Problem Set 04 - Department of Physics and Astronomy

Astronomy 362 Problem Set #4
Due Friday, February 14 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 Chapter 24 for homework (if you haven’t
already).
1. [C&O 24.1] Approximately how many times has the Sun circled the center of the
Galaxy since the star's formation? Clearly explain your reasoning.
2. [C&O 24.3 (tweaked)] The globular cluster IAU C0923-545 has an integrated
apparent visual magnitude of V = +13.0 and an integrated absolute visual
magnitude of Mv = –4.15. It is located 9.0 kpc from Earth and is 11.9 kpc from the
Galactic center, just 0.5 kpc south of the Galactic midplane.
a. Estimate the amount of interstellar extinction between IAU C0923-545
and Earth.
b. What is the amount of interstellar extinction per kiloparsec along this line
of sight?
3. [C&O 24.5 and 24.6 (heavily modified)] A
question analyzing the hypothetical
differential star counts in Table 24.2.
Assume that all stars included in the
differential star counts are main-sequence A
stars of absolute visual magnitude MV = 2
that were observed in a single selected field
of fixed solid angle.
a. Briefly state what the differential star
count AM represents in this case in
English.
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b. Re-write the expression
ln10
AM ( M ,S,Ω, m ) =
ΩnM ( M ,S )10 3( m−M +5 )/5 (equation 24.5)
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as an expression for log10 AM as a function of m for stars of the same
absolute magnitude and M-K spectral classification, assuming a constant
stellar number density. Show that this implies an expected slope of plot of
log10 AM vs. m of 0.6.
c. Plot log10 AM as a function of V for the hypothetical data given in Table
24.2.
d. Assuming a constant density of stars out to at least V = 11, how much
interstellar extinction is present to that limit (express your answer in
magnitudes)? Hint: Consider the slope of the curve and note that equation
24.5 was derived assuming NO interstellar extinction was present.
e. What is the distance to the stars corresponding to V = 11?
f. If the solid angle over which the data were collected is 0.75 square
degrees, or 2.3×10-4 sr, estimate the number density nM(M, S) of A stars
out to V = 11. HINT: Look over your result from part (b)
g. Give two possible explanations for the change in slope between V = 11
and V = 15. Explain your reasoning.
4. [C&O 24.13 (tweaked)] The local standard of rest
(LSR) is the mean velocity of the material in the
solar neighborhood of the Milky Way. Consider a
sample of stars lying in the Galactic midplane and
distributed around the Sun in a circle (as shown
below). For the purpose of this problem, assume all
the stars are at rest with respect to the LSR; an
unrealistic situation, but useful for this problem. It
turns out the Sun is moving at about 16.5 km/s
relative to the LSR, as indicated in the figure.
a. On the diagram (an extra, much larger, copy is
attached to this handout), sketch the velocity vectors associated with the
apparent motion of each star, as seen from the Sun. Label the “apex” (the
location the Sun is moving toward) and “antapex” (the location the Sun is
moving away from).
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b. Sketch the radial velocity and tangential velocity components of each
star’s apparent motion on the diagram. NOTE: Tangential velocity is the
component perpendicular to the radial velocity.
c. Describe how you might locate the apex of solar motion given radial
velocity data for a large sample of stars in the solar neighborhood.
d. Describe how you might locate the apex of solar motion given proper
motion data for a large sample of stars in the solar neighborhood. NOTE:
“Proper motions” are motions in the plane of the sky, essentially reflecting
tangential velocities.
e. (Extra Credit) Given that we are in a differentially rotating galaxy, what
is unrealistic about the motion of these stars? Does this invalidate your
answers for (c) and (d) as ways to determine the direction of the Sun’s
motion?
5. [C&O 24.19] An application of Oort’s constants.
a. Estimate dΘ/dR in the solar neighborhood, assuming that the Oort
constants A and B are +14.8 and –12.4 km s-1 kpc, respectively. What does
this say about the variation of Θ with R in the region near the Sun? HINT:
Start with the definitions of the Oort’s A and B constant in terms of dΘ/dR
and Θ0/R0 as shown in equations (24.39) and (24.40).
b. If A and B were +13 and –13 km s-1 kpc, respectively, what would the
value of dΘ/dR be? What would this say about the shape of the rotation
curve in the solar neighborhood?
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6. [This is a heavily modified version of problem 2.15 from Sparke &
Gallagher] Let’s work on trying to interpret the meaning of the l-vLSR diagram
shown for neutral hydrogen (HI) gas in the Galaxy (attached to this homework).
We will do this by modeling what we will see for a very simple model of the
Galaxy with us a distance R0=8 kpc from the Galactic center and all the gas
orbiting a constant rotational velocity of Θ(R)= Θ0= 220 km s−1 at all radii.
HINT: Before tackling this problem, review the laws of sines and cosines, they
are your friends.
a. Consider the sketch shown here labeling the sun (S),
the Galactic Center (GC), and some point P. Find an
expression for the distance d(l,R) for the gas in an
orbit with R at point P at galactic longitude l. Note,
you will need to solve a quadratic equation to
determine the distance. This means there may be two
solutions for the distance to the gas for any given
galactic longitude l and galactocentric radius R.
b. Clearly explain what the multiple distance solutions
in the last part physically correspond to. Also
explain clearly under what conditions there will be
only one solution for the distance and verify the
“single solution” case corresponds to when our line
of sight is tangent to the circular orbit of gas a
distance R<R0 from the center of the galaxy.
Parts (a) and (b) suggest it is not a simple matter to look at gas at some
position in the sky and unambiguously determine its distance. Therefore, let’s
tackle the problem in reverse. Let’s simulate gas in different positions around
the galaxy and determine where it appears in the l-vLSR diagram. To do so, we
first need to figure out a couple of expressions.
c. First, find the expression for the Galactic longitude l(θ,R) to gas at
Galactocentric azimuth θ and Galactocentric radius R.
d. Next, find the radial velocity vr(l,R) (which we will treat as being
approximately the LSR velocity) for gas in circular orbits as a function of
galactocentric radius R of the gas’ orbit and galactic longitude l of our line
of sight to the gas by starting with equation (24.37) and inserting all the
values we know. Remember, all the gas in this model has the same orbital
velocity, Θ(R)= Θ0= 220 km s−1.
e. Make a model l-vLSR diagram similar to figure shown, but for gas in
circular rings at R = 4, 6, 10, and 12 kpc. Do this by computing l and vr for
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gas at these radii at regular intervals of Galactocentric azimuth θ around
each ring (say at 30˚ intervals). Remember, we will be assuming vLSR ~ vr.
Then plot up l versus vLSR, preferably with a different symbol for each
ring. Limit your plot to views toward the Galactic Center (l>270˚ or
l<90˚) since the formulas you used will limit their solutions to these
values.
To make your life easier, I have come up with an Excel spreadsheet that
has most of the work done.
That spreadsheet (called
Model_lv_diagram.xls) is available for download with the homework.
However, you will have to fill out the formulas for computing l and vr for
each of these (θ,R) positions. Remember, Excel can only do trigonometric
calculations such as SIN() or ASIN() in radians, so convert any degrees
into radians (by multiplying any degrees value by PI()/180.0) and
when needing degrees from the result of a trigonometric function (like
when computing the galactic longitude), convert the radians back into
degrees (by multiplying radians by 180.0/PI() ).
f. Now examine the real l-v diagram at the end of this homework. Based on
our simple model, explain where the gas lies (that is, in which general
direction in the galaxy, and is it in an orbit larger than our own or smaller
than our own) that corresponds to (l∼50˚,V>0); (l∼50˚,V<0); (l∼120˚,V<0);
(l∼240˚,V>0); (l∼300˚,V>0); and (l∼300˚,V<0). And finally, a challenging
case, where is the gas at (l∼120˚,V>0)?
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Extra Copy of Figure for C&O 24.13 (Tweaked)
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