Implications of Atmospheric Differential Refraction for Adaptive Optics Observations

13 Pages·0.93 MB·English

by  Henry G. Roe

Implications of Atmospheric Differential Refraction for Adaptive
Optics Observations1
Henry G. Roe,2
PASP, in press.
ABSTRACT
Many adaptive optics systems operate by measuring the distortion of the wavefront in one
wavelength range and performing the scientific observations in a second, different wavelength
range. Onecommontechniqueistomeasurewavefrontdistortionsatwavelengths<∼1µmwhile
operatingthescienceinstrumentatwavelengths>∼1µm. Theindexofrefractionofairdecreases
sharply from shorter visible wavelengths to near-infrared wavelengths. Therefore, because the
2 adaptive optics system is measuring the wavefront distortion in one wavelength range and the
0 scienceobservationsareperformedatadifferentwavelengthrange,residualimagemotionoccurs
0 and the maximum exposure time before smearing of the image can be significantly limited. We
2 demonstratetheimportanceofatmosphericdifferentialrefraction,presentcalculationstopredict
n theeffectofatmosphericdifferentialrefraction,andfinallydiscusstheimplicationsofatmospheric
a differential refraction for several current and proposed observatories.
J
Subject headings: atmospheric effects — instrumentation: adaptiveoptics
6
1
1. Introduction as the beamsplitter in order to send visible light

1 (<∼ 1µm) to the wavefront sensor and infrared
v Adaptive Optics (AO) has been used for astro- light(>∼1µm)to the science instrument. Exam-
3 nomical observations for more than a decade and ples oftelescopes with AO systems that canoper-
7 numerous medium-to-large telescopes around the
ateinthiswayinclude the3-mShanetelescopeat
2
world are now equipped with AO systems. If an
Lick Observatory (Gavel et al. 2000), the CFHT
1
AO system’s wavefront sensor operates at differ-
0 3.6-m telescope (Rigaut et al. 1998), the Gem-
2 ent wavelengths than are being observed by the ini North 8-m telescope (Graves et al. 2000), and
0 science instrument, and no correction is made for
/ atmospheric differential refraction, the target ob-
h
p jectwillappeartodriftwithrespecttothescience
- instrument. An example of the problem this phe-
o
nomenon can introduce is shown in Fig. 1 which
r
t is the difference of two images of Saturn’s moon
s
a Titan taken just 2.5 minutes apart while contin-
: uously tracking and correcting on Titan with the
v
i KeckIItelescope’sAOsystem. Themaximumex-
X
posuretimepossiblewithoutdegradingthespatial
r resolution of the data is significantly restricted if
a
no correction for atmospheric differential refrac-
tion is made.
Adaptive optics systems for astronomical ob-
servingoperatebysplittingtheincominglightinto
Fig. 1.— The difference between two raw images of
twobeams,oneofwhichgoestothewavefrontsen-
Titan taken just 153 seconds apart with NIRSPEC’s
sor of the AO system and the goes to the science
SCAM detector on the Keck II telescope while con-
instrument. Inmanycasesadichroicopticisused tinuously guiding/correcting with the AO system on
Titan. Each image is the result of three 10 second
1DatapresentedhereinwereobtainedattheW.M.Keck exposures. At the start of the first sequence of expo-
Observatory, which is operated as a scientific partnership sures Titan was at an elevation of 57◦.86 and an hour
among the California Institute of Technology, the Univer- angleof2.25; 153secondslaterwhenthesecondsetof
sityofCalifornia,andtheNationalAeronauticsandSpace exposures started Titan was at an elevation of 57◦.26.
Administration. The Observatory was made possible by Titan’s declination was 16◦.73. The 1-pixel wide slit
the generous financial support of the W.M. Keck Founda- ofNIRSPEC’sspectrometerisacrossTitan’sdiskand
tion. the apparent motion of Titan due to the effect of at-
2Department of Astronomy, 601 Campbell Hall, mospheric differential refraction is obvious along the
University of California, Berkeley, CA 94720-3411. elevation vector.
([email protected])
1
both of the W.M. Keck Observatory’s 10-m tele-
scopes (Wizinowich et al. 2000). The wavefront
sensor measures the distortions to the wavefront
anda correctionis calculatedandappliedtoade-
formablemirror. Thefirstorderofdistortionthat
an AO system is called upon to correct is simply
image motion, commonly known as ‘tip/tilt’. If
significant tip/tilt residuals remain after AO cor-
rection then the resulting data will be of lower
spatial resolution, regardless of how well the AO
system corrects for the higher order terms of fo-
cus, astigmatism,coma, and so forth. Minimizing
tip/tilt residuals is critical to achieving optimum Fig. 2.— The refractive index of air as a function of
wavelength across the visible and near-infrared spec-
performance from an AO system. The effect of
trumatstandardtemperatureandpressureintheab-
atmospheric differential refraction is to introduce
sence of water vapor.
a systematic tip/tilt errorthat anAO system will
not correct for unless specifically accounted for in
theAOcontrolsoftware. Foreaseintheremainder ADRforAOobservingwithcurrentandcurrently
of this report we will refer to atmospheric differ- proposed telescopes. The code used in these cal-
ential refraction as “ADR”. culationsisavailablewiththeelectronicversionof
this paper or by request from the author. While
Problems due to ADR arise when the AO sys-
thecodeiswritteninthecommonlyusedIDLpro-
tem is correcting on visible wavelength light and
gramming language, it could easily be translated
the science instrument is observing at infrared
to otherlanguages. We encourageothers toinves-
wavelengths. Fromvisibletoinfraredwavelengths
tigate the implications of ADR for their favourite
the refractive index of air decreases sharply, as
telescope, site, target, observing strategy, etc..
shown in Fig. 2. A star’s visible pointing center
always appears at higher elevation than a star’s
2. Theoretical Calculations
infrared pointing center, except when the star is
at the zenith. That the two pointing centers do
The equations necessary for calculating the ef-
not coincide is not unto itself a problem for the
fectofADRaregivenbySchubert&Walterscheid
typical AO system, however the offset between
(2000). FromSchubert&Walterscheid(2000)the
the two pointing centers is not constant. With-
refractive index of air n at pressure p millibar,
out some consideration for the effect of ADR, a
temperature T Kelvin, partial pressure of water
properly performing AO system will hold the vis-
vapor p millibar3, and wavelength λ microns is
w
ible pointing center of the star fixed, relative to
boththewavefrontsensorandthesciencecamera.
n(λ,p,T,p )=1+ (1)
w
However as time progresses and the star moves
29498.1 255.4 pT
in elevation, the infraredpointing center will drift 64.328+ + s 10−6
(cid:20) 146−λ−2 41−λ−2(cid:21)(cid:20)p T(cid:21)
with respect to the visible pointing center, and s
thus in the infrared the star appears to drift with −43.49 1− 7.956×10−3 pw10−6,
respecttothescienceinstrument. Observersmust (cid:20) λ2 (cid:21) ps
consider this effect in determining the maximum
where p is 1013.25 millibar and T is 288.15 K.
exposure times in order to avoid ‘trailed’ images, s s
For a given refractive index n the angle between
unless some other compensation is made.
thetruezenithdistancez andtheapparentzenith
In this paper we present calculations demon- t
distance z is well approximated by
strating when and how much of a problem ADR a
can be for an AO system. Using data taken with n2−1
the AO system on the Keck II telescope we show R≡zt−za ≃206265(cid:18) 2n2 (cid:19)tanzt arcsec
that in a long exposure, or sequence of exposures,
(2)
the most significant uncorrected tip-tilt motion is
due to the effect of ADR. We further show that 3Schubert & Walterscheid (2000) contains a typo that re-
the effect of ADR is dependent on the spectral sultedinamissingfactorof10−6 forthewatervaporcor-
type, or color, of the star being used as a refer- rection. Equation2hereiscorrect.
encesource. Finally,wediscusstheimplicationsof
2
◦ Y
forzenithdistanceslessthanabout80 . Observers
East
typically work at much more modest zenith dis-
tancesofz <40◦−50◦ wherethis approximation
t
is extremely good.
g
Atmosphericdifferentialrefractionbetweentwo An
s to Zenith
wavelengths (e.g. λ of an AO wavefront sensor o
vis P
and λ of a science instrument) is then
ir
R −R = (3) X
vis ir
n2 −1 n2 −1 ParAng
206265 vis − ir tanz arcsec
(cid:18) 2n2vis 2n2ir (cid:19) t North
Note that equation 4 is written to give a posi-
Fig. 3.— Schematic clarifying definition of position
tive angular distance since the zenith distance of angle and parallactic angle. The parallactic angle
the visible pointing center, z , will be less than (ParAng) is defined as the angle measured from the
vis
that of the infrared pointing center, z . The true vector pointing north on thesky counter-clockwise to
ir
the vector pointing to the zenith. The position angle
zenithdistance(z )isrelatedtolatitudeoftheob-
t (PosAng) is defined as the angle measured from the
server (φ), declination of the target (δ), and hour
positiveY-directiononthedetectorarrayclockwiseto
angle of the target (H) by thevector pointing north on thesky.
cos(z )=sin(φ)sin(δ)+cos(φ)cos(δ)cos(H)
t
(4) suchthatinstrumentcoordinatesremainfixedrel-
ativetorightascensionanddeclination. Therefore
To relate the ADR offset of equation 4 to the
the instrument coordinate system rotates relative
(x,y)coordinatesofadetectorarrayorientedarbi-
to elevation and azimuth. The important point
trarily with respect to the sky, we introduce sev-
is that the instrument coordinate system rotates
eral angles. As shown in Fig. 3 the parallactic
about the visible pointing center. Thus, the ef-
angle (ParAng) is the angle from the sky North
fective motion of the infrared pointing center in
vector counter-clockwise to the elevation-up vec-
instrument coordinates is not along the elevation
tor. Position angle (PosAng) is the angle from
vector except when the parallactic angle is un-
the increasing Y-direction of the detector array
changing. In the images of Titan differenced in
clockwisetotheskyNorthvector. Finally,thedif-
Fig. 1 the parallactic angle was nearly unchang-
ference of PosAng minus ParAng gives the angle
ingandthereforethe motiondue toADRappears
from the increasing Y-direction clockwise to the
along the elevation vector. In later examples the
elevation up vector. In the (x,y) coordinates of
parallactic angle is changing rapidly and then the
the detector array the offset from visible pointing
apparentmotionis notalongthe elevationvector.
center to infrared pointing center is then
Equation 4 gives the instantaneous offset be-
R −R
X = ir vis sin(PosAng−ParAng) tween visible and infrared pointing centers, but it
offset PlateScale is the first derivative of R −R with respect
ir vis
(5)
R −R to time that causes the problem addressed in this
Y = ir vis cos(PosAng−ParAng), paper. In the extreme case of an AO equipped
offset
PlateScale
telescope atthe SouthPole,ADR, asdescribedin
(6)
this current report, is not a problem since targets
wherePlateScaleisarcsecondsperpixelofthear-
do not move in elevation angle.
ray. The parallactic angle (ParAng) is a function
of hour angle, declination, and latitude, Inordertofindtheinstantaneousrateofimage
motion due to ADR we define the constant
sin(H)
tan(ParAng)= cos(δ)tan(φ)−sin(δ)cos(H). β = 206265 n2ir −1 − n2vis−1 (8)
(7) PlateScale(cid:18) 2n2 2n2 (cid:19)
ir vis
In the coordinate system of elevation and az-
and take the partial derivative of X and
offset
imuth, the motion of the infrared pointing center
Y with respect to hour angle.
offset
relativeto the visible pointing center due to ADR
is always along the elevation vector. However,
most observations are made using a field rotator
3
∂
X = (9)
offset
∂H
∂z
βsin(PosAng−ParAng)sec2(z ) t
t
∂H
∂ParAng
−βtan(z )cos(PosAng−ParAng)
t
∂H
∂
Y = (10)
offset
∂H
∂z
βcos(PosAng−ParAng)sec2(z ) t
t
∂H
∂ParAng
+βtan(z )sin(PosAng−ParAng)
t
∂H
The partialderivativesofz andParAng with re-
t
spect to H are
∂
z = (11)
t
∂H
cos(δ)cos(φ)sin(H)
Fig. 4.— (a) predicted image motion and (b) maxi-
1−(cos(H)cos(δ)cos(φ)+sin(δ)sin(φ))2 mumexposuretimesforanobservationat1.65µm(H-
q band) of a target at declination +30◦ with the Keck
10-meter telescope assuming an effective wavelength
∂ of 0.65 µm for the wavefront sensor of the AO sys-
ParAng = (12t)em. Each hourangleis markedbyacircle anda line
∂H
pointing toward the zenith at that moment. The mo-
cos(H)cos(δ)tan(φ)−sin(δ)
tionduetoADRisnotnecessarilyalongtheelevation
sin2(H)+(cos(H)sin(δ)−cos(δ)tan(φ))2 vector. Note that (a) does not depend on telescope
aperture,butdoesdependonwavelengthsofobserva-
The combination of eqns. 8, 12, and 13 with tion,declinationoftarget,andlatitudeofobservatory.
The origin of the coordinate system in (a) lies at the
d π radian visiblewavelength pointingcenter,which theAOsys-
H = (13) temisholdingsteady. Maximumexposuretimein(b)
dt 1800 minute
iscalculatedfora10-meterdiametertelescopeandas-
into eqns. 10 and 11 gives the instantaneous rate sumingmaximumallowedimagemotionof0.25ofthe
FWHM of the diffraction limited point spread func-
of image motion in the x/y-plane of the detector.
tion.
The IDL routine adr rateofmotion.pro included in
the electronic version of this paper calculates this
instantaneous rate of image motion. Output from these calculations for an observa-
Finding the maximum exposure time allowed tion with the Keck telescope is shown in Fig. 4.
before the image motion exceeds some givenlimit The predicted motion shown in Fig. 4a depends
is not as straightforward as simply dividing the on site, but not diameter of telescope, while the
desired image motion limit by the instantaneous maximum exposure times shown in Fig. 4b are
rate of image motion at the start of the expo- based on the 10-meter diameter of the Keck tele-
sure. This is because the image motion is usu- scopes. A smaller telescope has a coarser diffrac-
ally curved in the x/y-plane and because the tionlimitandthereforeislessaffectedbytheissue
second partial derivatives of X and Y ofdriftduetodifferentialrefraction,howeversome
offset offset
with respect to time are not constants. There- smallertelescopesareatlowersites(e.g.theShane
fore, the most direct method of determining the 3-mofLickObservatory)wheredifferentialrefrac-
maximum exposure time is to calculate X tion is greater. A smaller telescope ata higher al-
offset
and Y with fine time sampling and simply titude, higher latitude site suffers less from ADR
offset
search for the point at which the maximum de- than a larger telescope at a lower altitude, lower
sired image motion has been exceeded. The IDL latitude site. The implications ofADR for several
routine adr maxexptime.pro, included in the elec- current and proposed telescopes are discussed in
tronic version of this paper, calculates the maxi- section 4.
mum allowed exposure time.
4
Acriticalparameterincalculating the implica-
tions of ADR is the effective wavelength of the
reference source on the wavefront sensor. The
wavefrontsensorof an AO systemis often photon
limited and therefore designed to have as broad a
wavelength bandpass as possible. In the case of
a broad wavefront sensor bandpass, the effective
wavelength of the reference source on the wave-
front sensor depends on the color of source, as we
will show in section 3 using observed data. Usu-
ally of less importance is the effective wavelength
of the target on the science instrument since the
variation of air’s refractive index with wavelength
is much less in the near-infrared than in the visi-
ble.
3. Data Reduction and Analysis
All the data presented here were taken using
the W.M. Keck Observatory’s near-infrared spec-
trographNIRSPEC behind the AO systemonthe
Keck II 10-meter telescope. NIRSPEC contains
two infrared arrays: a 1024 × 1024 InSb AL-
ADDINforspectroscopyanda256×256HgCdTe
PICNIC array as a slit-viewing camera (SCAM).
The images of Titan shown in Fig. 1 were taken
2001 January 11 (UT) during high-spectral reso-
lution long-exposure spectroscopy for a project of
Eliot Young’s. This observing run was the first
time we noticed the problems presented by atmo- Fig. 5.— Example images of HIP 110 and HIP
sphericdifferentialrefractionwhenobservingwith 13117. These images have been processed using stan-
dard near-infrared techniques of sky-subtraction and
AO.TheimagesshowninFig.1are30-secondex-
flat-fielding. Bad pixels have been replaced with the
posures taken just 2.5 minutes apart and clearly
median of their nearest 4 to 8 good neighbors.
show a movement of ∼1 pixel, or ∼1/2 the full
width at half maximum (fwhm) of the diffraction
limit. 3.1. CalibrationofPlatescale andPosition
Angle
During the nights of 2001 August 20 and 21
(UT) James Lloyd and James Graham observed
The platescale of SCAM behind AO was de-
several binary stars as part of an ongoing search
signed to well sample the core of a diffraction
for low-mass companions. Images were taken of
limited point spread function (PSF) in the near-
eachfieldnearlycontinuouslyfor30to60minutes infrared with a pixel size of approximately 0′.′017.
while neither adjusting the parameters of the AO
InordertodeterminethetrueplatescaleofSCAM
systemnoroffsettingthepointingofthetelescope.
behind AO and confirm the accuracy of the
We first discuss platescale and position angle cal-
instrument-reported position angle, four Hippar-
ibration using several Hipparcos binary star sys-
cos binary systems chosen for their small uncer-
tems. We then focus on data on two of Lloyd &
tainty in separation and small parallax were ob-
Graham’sstars(HIP110andHIP13117)toshow
served earlier on the evenings of 2001 August 20
that on multi-minute time scales atmospheric dif-
and 21 (UT) by de Pater et al.. The details of
ferentialrefractionisthedominantimageblurring
these plate scale calibrations are givenin Table 1.
effect that is uncorrectedby the AO system. Fur-
Each Hipparcos field was imaged at a number of
ther, we use these data to show that the spectral
positions aroundSCAM’s field-of-view in orderto
type or color of the star is important when con-
check for distortions. As noted in Table 1, three
sidering how to correct for ADR.
of the four fields were imaged a second time af-
ter a 90◦ rotation of SCAM’s field-of-view. All
5
Table 1
Details of plate-scale calibration.
Coadds× Assumed SCAM Hipparcos Measured Derived
Star #of Exposure P.A.of Reported Measured Binary Separation PlateScale
Name images Time Binarya P.A.b P.A.c Separation (pixels) (mas/pixel)
HIP83634 14 10×0.1s 306◦.4 92◦.0 92◦.13±0◦.23 1′.′436±0′.′007 87.03±0.51 16.50±0.013
HIP83634 17 10×0.1s 306◦.4 2◦.0 1◦.95±0◦.29 1′.′436±0′.′007 87.24±0.49 16.46±0.012
HIP89947 6 50×0.2s 340◦.1 92◦.0 92◦.33±0◦.53 1′.′642±0′.′006 98.67±0.30 16.64±0.008
HIP91362 10 50×0.2s 223◦.2 82◦.0 82◦.63±0◦.35 1′.′050±0′.′009 61.88±0.52 16.97±0.020
HIP91362 10 50×0.2s 223◦.2 -8◦.0 -7◦.21±0◦.34 1′.′050±0′.′009 62.29±0.31 16.86±0.017
HIP100847 10 10×1.0s 129◦. -8◦.0 -8◦.28±0◦.24 0′.′871±0′.′010 51.59±0.23 16.88±0.021
HIP100847 10 10×1.0s 129◦. -98◦.0 -97◦.87±0◦.55 0′.′871±0′.′010 51.93±0.36 16.77±0.022
aPositionangleofsecondarystarrelativetoprimarystarasreportedintheHipparcoscatalog(ESA 1997).
bPositionangleoftheSCAMdetectorreportedbytheinstrumenthardware. SeeFig.3fordefinition.
cPosition angle of the SCAM detectordeterminedfromthe data assumingthe position angleof thebinary pair givenin ESA
(1997)isperfect.
of these data were taken in the H-band filter and 3.2. Measurement of Differential Refrac-
wereprocessedusingstandardinfraredtechniques tion
of sky subtraction and flat-field correction using
The relevant ephemeris data and details of the
data taken on the twilight sky. To measure the
observationsofHIP110andHIP13117areshown
x,yoffsetbetweentwostarsonthesameimagewe
in Tables 2 and 3. Note that two separate se-
rebinnedthe imageby afactorof8 (64newpixels
quencesofdataweretakenonHIP110. Allofthe
for each original pixel) using sampling and then
stellar data were taken in the K-prime filter, each
calculated the autocorrelationfunction
image containing 100 coadds of 0.40 second ex-
A=FFT−1(FFT(R)Conjugate(FFT(R))), posure. Each pair of binary stars were aligned on
thechiptoavoidregionsofbadpixels. Theimages
(14)
wereprocessedusingstandardinfraredtechniques
where R is the rebinned image, FFT represents a
forward fast fourier transform, and FFT−1 repre- of bias subtraction and flat-field correction using
data taken on the twilight sky. Those pixels that
sents an inverse fast fourier transform. The posi-
tion of the secondary maxima in A gives the x,y were flagged as bad were replaced with the me-
dianoftheirnearest4-8goodneighbors. Example
offset between the two stars. Table 1 gives the
frames from both fields are shown in Fig. 5.
Hipparcos measured position angle (PA) of each
binarypair,the PAofthe detector asreportedby The goal of this exercise is to track the motion
theinstrumenthardware,andthePAofthedetec- ofastaracrosstheseveraldozenimagesthatmake
tor measured from the data assuming no motion up an exposure sequence. To do this we extract a
of the stars since the epoch of the Hipparcos ob- regionof roughly±50pixels aroundthe starfrom
servations. Further, Table 1 gives the Hipparcos each image of the sequence. In essence we then
measurement of angular separation for each pair, oversample by a factor of 4 and find the peak of
ourmeasuredseparationinpixels,andtheimplied the cross-correlationfunctionfor everypairofim-
plate scale for our detector. For the current work ages in this extracted stack. From this matrix of
weadoptaplatescaleof0′.′0167±0′.′0002perpixel, quarter-pixelresolutionoffsets betweeneverypair
although we note that most of the uncertainty in of images we calculate a least-squares fit for the
thisdeterminationofplatescaleappearstobedue relativeoffsetsoftheimages. Sinceeachfieldcon-
to inaccuracies in the ‘known’ separation of the tains two stars we can do this for each star and
binaries in the Hipparcos catalog,suggesting that obtain an internal cross-check on our method of
these binaries have moved slightly in separation tracking image motion. Figure 6 shows the im-
since the epoch of Hipparcos (1991.25). The posi- agemotionduringthesecondsequenceofHIP110
tionangledeterminationshowninTable1demon- observations as measured from each of the com-
strates that the instrument-reported position an- ponents of HIP 110, which agree well with each
gle is accurate to better than ∼0◦.5. other. This good agreement gives us confidence
that our method of detecting image motion is not
6
Table 2
Binary stars used for fitting for the effects of ADR.
Parallax Separationa Spectral
Star mV (milliarcsec) B−Va V −Ia ∆HIPmaga,b (arcsec) Typec
HIP110 8.61 20.42±1.91 0.787±0.003 0.820±0.007 0.64±0.03 1.197 GV
HIP13117 11.69 29.67±9.34 1.460±0.022 1.85±0.10 0.64±0.22 1.679 MV
aFromtheHipparcoscatalog(ESA 1997).
bMagnitudedifferencebetweenthetwostarsofthebinaryintheESA (1997)definedpassband.
cSpectraltypeandclassestimatedfromthetablesofDrilling& Landolt(2000) usingmV,parallax,B−V,and
V −I.
length of the star on SCAM. The parameter of
interest being fit for is the effective wavelength of
the reference star on the wavefront sensor. Also
allowed to vary is the (x,y) position where the
starwouldappearonSCAMattheeffectivewave-
lengthofthewavefrontsensor. This(x,y)position
is constant within an observing sequence. The re-
sultsofthesefitsareshownwiththeircorrespond-
ing observations in Fig. 7 and are summarized in
Table 3. For the purposes of this fitting we as-
sumed a typical atmospheric pressure and tem-
perature for Mauna Kea of 456 mm Hg and 2◦C
(Cohen&Cromer1988). Wealsoassumedawater
vapor partial pressure of zero, although we found
novariationinourresultsoverareasonablerange
of watervapor partialpressuresfor the summit of
Mauna Kea.
Athoroughexaminationofthe uncertaintiesin
our determination of effective wavelength on the
Fig. 6.— Image motion during second sequence of
wavefrontsensorisdifficult,primarilybecausethe
HIP 110 observations as derived independently from
residuals of the fits (dashed lines in Fig. 7) are
each component of theHIP110 binary.
clearly not randomly distributed. While most of
theresidualimagemotionisattributabletoADR,
affectedby issues suchasbadpixels, residualflat- thereappeartobeotherphenomenacausingresid-
field noise, read-noise, etc. that would differ be- ual image motion at the several pixel level.
tween the two stars.
One method for better understanding the un-
For each of the six observed sequences (2 com- certaintiesinbest-fitλ istorefitλ toaran-
eff eff
ponents of HIP 110 observed in two separate se- domly chosen fraction of the observations. For
quences and 2 components of HIP 13117) we use each observational sequence we randomly chose
the downhill simplex method ‘amoeba’ of Press, one half of the observations and refit for λ .
eff
Teukolsky, Vetterling, & Flannery (1992) as im- By repeating this procedure 1000 times for each
plemented in the IDL software package to fit for observational sequence we made the error esti-
the effect of ADR. The fixed parameters are: the mates shownin Table 3. The majorsource ofsys-
zenithdistanceofeachobservation,theparallactic tematic uncertainty in λ is the uncertainty in
eff
angle of each observation, the relative (x,y) off- platescale, which adds an additional ±0.004 µm
set of each observation, the platescale, the posi- uncertainty to the values of λ in Table 3. This
eff
tion angle of the SCAM array, the atmospheric is not a truly rigorous exercise in error analysis,
pressure and temperature, the partial pressure of howeveritdoesshowthattheuncertaintiesinλ
eff
atmospheric water vapor, and the effective wave- for HIP 110 and HIP 13117 (∼<0.025 µm) are
7
Fig. 7.— Image motion for the brighter component of each binary pair. (a) is the first sequence of HIP 110
observations, (b)is thesecond sequenceof HIP110, and (c) is HIP13117. Thetriangles mark therelative positions
of thestar from image to image. The solid line is the best-fit for themotion dueto ADR.The dashed lines connect
each observation (triangle) with its corresponding point along the best-fit solid line. As shown in Fig. 6 the image
motion derived from each binary component is nearly identical and therefore the fits to the secondary components
are not shown in this figure, but the resulting best-fit effective wavelengths are included in Table 3. Also shown on
(a),(b),and(c)aregrayarrowspointingtothezenithatthestartandendofeachsequenceofobservations. (d),(e),
and (f) show the image motion residuals of the observations in (a), (b), and (c) after subtracting themotion due to
ADR.NotetheasymmetryoftheresidualsinthecaseofHIP110in(d)and(e). Thisisduetothecloseseparationof
HIP110influencingtheAOcorrectionasdescribedinthetext. Theorientationofthebinarypairineachobservation
sequenceis indicated by an arrow.
8
muchlessthanthedifferencebetweenλ forHIP implemented this type of correction. However,
eff
110andHIP13117(∼0.11µm). As expected, we in some cases for practical reasons implementing
find that HIP 13117, an MV star, has a redder this correctionmaybe difficult, or λ ofthe ref-
eff
λ than HIP 110, which is a GV star. Our de- erence source on the wave-front sensor may not
eff
tection of the color difference between these two be well known. Knowing λ of the science tar-
eff
starsisrealandshowsthattofullycorrectforthe get on the science instrument is less critical since
effectofADRthecoloroftheAOcalibratorsource the index of refraction of air varies much more
must be taken into account. gradually at near-infrared wavelengths than visi-
Figure 7 shows that the largestsource of resid- ble wavelengths. In the following sections we in-
ual image motion is accounted for by ADR, how- vestigate both of these cases for several existing
ever residuals of up to 1 to 2 pixels (0′.′017 − andproposedtelescopes. Analternativetechnique
−0′.′034) remain, which are significant given that forADRcorrectionistoinsertanatmosphericdis-
the diffraction limit resolution is 0′.′046 at 2.15 persion corrector between the telescope and AO
µm. The source of the largest remaining resid- system. Thisapproachwouldbeeffectiveinmany
uals is most likely due to the binary nature of situations, although it does invariably lead to at
these stars. The AO system uses light from the leastsomelossofthroughputandincreaseinther-
brighterofthetwostarstomeasurethewavefront mal background. Any increase in thermal back-
distortion. In these observations a 1” field stop is ground is detrimental to observations at longer
employed in front of the wavefront sensor in or- near-infraredwavelengths. Wealsodiscusstheim-
der to block the light of the dimmer star from portance of considering the effect of ADR when
reaching the sensor. With binaries of closer sep- attempting slit-spectroscopy with an AO system.
aration, moments of worse seeing can cause light
4.1. Maximum Exposure Times
from the dimmer star to leak in through the field
stop. During these moments the AO correction Inthefollowingweadoptthemaximumaccept-
is worse and the residual tip-tilt error in the di- abledriftinasingleexposuretobe0.25ofthefull
rection of the position angle of the binary will be width half maximum (FWHM) of the diffraction
larger. Insupportofthisargumentthatthelarger limitedcoreofthescienceinstrumentpointspread
tip-tilt residualswe observearedue to this binary function (PSF), i.e. 1.04 λ/D radian, where λ is
effect, we show in Fig. 7d-f the residuals of tip- the wavelength of the science observation and D
tilt image motion after subtracting the effect of is the diameter of the telescope aperture. These
ADR.ForbothobservationsofHIP110theseresid- calculationswereperformedforaλ ofthewave-
eff
uals are extremely asymmetrical in the x-y plane, front sensor of 0.65 µm and a science wavelength
with the larger residuals parallel to the position of 1.65 µm, which is roughly the middle of H-
angle of HIP110. In these data the position an- band. Foralloftheseexamplesweusethelatitude
gle of HIP110 is approximately orthogonal to the of Mauna Kea observatory in Hawaii (19◦.826 N),
direction of motion due to ADR. The residuals in but clearly if the sign of the target declination is
the direction of motion due to ADR are approx- reversed then these figures would be correct for
imately gaussian distributed with σ = 0.3 pixels. an observatory at 19◦.826 S latitude. In general,
The stars of HIP13117are more widely separated an observer closer to the equator will be affected
andthus the AO correctionis less affected by this morebyADRthananobserveratoneofthepoles,
binary star issue. This is apparent in that the who will not have to contend with the problems
residuals ofimage motion measuredfor HIP13117 discussed here. All of these calculations assumed
are roughly symmetrically distributed in the x-y zero water vapor. In generalreasonablevalues for
plane with a standard deviation of 0.3 pixels. the partial pressureof water have little impact on
these calculations,but the implications ofvarying
4. Implications for Observing watervaporshouldbeconsideredifextremelyhigh
precision is sought. See Section 2 for further de-
Ideally one would know λ of the reference
eff scriptionofthesecalculations. We aremakingthe
source on the WFS and λ of the science target
eff IDL code used in this work publicly available for
on the science instrument. The AO system could
anyone to use to examine the implications of at-
then continuously correct for the calculated effect
mospheric differential refraction for their favorite
of differential refraction by inserting tip-tilt mo-
telescopeandtarget. Afinalnoteisthatnearlyall
tions inorderto keepthe sciencetargetsteadyon
modern telescopes are built on altitude-azimuth
the science instrument. Some AO systems have
mounts which often have a zenith
9
Table 3
Details of observations and results of fitting for atmospheric differential refraction.
Star UTTime Elevation #of Best-fitλeff
Name Range Range images (µm)a
HIP110a(#1b) 12:28-13:06 70◦.27-68◦.68 40 0.611±0.016
HIP110b(#1b) 12:28-13:06 70◦.27-68◦.68 40 0.608±0.014
HIP110a(#2b) 13:16-13:55 67◦.71-62◦.84 41 0.623±0.008
HIP110b(#2b) 13:16-13:55 67◦.71-62◦.84 41 0.623±0.007
HIP13117a 14:07-14:46 64◦.15-68◦.98 31 0.735±0.018
HIP13117b 14:07-14:46 64◦.15-68◦.98 31 0.730±0.019
aSeetextfordetailsoffitting. Thisistheeffectivewavelengthofthewave-
frontsensoroftheAObenchwhenobservingthisstar.
bThesenumbersrefertothefirstorsecondsequenceofobservationsonHIP
110.
Fig. 8.— Maximum exposure time due to the effect of atmospheric differential refraction. All calculations assume
λ of the wavefront sensor is 0.65 µm, science observations are at 1.65 µm, observatory latitude is 19◦.826 N,
eff
maximum allowed image motion is one quarterthe FWHM of the diffraction limited PSF at the science wavelength
of1.65µm,andthatthereisnowatervapor. (a)isforthecaseofa4-metertelescopeatsealevel. (b)isforthecase
of a 4-meter telescope on Mauna Kea, similar to NASA’sIRTF3-meter telescope or theCFHT 3.6-meter telescope.
(c) is for the case of a 10-meter telescope on Mauna Kea, such as the Keck 10-meter telescopes. (d) is for the case
of a 30-meter telescope on the summit of Mauna Kea. Note that maximum exposure time is not symmetric about
transit.
10