Download 1 Ay 122a Fall 2015/16 – HOMEWORK #3

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Ay 122a Fall 2015/16 – HOMEWORK #3
Due Tuesday, December 1, 2015, 5:00pm
1. MOSFIRE Exposure Time Calculator: Direct Imaging
At this link you can find all of the pieces of information you will need to design an exposure
time calculator (ETC) for an “as-built” instrument called MOSFIRE, which works in the near-IR
(0.95-2.5 µm), for both imaging and spectroscopy:
Included among the files are the typical atmospheric transmission at the relevant wavelengths
on Mauna Kea, the filter bandpasses for each of 4 imaging filters (Y,J,H,Ks centered at 1.05, 1.25,
1.65, and 2.15 µm respectively plus a broader K filter used for spectroscopy– see “allfilters.pdf”
for a plot), the internal transmission of all of the glass in the optics, the total transmission of all
optical anti-reflection coatings, and the near-IR sky background in units of photons m−2 µm−1 for
a typical night on the summit of Mauna Kea. MOSFIRE is mounted at the Cassegrain focus of
the Keck 1 10m (diameter) telescope; you may assume that the reflectivity of the telescope mirrors
(primary and secondary) are each 95%, and that the detector quantum efficiency (the fraction of
photons incident on the detector that are recorded) is 90% at all wavelengths.
Using the software package of your choice (e.g., MatLab, IDL, python, etc.), design an application that calculates the total system efficiency of the telescope+instrument as a function of
wavelength (interpolation of the tables provided is fine for this purpose). Your program may assume we are interested in point source sensitivity for imaging, and for the time being that the
detector noise is negligible. The program should take as input the desired apparent magnitude (on
the AB system is fine) of a target source and the target S/N for a measurement, and calculate
the total integration (exposure) time needed to reach that apparent magnitude, as well as handle
a case where the exposure time is given and the resulting S/N is desired. For all calculations you
may assume that the image quality is FWHM= 0.5 arcsec; it is OK to assume that the effective
“footprint” of the image on the detector is defined by the region within the aperture defined by
r = 0.5 arcsec (i.e., an aperture diameter of 1.0 arcsec).
The plate scale at the MOSFIRE detector is 100µm per arcsec, and the physical pixel size is
18 µm.
Use your ETC to estimate the total integration time needed for the following:
a) Detection with S/N = 100 of a star with Ks (AB) = 23.0 (i.e., in the Ks filter). What is the
estimated sky background in mag arcsec−2 ?
b) The S/N that would be achieved on an AOV star (i.e., like Vega) of apparent magnitude
Ks = 20 (on the Vega system; this is Ks = 21.85 on the AB system) in a 60-second integration
in the Y, J, and H bands. Use your program to estimate the sky background (AB mags arcsec−2 )
in each band, and compare with the typical numbers listed in the class notes. If the “full well” of
the detector pixels is ' 100, 000 electrons, what is the maximum exposure time you can use before
the star becomes “saturated”? How long could you integrate if all of your objects are much fainter
than the background (and therefore you would only need to worry about keeping the background
from going non-linear)?
c) The difference in integration time required to detect the star using the K filter rather than
Ks – note that the Ks filter was designed to cut off at shorter wavelengths than K, to avoid some
of the thermal emission that is increasing very rapidly with increasing wavelength beyond λ = 2.3
µm.
Hints: The magnitudes here are all “above the atmosphere”, and so you will need to convert
mags to flux density units, then apply the atmospheric attenuation averaged over the relevant band-
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pass, to calculate the rate of source photons incident at the telescope. Apply the efficiency numbers
to both the source photon rate and the background photon rate to calculate detected photon rates
from each.
2. MOSFIRE Exposure Time Calculator: Spectroscopy
Now, consider the differences when using the instrument for spectroscopy (in the same bands).
First, you will need the curves for the diffraction efficiency of the reflection grating in each band to
adjust the total instrumental throughput (also included among the files at the link above). While
the background against which you are detecting your sources is the same, it is now being dispersed
(along with the sources, of course) so that the count rates per pixel are reduced by a large factor.
You may assume that the spectral resolving power achieved in each band is R = 3600, sampled
with ∼2.8 pixels per resolution element for a slit with angular width 0.7 arcsec on the sky1 You may
also assume that the dispersion per pixel is constant across each band, to a first approximation.
Add to your ETC in problem 1 the capability to estimate S/N for spectroscopic observations. It
is conventional to esimate “S/N per resolution element”, which in this case would be a “footprint”
at the detector of 2.8 pix (FWHM of a spectral resolution element) by 0.7/0.18 ' 3.9 pixels (the
space occupied by the profile of the object along the slit (spatial direction). The total count rate
of the background per spectal resolution element is then equivalent to the background expected for
an aperture of ∼ 0.7 arcsec by 0.7 arcsec, but now the count rate is reduced by lower instrumental
throughput (which is wavelength dependent) and by the ratio between the bandwidth of a filter
(in the imaging case) and that of a spectral resolution element (e.g., λ/∆λ = 3600, i.e., ∆λ '
21500/3600 ' 6 Å per resolution element in the middle of the Ks band. )
Whereas for broadband imaging it was safe to neglect detector noise, it may become a significant
factor for spectroscopy. For this calculation, include a placeholder for the effective read noise per
pixel per readout and assume that you must read out the detector every 180 seconds, independent
of the total integration time needed to reach the desired S/N. This means that each 180s integration
must “pay” the read noise penalty.
a) Use your ETC to estimate the total integration time needed to reach mAB = 23 at S/N= 10
per resolution element in each of the 4 near-IR bands. Note that this will be strongly wavelength
dependent due to OH emission from the sky– we are interested in numbers typical for regions free
of strong OH lines in each band.
b) At what read noise do the data become comfortably “background-limited”, where the noise
from photon counting statistics from the background signal is large compared to that from read
noise in the darkest regions of the background spectrum?
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Because of the anamorphic magnification of the grating, the spatial pixel size at the detector remains 0.18 arcsec
per 18µm pixels in the spatial direction (i.e., along the slit) but in the dispersion direction each pixel subtends ∼0.243
arcsec at the slit.