142 Astronomy Workshop #8: Problems
1. The epicyclic frequency. Consider a star in a nearly circular orbit in a galaxy. We can think of the motion of the star as a radial oscillation about a circular orbit of radius R0 . The frequency of radial oscillations is called the epicylic frequency. In this problem we are going to calculate this frequency as a function of radius for different gravitational potentials. In terms of the radial and tangential velocity components and the gravitational potential, the energy divided by the mass of the star is
2 v
E=
v2 + R + ( R ) 2 2
a.
What is the angular rotation rate ( R ) of a star in a circular orbit? What is the angular momentum, L, of a star in a circular orbit? If the gravitational potential is axisymmetric, then the angular momentum is conserved. Show that you can write the energy of a star in a non-circular orbit as
b.
c.
E=
v2 + R + ( R ) 2 2 R2 L2
d. Let R = R0 + r . Assume that r R0 . Expand the energy to second order in r. Show that the energy can be written as
2 3L2 d 2 ( R0 ) vr 4 + + R dr 2 2 0
E = constant + = constant +
r2 2 2
2r2
v2 + r 2
for 2 = 3( R0 )2 +
d 2 ( R0 ) dr 2
is the epicyclic frequency.
e. For the Keplerian potential ( R ) =
GM GM show that ( R ) = , R R3
rotation curve
( R ) = ( R )
f.
For
a
galaxy
with
a
flat
2 ( R ) = vc ln( R )
show
that
( R ) = vc /R ,
( R ) = 2 ( R)
2.
Similarity between electromagnetic and gravitational forces. We have two central, 1 r 2 force laws in physics, namely gravitation and electrostatics:
Gravity: Fab =
Gma mb
2 rab
rab
; Electrostatic: Fab =
Kq a qb
2 rab
rab
;
recall that masses are always positive, but charges can have either algebraic sign. In the electrostatic force law, K = 1 in cgs units and K = 1 4 0 in MKS units. Corresponding to these forces are fields: the electric field, E = F q , and the gravitational field (or acceleration), g = F m . The fluxes of these fields are related to sources of the fields (masses or charges) by Gausss Law:
Gravity:
g dA = 4 GMenclosed
; Electrostatic:
E dA = 4 KQenclosed
.
Consider two situations: a point mass m lying a distance z from an infinite x-y plane with mass per unit area , and a point charge q lying a distance z from an infinite x-y plane with charge per unit area . Show, by either means you want, that the forces exerted by the infinite planes on the point sources are
Gravity: F = 2 G mz ; Electrostatic: F = 2 K qz .
Techniques you have learned in electrostatics can often be applied to gravitational systems. 3. Web assignment. On using an exposure time calculator for a world class telescope. How distant can a Milky Way sized galaxy be and still be detected by Advanced Cameras for Surveys (ACS) on board HST?
Image of Q1634+267A and B. Horizontal bar shown is 2" in length, which corresponds to roughly 16zh65 kpc, for an Omega = 0.1 open cosmology. This image was taken with the NICMOS camera on board HST and is at 1.6 microns (H band). See Peng et al. 1999 for more information.
1
2
Motivation: Q1634+26A,B are two Quasars only separated by 3.8" on the sky. Mysteriously, both quasars have the same redshift and very similar spectra. Either binary Quasars exist and somehow form, or there is a foreground object gravitationally lensing the background quasar and causing two images of the Quasar. Perhaps we can search and find the lensing object (presumably galaxy). If we detect nothing, then either there are dark massive galaxies out there or there are binary quasars. Pretend that you are considering writing a proposal to the Hubble Space Telescope to address this science question.
The Quasar is at a redshift of 2 so the lensing galaxy could be really far away (like at a redshift of 1). Our proposed experiment is to get a really deep HST image to see if we can find the hypothetical lensing galaxy.
How faint a galaxy can we detect in one orbit with ACS (advanced camera for surveys) on board HST (Hubble Space Telescope)? To do this assignment, go to: http://etc.stsci.edu/webetc/index.jsp
Click on ACS Imaging ETC This is the exposure time calculator for the ACS (Advanced Camera for Surveys) Instrument on board HST (Hubble Space Telescope). This web site is used by observers to figure out how long they need to observe to detect (or sufficiently well detect) their particular astronomical objects to accomplish their science experiments.
1.
Camera and filters.
Pick: HRC (high resolution camera) Not the WFC (Wide field Camera). Pick a really broad filter: F606W W stands for Wide. 606 stands for centered at 606nm. For this project we don't need a wide field of view so if we detect anything it would be nice to have high angular resolution. (So we choose HRC). We want photons because we are trying to detect a possibly faint galaxy so we want a broad filter.
2.
Exposure parameters
Pick: Time needed read to S/N ratio = 100 S/N is signal to noise ratio. We want enough signal so we can tell if its a galaxy and not something else like an unresolved star or a cosmic ray hit. 3
Choose point source and default aperture. This is fine since we are going to estimate exposure time in the simplest way with a point source. If we chose to assume the source extended then the aperture where we measure flux would be important.
3.
Spectral energy distribution
Pick: Non-stellar object, choose spiral galaxy This is what we think is most likely because there are a lot more spiral galaxies than ellipticals. It would also be easier to see an elliptical galaxy because they are brighter (so it should have already been seen). Choose a possible redshift for the galaxy. Based on theory of gravitational lenses we might expect the galaxy to be at redshift of something like 1. The source redshift will affect the model spectrum. Distance galaxies are redshifted, hence redder than nearer galaxies. It would be a bad idea to pick a very blue filter for this proposed observation as we would not be able to see a source at moderate redshift.
4.
Normalize the target's flux
Enter a V band magnitude of 24 Let E(B-V) = 0 (don't worry about extinction for the moment, assume the quasar is at high galactic latitude).
5.
Background.
Background often limits your ability to detect anything. Let the Zodi be normal. The Zodiacal cloud is a bunch of dust particles in the solar system which scatter light from the sun.
6.
CCD parameters
CR-split (CR means cosmic ray) must be at least 2 otherwise we are never going to be able to get rid of cosmic rays. By CR-split =2 means we are going to get two images. If you read out the chip more often then you are adding more read noise making it harder to detect your object. Also it can take a while, wasting valuable HST time. So you want to read out as few times as possible (but > 1). Leave CR-split equal to 2. (Can you tell that I am annoyed with observers who have set CR-split=1 leaving the archive full of images with cosmic rays that are impossible to get rid of reliably?)
Now for the excitement. Click: Submit simulation
You should have found out how long you need to expose the camera to get a S/N=100 observation of a 24th magnitude source. a) How long does it take to detect a V=24 mag source at S/N=100? b) Suppose we integrate for 1 whole orbit (about an hour) how faint an object could we detect with S/N=100? S/N=10? c) Try using a narrower filter like F555W. How faint an object can you detect in one orbit (at S/N=10)? d) The Milky Ways absolute Vband magnitude is about -21.4. How bright would the Milky Way be at a redshift of z=1? You can use http://www.astro.ucla.edu/~wright/CosmoCalc.html to get whats called the Luminosity distance (used like a distance to get the Distance Modulus).
4
e) Could you have detected a Milky Way size galaxy at z=1 in one HST orbit (at S/N=1...