Astronomy 4100 / 6100
Homework 3, Due Mon Feb 20 at 1:30pm
1. You have measured a flux of 5x10-15 erg s-1 cm-2 Å-1 at 5483 Å from a faint star.
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a.) What is the V magnitude of the star? (You may find Table 1 handy here:
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http://www.stsci.edu/hst/observatory/documents/isrs/scs8.rev.pdf ) (10 points)
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b.) What is the ST magnitude of the star in V?
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c.) What is the AB magnitude of the star in V?
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Note that a, b, and c are all the same! That is by design.
d.) If you also measure a flux of 5x10-15 erg s-1 cm-2 Å-1 at 8637A, what is its
I-band magnitude? ST mag in I? AB mag in I?
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Using the same logic as above: mI = 13.18 mST (I) = 14.65 mAB (I) = 13.66
Note that the ST mag is the same as in part b (by definition). But the
others are significantly different.
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e.) If the star is located at a distance of 1.05 kpc, what are its absolute V and I
magnitudes? (Use the Vega scale for this, not ST or AB).
2. Determine how to propagate your errors for the following formulae. Give your
answers in their simplest form. (10 points)
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Follow the rules in the lecture notes to write down. Simplest form is
important here, because it cancels out a lot of terms (also in part b).
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break into pieces: !
, with measured quantities a, b, and c
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Astronomy 4100 / 6100
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3. You wish to observe a 20th magnitude A0 star in two different filters: the optical B
filter (λc = 0.44 μm; dλ/λ = 0.22) and the near-infrared K filter (λc = 2.22 μm; dλ/λ=
0.23). You have been assigned time on a 3.5-m telescope and have optical and nearinfrared detectors available to make your measurements. (20 points)
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a.) What is the flux in photons at the top of the atmosphere in B and K for the
20th magnitude star? (Note that Vega is also an A0 star).
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Vega has zero color by definition, which means m(B)=m(K). Use the table
from Lecture 8 (“Errors & Uncertainties”) and refer to the in-class
worksheet that we did (answers in the lecture slides)
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f luxB = 4.26 × 10−5 Jy = 62.27 photons s−1 m−2
f luxK = 6.70 × 10−6 Jy = 51.66 photons s−1 m−2
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(NOTE: I did not round off any of these numbers until the very end)
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b.) What is the total number of photons per second from the A0 star incident on
your detectors in B and K? Assume that you have average atmospheric
conditions, with transmissions of 60% in B and 80% in K. The telescope has a
Cassegrain design. Assume a mirror reflectivity of 92%, and an average
transmission of 80% for the filters. Assume there are no additional optics in the
system.
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Remember that a Cassegrain design has two mirrors, and each will
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suffer from imperfect reflectivity.
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c.) Your observing time is scheduled during the new moon phase, so you
estimate the background sky has a surface brightness of 22 mag arcsec-2 at B
and 12.5 mag arcsec-2 at K. Determine the total number of photons per second
from the sky incident on your detectors in B and K within a 3 arcsec diameter
circular aperture.
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Use the same logic as parts a and b for most of this problem, but keep in
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mind that photons reflecting off the atmosphere into your telescope donʼt
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have to go through the atmosphere from outer space, so you would ignore
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B:
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K:
Astronomy 4100 / 6100
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the atmospheric transmission term here. To cancel out the arcsec-2 term,
you need to account for the area of the aperture on the detector. But you
canʼt do this right away -- youʼll get a huge magnitude (150 or so) that
makes no physical sense. Wait until you have units of fluxes before
accounting for the area on the detector.
skyf luxB = 6.75 × 10−6 Jy arcsec−2 = 9.87 photons s−1 m−2 arcsec−2
skyf luxK = 6.70 × 10−3 Jy arcsec−2 = 51657 photons s−1 m−2 arcsec−2
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B sky: !
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d.) The optical detector has quantum efficiency of 85% and readout noise of
10e-/pix. How long will your exposure time in B need to be to achieve a
signal-!to-noise ratio of 100? Assume that each pixel subtends an angle of
0.1arcsec2 .
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Your circular aperture has an area of
arcsec2 , so the number
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of pixels in your circular aperture is
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Use the fluxes from above and the whole S/N equation and solve for t.
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Remember that the sky flux you calculated above is actually the sky flux
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integrated over the aperture on the detector, so it is already Rsky * npix.
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No dark current was mentioned in the problem, so you can assume D=0.
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e.) What signal-to-noise ratio will you achieve in K with the exposure time you
found in (d) if the near-infrared detector has a quantum efficiency of 75% and a
readout noise of 20e-/pix? What exposure time would you need to achieve
S/N=100 at K? Again assume that each pixel subtends an angle of 0.1arcsec2 .
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f.) If the saturation level of the optical detector is 30,000 counts and the detector
has gain = 2, what minimum exposure time in B would cause you to saturate the
detector? Assume that the detector responds linearly up to the saturation level.
For simplicity, assume that the flux from the star is spread evenly over the !2
arcsec diameter circular aperture. (Donʼt forget about the contribution from the
sky!)
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K sky:
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t = 150 s
in 150 s you will get S/N = 1.85 in K
for S/N=100 in K, you will need t = 438,000s (5.1 days)
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Saturation levels are always quoted per pixel, so you need to determine
the other flux contributions per pixel for the equation below, paying
attention to how many pixels are in a 2 arcsec diameter circular aperture.
Astronomy 4100 / 6100
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Remember that you must read out the detector to know how many counts
you have, so you need to include read noise contribution as well.!
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g.) What magnitude object would you need to observe in B with this same
equipment and under these same conditions to saturate the pixels in your 2
arcsec diameter circular aperture with an exposure time of 10s? Of 1s?
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Use the equation above with t = 10s and the same skyflux and read noise,
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but solve for the object flux.
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t = 5000 s to saturation (about 1.4 hours)
Then, use the magnitude equation to compare the object flux with that
determined for the original !object (with m=20). Be sure you are
comparing the same flux units (either photons per second per pixel, or
photoelectrons per second per pixel, or whatever you like -- just be sure
theyʼre the same). Solve for the unknown magnitude. Otherwise you can
reverse all the steps (mirror reflectivities, atmospheric !transmission, etc)
and go all the way back to the very beginning to solve for magnitude, but
this takes longer.
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Saturate in t = 10s for m = 12.6 mag
Saturate in t = 1s for m = 10.1 mag
In reality, these numbers are a little surprising given the small size of the
pixels that we assumed. For reference, the SDSS 2.5m telescope images
(with effective exposure times of 54s) saturate at g=14mag. However, we
have assumed really good !transmission properties for all the optics in the
system, while also assuming very few optics. Usually, you will have others
in addition to what we considered here (such as a “window” between the
detector and the telescope optics), and each additional component in the
optical path will further reduce the number of photons that reach the
detector.
ASTR 4100 students -- stop here!
ASTR 6100 students -- continue to 4
4. Globular clusters are dense, gravitationally-bound clusters of stars that were long
thought to be made up of single populations of stars that all formed at approximately the
same time. However, recent research has shown that at least some globular clusters
show evidence for multiple distinct populations of stars (e.g., Bedin et al. 2004, ApJ,
605, L125).
Suppose that you would like to improve on the experiment of Bedin et al. by observing
Omega Centauri with the Hubble Space Telescope and the new Wide Field Camera 3.
Use the WFC3 information and exposure time calculator (found online at www.stsci.edu/
hst) to design your experiment. Turn in a 1-page discussion of your experiment design,
Astronomy 4100 / 6100
including instrument setup, exposure times, expected signal-to-noise ratios, and other
pertinent details. Assume that your ability to fully justify your setup and how that setup
will allow you to make the required measurements will determine whether you are
allowed to carry out the experiment or not. (10 points)
Points were awarded based on method. The first thing to do is to read the Bedin
article and notice the color-magnitude diagrams that show the different mainsequences that are the evidence for distinct populations.
Determine the
apparent magnitudes of the stars in those individual main-sequences in the
particular filters, and use them in the exposure time calculator to plan your
exposures (not the integrated magnitude of all the stars in the globular cluster -you want to focus on the individual stars here). Keep in mind that Omega Cen is
very extended compared to the field of view of WFC3, you may want more than
one pointing to compare stellar populations in different areas of the cluster.
Some discussion of how reasonable you thought your time request turned out to
be was necessary as well.