance in million km
inherent astrometric accuracy of reference stars to set the
(d ) or in AU (u, as in eq. 2), respectively. This relation-
6
absoluteastrometricgrid. AsfortheHSTGuideStarCat-
ship is displayed in Fig. 10, for values of θ = 30 and
mas
alogGSC-IIusedhere,currentlyoneofthemostpopulated
300 mas.
source of data across the sky for astrometric studies, this
According to the figure, one can hope to optically pin-
limit turns to be about 300 mas (Lasker et al., 2008;Buc-
point deep-space probes around Mars well within a few
ciarelli et al., 2008).9
tens of km. When complemented with radar informa-
On the other hand, multiple measurements are pos-
tion(moreeffectiveinrangingmeasurementsandDoppler
sible of the target differential position with respect to a
velocity shift), the optical output could therefore be ex-
number of reference (astrometric) stars across the field of
tremely valuable foraneffective 3Dlocationofman-made
view, and this eventually leads to improve target astro-
probes at interplanetary distances.
metric accuracy by a factor √N . In addition, further
stars Quite interestingly, Fig. 10 also suggests that, under
improvements could be envisaged in case a series of in-
appropriate observing conditions (i.e. by masking Moon’s
dependent observations could be collected. Definitely, our
overwhelmingluminosity),Moon-orbitingspacecraftcould
be accurately locatedwithin a few hundred meters,a pre-
cision that could even raise to about 10m in the local
9TheGSC-IIprovidesabout6referencestarspersquarearcmin,
down to V 20, although the reported accuracy refers only to the framework, for instance when considering station-keeping
∼
brightest stars (V 18.5) eventually usedinour study. Thisfigure maneuvering of Earth artificial satellites in geostationary
≤
is expected to drastically improve (down to 30µas for V 15 orbit (Montojo, Lo´pez Moratalla, & Abad, 2011).
∼ ∼
stars)intheforthcomingyears,justrelyingontheGaiastarsurvey
(deBruijne,2012).
7
Furthermore, our results also point to a much fainter
apparentmagnitudethanoriginallyexpectedforthespace-
craft (namely R 17-18, according to Altmann et al.,
c
∼
2011),based on the direct experience on previous L2 mis-
sions (especially Wmap and Planck). Yet to a more up-
dated analysis (Altmann et al., 2014), the reason of this
discrepancy remains unclear. Evidently, Gaia’s faintness
and its exceedingly red color have to be related to more
inherent reflectance characteristic of the spacecraft struc-
ture, especially dealing with the extended Kapton multi-
layerinsulation(MLI)blanketsthatcovermostofthesun-
shield.
Tohelpbetter investigatetheproblem,duringthe pre-
liminary station-keeping maneuvers of Feb 27 and March
Figure11: TheapparentmagnitudeofGaiaindifferentphotometric
7, 2014, the probe has repeatedly been re-oriented, from
bands alongthe Oct2014 andMarch2015observingruns. Inaddi-
its nominal attitude configuration at solar aspect angle11
tion to Rc (dots) and V (square) Johnson-Cousins bands, “white”-
light(triangles) andGunn z (rombmarker)observations havebeen ω =45o, such as to have its sun-shield directly facing the
convertedtotheRcmagnitudescaleasdiscussedinthetext. Dashed Sun, that is with ω =0o. This greatly brightened Gaia’s
linesmarkthemeanRc andV magnitudelevels. apparentluminosity up to Rcpeak =14.5±0.2 (James, 2014;
Jacquesetal.,2014;Dupouy&deVanssay,2014;Dupouy
3.2. Photometry & Laborde, 2014).
Both for the 2014 and 2015 observing sessions, pho- The“face-on”experimentclearlyindicatedthatGaia’s
tometry in V and R bands has been directly calibrated sun-shieldreflectancedisplaysanimportantdirectionalpat-
c
to standardmagnitudes by observingtwo closeby Landolt tern (an effect often referredto as the “oppositionsurge”,
(1992) fields. Typical rms uncertainty for these observa- e.g. Warell, 2004) with spacecraft luminosity increasing
tionsis0.2mag,assummarizedinTable1. Althoughwith by ∆mag = 20.8 14.5 = 6.3±0.3 mag when changing
−
a much broader passband, the effective wavelength of the ω from 45o to 0o. This figure has to be compared with
CCD response curve happened to closely match that of the straight Lambertian prediction, that leads to a much
the R filter (λ 6470 ˚A), and this eased a rough but shallower magnitude brightening of
c o
≃
stillconvenienttransformationofthe“white”instrumental cos45o
magnitudesoftheOct2014sessionintopseudo-equivalent ∆mag= 2.5log 0.4 mag. (14)
− cos0o ∼
R figures (called w in Table 1). The whole conversion (cid:18) (cid:19)
c R
procedure reliedona gridofseveralfieldstars incommon
3.2.1. Bolometric correction and spacecraft albedo
with the R and “white”-light images, and eventually led
c According to eq. (2), the expected bolometric magni-
to a total error of 0.3 mag (rms) for the w magnitudes.
R tude of Gaia at the relevant distance of our observations
Just for the sake of comparison, a similar procedure was
is Bol = 16.9 mag.12 As from eq. (4), this figure can be
also applied to the unique Gunn z frame of the 2014 ses-
linkedtothe apparentR magnitudeby correctingforthe
c
sion, leading to a coarser Rc proxy of (i.e. zR in Table 1), appropriate bolometric correction (BC′ ) and the space-
witha0.4magoverallerror,givenalargedifferenceofthe R
craft albedo (α). Both these quantities directly relates
z band effective wavelength (λ =9040 ˚A).
o to the reflectance spectrum of the probe along the entire
A plot of all the entries of Table 1 is shownin Fig. 11,
wavelengthrange,afeaturethatcanonlybepoorlyknown
versusobservingtime. Withinthephotometricaccuracyof
aprioriandneedstobeconvenientlyassessed“onthefly”.
our observations,along the 2014sessionGaia displayeda
For this task,asa partofthe characterizationchecksdur-
constant apparent luminosity with an average magnitude
ing the early phases of the Gaia mission (Altmann et al.,
hRci = 20.8±0.2 and hVi = 22.0±0.2. These results are 2014), narrow-band multicolor photometry was acquired
basically confirmed also by the 2015 observations, taken
in early 2014 with the Grond imager at the ESO 2.2m
atthesametopocentricdistanceandphaseangle,10 which
telescope in La Silla. This allowed us to reconstruct the
indicate hRci = 20.7±0.2 and hVi = 21.7±0.2. Overall, spacecraft reflectance spectrum along the 4500-13000 ˚A
the observations point to a color of (V −Rc) = 1.1±0.2, wavelengthrange andtherefromderive the spacecraftcol-
much redder than the solar value of (V − Rc)⊙ = 0.4, ors during the “face-on” experiments and later along the
as estimated from Pecaut & Mamajek (2013). By itself,
standard operation phase.
this difference suggests that Gaia seems to drastically re-
process the incident solar flux.
11This is the angle of the incident solar flux with respect to the
normalofthesatelliteplanarsurface.
10The phase angle is the angular distance between Sun (S) and 12We adopt s = 886 cm as Gaia’s relevant size to be adopted
d
Earth (E), as seen from the probe (P), that is φ = SPE. For our in eq. (2). See: http://www.esa.int/Our Activities/Space Science/
2014and2015observations φ 7o Gaia/Gaia factsheetformoredetails.
≃
8
Figure 13: The observed Gaia (V Rc) color is contrasted in the
−
Figure 12: The Gaia color properties, as derived from the Grond plottoconstrainthespacecraftbolometriccorrection. AsforFig.12,
reflectancecurvesofAltmannetal.(2014),areassessedinthe(B the spacecraft location is compared with the stellar locus for dwarf
V)vs.(V Rc)plane. Thespacecraftlocationiscomparedwithth−e and giant stars of different spectral type (as labelled on the plot),
stellar loc−us for dwarf and giant stars of different spectral type (as according to Pecaut & Mamajek (2013) and and Houdashelt et al.
labelled on the plot), according to Pecaut & Mamajek (2013) and (2000), respectively, and with the relevant points for the Sun and
and Houdashelt et al. (2000), respectively. The shaded triangular Planckprobe.
region on the plotedges the allowed color range of Gaia(including
the experimented “face-on” orientation, as discussed in the text),
accounting for thereported variabilityof the spacecraft reflectance. tions, eq. (4) takes the final form as
As a reference, the Sun is also marked on the plot, together with
thederivedcolorsofthePlanckprobe,accordingtoAltmannetal. 20.8±0.2 0.4±0.3 =16.9+1.1±0.2 2.5 logα, (15)
(2014). − −
where the apparent R magnitude in the l.h. term of the
c
equation has been corrected13 to ω = 0o, according to
As far as the (B V)and the (V R ) colorsarecon-
− − c eq. (14). This relationship eventually leads to a nominal
cerned,thespacecraftlocationiscomparedinFig.12with
value of the Gaia albedo α=0.11±0.05.
the Sun and the locus for dwarf and giant stars of differ-
Strictly speaking, however,this estimate is most likely
ent spectral type. The empirical compilation of Pecaut
to be takenas a lower limit for the true Bondalbedo, due
& Mamajek (2013) for MK-class V stars is adopted, for
to sun-shield reflectance anisotropy,as we were discussing
this comparison, together with the theoretical models of
before. In fact, by applying our arguments to the real
giant stars from Houdashelt et al. (2000). As a further
“face-on” spacecraft configuration, one is led to conclude
iPnltearnesctkin,gasmdaetrcihv,edinfrtohmefitghueroerwigeinaallsoreafldedcteadntcheespcoelcotrrsumof that Gaia could not be brighter than Rcpeak ≥ 16.9±0.3+
of Altmann et al. (2014). Contrary to Planck, which 1.1±0.2 =18±0.4,asoriginallyestimatedbyAltmannetal.
(2011), but evidently at odds with the observed evidence.
resulted to be a quite effective proxy of the Sun, Gaia
roughly behaves like a red dwarf of very late M spectral
type, with an even more depleted B luminosity (i.e. “red- 4. Probing interplanetary distances
der”B V color). Inaddition,onehastoreportfromthe
− Whencoupledwithprevioustheoreticalarguments,the
Grond data, an important (and yet partly unexplained)
Gaia observationsprovideus with a useful tool to consis-
spectralvariabilityofthe probe,withits opticalcolorsac-
tently size up the requiredtelescope performance in order
tually cornered within the wide region marked in Fig. 12.
to detect deep-space probes at even larger distances from
Onthe basis ofthe reference (V R ) color,inFig. 13
c
− Earth, for instance when heading toward other planets of
wetrytoenvisageasuitablebolometriccorrectionbycom-
thesolarsystem. Ifwesetaminimum(S/N)thresholdfor
paring, again, with the Pecaut & Mamajek (2013) and
target detection, then eq. (6) (with “on-target” telescope
Houdashelt et al. (2000) stellar calibrations. Differently
tracking,thatisfor∆µ =0)canbesolvedtoobtainthe
from Planck, Gaia’s color properties point to a much sat
faintest possible magnitude we couldreachand, by means
larger correction, that we can tentatively place around a
value of BC′ 1.1 0.2 mag, although with a large
R ≃ − ±
uncertainty, further magnified by the spacecraft spectral
13Theerrorbarontheeq.(14)magnitudecorrectionaccounts for
variability. the 15o maximumphaseangleexcursionofthespacecraftalongits
±
With the relevant figures, under Lambertian assump- librationorbitaroundL2,asseenfromEarth.
9
Figure14: Visibilitymapsofdeep-spaceprobes,asopticallytrackedfromEarthwith1hrCCDexposurebydifferenttelescopes. Skyconditions
assumeaseeingFwhm=1arcsecandµR =20.5magarcsec−2. TheGaiaeffective area =105 cm2 (that isbyassumingα=0.11as a
sky A
conservative estimate of the true Bond albedo) has been taken as areference. The marked “horizons” in all panels assumeto observe with
a2m(yellow region),8m(red),and40m (pink)telescopes, at(S/N)>5detection threshold. ThesmallorangedotmarksEarth’sinfluence
sphere(edgingtheLagrangianL1andL2points),throughout. Leftpanelreferstothecaseofasphericalspacecraft,whilerightpanelassumes
aprevailingplanarstructureofthetarget. Mercury,Venus,Earth,andMarsorbitsaresketched, asaguideline.
of eq. (4) and (2), its corresponding maximum geocentric Even 2m-class telescopes could however usefully track
distance. deep-spaceprobes,forinstanceduringtheirflybyapproach
AsprobeneedstobeunderSun’silluminationtobede- toEarthforgravityslingtowardexternalplanets. Success-
tectedfromEarth,thisalsosetsaconstrainttotheallowed ful observations of the Rosetta spacecraft (Glassmeier et
rangeofthephaseangle,dependingonspacecraft(geocen- al., 2007), along its 2005 and 2007 gravity-assist maneu-
tric)distanceandphysicalstructure. Ifplanarcomponents vers (e.g. Bedient & Hutsebaut, 2005; Dillon, 2005; Hill
(i.e. large solar panels etc., either facing Earth or Sun) et al., 2005; James, 2005; Juels & Holvorcem, 2005; Kus-
are the prevailing features of the probe, then the simple nirak, 2005a,b; Manca & Cavagna, 2005; Stevens, 2005;
geometrical arguments outlined in Sec. 3.2 for the case of Ticha & Tichy, 2005; Birtwhistle, 2007a,b,c; Bittesini et
Gaiaindicatethatitsapparentluminosityunderdifferent al., 2007a,b,c; Donato et al., 2007; Hill et al., 2007a,b;
view angles scales as cos(φ). This evidently implies that Kowalski et al., 2007)14 are a relevant example of such
φ 90o fortheprobetobevisible. Muchlargervaluesof a kind of applications. The current Hayabusa 2 mission
| |≤
φmay,however,beallowedtoroughlysphericalshapes,as (Kuninaka et al., 2013) could soon provide another inter-
a cardioid-shaped luminosity law, as [1+cos(φ)]/2, (e.g. sting case to benchmark the method.
Meeus, 1998) holds in this case. According to Fig. 14, optical tracking is of lesser point
The visibility map for the illustrative case of either a for any space mission toward inner planets (i.e. Venus
spheric or straight a planar satellite geometry is sketched and Mercury). This is because the orbit geometry leads
inthetwopanelsofFig.14,respectively. Calculationshave probesusuallytoappearfromEarthatlargephaseangles,
been carried out for the Gaia’s effective area = αs2 thus poorly reflecting Sun’s light in our direction. An en-
A ≃
105 cm2 (which assumes a conservative albedo value α = hanced spacecraft effective area– either in terms of better
0.11) supposing to detect the spacecraft at a (S/N) ratio reflectancetoimprovethe albedoordirectlyby increasing
better than 5. With the 8m-class (or bigger) telescopes the physical size of the target, as likely the case of the
currently in use, one sees thatan importantregioncanbe forthcoming(manned)missionstoMars–couldbe the ob-
explored, well beyond the Earth’s influence sphere (that vious issue in this case, as we show in the illustrative case
embodies the Lagrangian L1 and L2 points) reaching, for of Fig. 15. By the way, a closer look to Fig. 14 also shows
instance, distant spacecraft along their initial Hohmann
transfer to Mars, up to geocentric distances of some 30
million km (roughly 0.2 AU). 14DASO Circulars are made available in electronic form at the
URL:http://www.minorplanetcenter.net/iau/DASO/DASO.html.
10