Problem Set 05

Astronomy 362 Problem Set #5
Due Friday, February 28 at 12 noon
In the homework problems below, “C&O” refers to your textbook. I have selected the
problems with the expectation that they will take up to 1 hour of hard thinking/calculating
each. If you are taking longer than this, feel free to come by my office to hash out where
you are stuck.
Note on collaborating: You may work together on this problem set, but all work
presented here must be your own. You must clearly acknowledge any people you
collaborated with.
ASSUMED READING: Please finish C&O Chapters 24 and 25 (Sections 1 and 2) for
homework (if you haven’t already).
BEFORE YOU BEGIN THIS PROBLEM SET!
Download the
Synthetic_Spectra_Files.zip file.
It contains two IPython notebooks
(endinging in .ipynb) and a file containing a description of the model galaxy (ending
in .csv). The initial model galaxy is actually a reasonably accurate model of the Milky
Way. A backup copy of this file is provided since you will be changing the galaxy
model during this problem set.
Unless you are running Anaconda Python, you will likely not have the astropy
Python library necessary to run these simulations pre-installed. An easy way to have a
compatible environment is to run it online! Create an account on the http://
wakari.io/ service (the account is free). This is an online service that lets you run
IPython notebooks on their servers! It has the advantage that all the necessary Python
libraries are already loaded for you. Upload the IPython notebook files and the galaxy
description CSV file onto Wakari and you will be ready to go.
NOTE: For this problem set, unless directed otherwise,
please assume the IAU values for Sun’s rotational velocity,
Θ0 = 220 km/s, and galactocentric radius, R0 = 8.5 kpc.
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1. [Variant of Problem
2.19 from Sparke
and Gallagher] The
G a l a x y ’s n e u t r a l
hydrogen (HI) disk
extends outward to
about 2.5R0. From the
attached rotation
curve of the galaxy,
show that the mass
M(<2.5R0)
≈
11
3×10 M⊙ [start with
equation (24.49)].
G i v e n t h e d i s k ’s
luminosity of about LB ∼ 1.8 × 1010 L⊙ (from Table 24.1) and the bulge luminosity
of LB ∼ 0.3 × 1010 L⊙ of that, show that the mass-to-light ratio of the Milky Way
has M/LB ~15. Recent observations suggest that the mass-to-light ratio out to the
solar circle (where R~R0) is about M/LB ∼ 3. What does this imply for where is
most of the Milky Way’s dark matter resides?
2. A Synthetic l-v diagram: As mentioned in class, we can observe neutral
hydrogen (HI) gas throughout the Galaxy since 21-cm radiation is not blocked by
interstellar dust. However, unlike stars, we have no easy way to determine
distance to a particular HI gas cloud we see, even though measuring the radial
velocity is trivial. In the last problem of the last homework, we worked through
the problem of computing the expected radial velocity vr and galactic longitude l
of a particular gas cloud if it was at a distance R from the galactic center in the
direction at galactic azimuth of θ. We used this to generate some l-v diagrams for
gas at various distances from the galactic center using Excel. We will now take
this a step further, using a Python program to generate synthetic l-v diagrams and
spectra for different distributions of gas in galaxies.
a. Open the Synthetic_LV_Diagram.ipynb notebook. Run the IPython
notebook to the end with the “default” galaxy model. Read the description
of what is going on in the IPython notebook. Print the image produced at
the end of a synthetic l-v diagram for this “Realistic” Milky Way model (a
png file is saved, which you should be able to download and print out).
Compare this synthetic l-v diagram to the real one that was attached to the
last homework. Does it seem to reproduce the major features? NOTE:
The “feature” in the real l-v diagram extending from l=+30˚, v=+100 km/s
to l=–30˚, v=–100 km/s, is often ascribed to the 3 kpc molecular ring
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around the galactic center, which is not a part of this model.
b. Now, make a different galaxy model where the rotation velocity drops as a
Keplerian model (r-1/2), this involves changing the .csv file. Explain how
you created the .csv file. By this I mean describe how you made the data
file from a mathematical point of view, not a technical point of view.
c. Using this alternative “Keplerian” model, run the IPython notebook to the
end. What changes do you see in the l-v diagram, compared to the
“realistic” Milky Way model results? Is it obvious that the Keplerian
model doesn’t work at reproducing the observed l-v diagram? Include
copies of the images (the IPython notebook does export the final image as
a PNG).
3. [C&O 25.4] Neglecting the effects of extinction and the K-correction (which
takes into account cosmological redshift changing the bandpass of the emitted
light versus the observed light), show that the surface brightness of a galaxy is
independent of its distance from the observer. HINT: Surface brightness is the
flux per solid angle. A solid angle is defined as the surface area subtended by the
solid angle divided by the distance squared (such that for a sphere the surface area
4πd2 divided by d2 gives 4π steradian total solid angle).
4. [C&O 25.8 (tweaked)] NGC 2639 is an Sa galaxy with a measured maximum
rotational velocity of 324 km s-1 and an apparent magnitude of B = 12.22 mag
(after making corrections for Galactic extinction).
a. Estimate its absolute magnitude in the B band from the Tully-Fisher
relation.
b. Determine the distance to NGC 2639 using its distance modulus.
c. What is the galaxy's radius (R25) at a surface brightness level of 25 B-mag
arcsec-2? BIG HINT: We didn’t dwell on it in lecture, but there is a
radius-luminosity relationship for early-type spirals given by equation
(25.11).
d. Find the mass of NGC 2639 that is interior to R25.
e. What is the luminosity of the galaxy (in L⊙) in the B band? (Note:
MB,⊙=5.47)
f. Calculate the mass-to-light ratio for NGC 2639 in the B band, interior to
R25.
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5. Synthetic Spectrum of Spiral Galaxy: Let’s now do a simulation of the 21-cm
line spectrum of a spiral galaxy.
a. Open the Synthetic_Galaxy_HI_Spectra.ipynb notebook. Run the IPython
notebook to the end with the “default” galaxy model. Read the description
of what is going on in the IPython notebook. Notice the final image
produced is automatically saved to a file, named based on your selected
inclination angle and redshift. Produce images at 5 different inclinations
including face-on (incl=0) and near edge on (incl=85). Save and
print those images.
b. What happens to the double horn profile in the unresolved spectra as the
galaxy is more an more inclined? Consider how Tully-Fisher method of
estimating a galaxy’s luminosity requires us to estimate the maximum HI
rotation velocity. How is the measurability of quantity affected by the
inclination of the galaxy?
c. Now consider the resolved HI velocity field, which shows what a VLA
might see when looking at this field. The lines are velocity contours.
Describe what happens to the velocity contours in the resolved 2-D image
as the galaxy becomes more and more inclined?
6. [C&O 25.20] According to the virial theorem, the central radial-velocity
dispersion is related to the mass and size of the galaxy by σ r2 ∝ M / R (see Eq.
25.13). Use arguments similar to those for the Tully-Fisher relation to show that
L ∝ σ r4 , which is the Faber-Jackson relation, Eq. (25.40).
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