Problems are from: Modern Astrophysics
by Carroll and Ostlie:
Look here for hints and solutions to homework problems.
HOMEWORK ASSIGNMENTS
4. From CH.16: Problems 5ab, 6, 7, 9, 10, 16 (assume theta = 90 deg)
Two Additional Problems:
2. The Law of Gravity expresses the force exerted
by one mass, M_1, on another, M_2, when separated
by a distance, r.
5. From CH.18: Problems 7, 9, 10, 13
6. From CH.18: Problems 12
Three Additional Problems
1. HD209458 is a G2V, solar-type star based on its spectrum. A planet resides in a circular orbit
0.046 AU from the star. The planet's mass is M=0.6 Mjup, R = 1.4 Rjup, and
it is composed primarily (98%) of hydrogen molecules and helium atoms.
Assume (correctly) that the star has L, Teff, and R of our Sun.
Do you expect the H_2 molecules to remain
bound to the extasolar planet during the star's lifetime, 5 Gyr.
2. a) Based on data in Apendix B, express the masses of various
Solar System bodies in units of the mass of Pluto:
b) In a sentence or two, would you characterize Pluto as a planet, or place it
in some other category, such as moon, asteroid, or comet?
3. Look up (in Appendix B) the density of Jupiter and the moon (glance at all other densities),
and also write down the density of rain drop (in the same units). 7. CH.19 : 6, 8, 11, 13, 16.
Additional Exercise for Non-Natives of Mars.
Identify any 15 geological regions of interest to you.
Now find those same regions on the topographcial map of Mars:
List your favorite regions and the associated altitude.
Hunt on the web to find out how a Laser altimeter works and describe
it in a few sentences.
Adopt a "model" for the atmosphere that consists of N layers,
each of which has optical depth of 1. Show that the total optical depth
is therefore, tau = N (Is optical depth really additive?)
Also assume that each layer has some temperature (constant
throughout that one layer), so that it radiates like a blackbody from its top and bottom.
Use energy conservation for each layer, as in problem 18.12.
7. From Ch.20:
3, 6 a and b (not c), 10, 13.
rho(r) = rho_c sin(kr)/kr
where rho_c is the density at the center of Jupiter, 4 gm/cm^3),
and k is another constant, given by: k = sqrt(2 pi G/K)
We define the gravitational potential from a point mass, M:
Phi = -GM/r
Now, if you place a mass, m, a distance r away from M,
the mass m has potential energy:
U = Phi m
Consider two point masses, m, separated by a distance, s,
which is small compared to the distance r from the origin to a point P
at which we want to know the gravitational potential, Phi.
The two masses are placed on the z-axis at z=+s/2 and
z=-s/2, i.e. on either side of the origin.
At P the potential is, of course:
1. From CH.4:: Problems 4.3, 4.6, 4.8, 4.11, 4.12
Problem 6:
You are to assess the energy requirements of a trip to Alpha Centauri
(a triple star) that is 4 light years away from Earth. Assume that your spacecraft
arrives, but does not decellerate upon arrival. It simply passess by.
To save energy, your spacecraft has an approximate
mass only that of an automobile.
a) If you travel at 0.01 c, how long does the trip take in years, and
how much energy is required? (Use estimates of the mass of an automobile.)
b) If you travel at 0.95 c, how long does the trip take? Answer for
both the clock on Earth and the clock on board the ship.
c) For part b) how much energy is required for the trip (be sure to
use relativistic formulas for kinetic energy)
d) Estimate the energy consumption of the entire United States
during one year. (You may estimate this energy use in any logical
way that is accurate to within a factor of 10 - i.e. each American
uses some amount of energy per year, multiplied by the number
of Americans.) How does a trip to Alpha Centauri compare
to our yearly energy usage?
2. From CH.12:: Problems 12.1, 12.2, 12.3, 12.5, 12.6, 12.9
For Problem 12.2:
Find the force due to radiation pressure on the particle.
Assume the dust particle has radius of 1 micron.
Hints: For 12.2, use the blackbody flux formula: F = sigma T^4.
For 12.3, use the Boltzmann Eqn. : Adopt 1/e as the fraction of
H atoms in the excited state.
For 12.5, you might need eqn 10.36.
3. From CH.12:: Problems 12.8, 12.10, 12.13, 12.15
Problem 5: Consider a region of space filled with small, stationary particles, having a
number density, n. A large object enters the region, having a cross sectional area, sigma,
and a velocity, v. Derive an expression for the number of collisions that the
large object makes with with small particles per second.
1. Photons have no mass. But construct a theory
in which a photon having energy, E = h nu,
has a hypothetical mass given by its energy:
h nu = m c^2 .
Using that mass, consider a photon leaving the surface of a
star having Mass, M, and radius, R.
Using the gravitational potential energy of a mass m,
U = -G Mm/r
find the fractional change in its energy, h delta(nu)/h nu,
when the photon reaches infinity far away.
This loss of photon energy actually occurs
and is called, "gravitational redshift".
F = G M_1 M_2 / r^2
You might wonder if this law has been tested in the regime
of larger and larger values of separation, r.
How far in r has the law of gravity been tested
for validity? Do a search on the WEB or use any other sources.
(Answer with a paragraph.)
Determine the following properties of the planet:
- surface temperature, Tpl
- escape velocity, Vesc, from the planet's surface
- RMS speed of hydrogen molecules, Vrms
- Jupiter
- Earth
- Io
- Europa
- Titan
- Triton
The masses of these three objects differ enormously, but
their densities are the same, within a factor of 5 (all are 1-5 g/cm^3)!
This is remarkable! The densities of gaseous planets, moon rocks, and frogs are
essentially the same! In a few sentences, explain from the microscopic physics
perspective why you think nature forces these objects to have the same density, despite
such different masses and chemical make-up.
Examine a (color) optical map of Mars at:
Optical Map of Mars
Topographical
Map of Mars from the "Mars Orbiter Laser Altimeter" (MOLA)
Hint for 19.6: Assume the atmosphere is optically thick, so that its top has a temperature
given by the usual equilibrium temperature for a blackbody heated by the host star.
Hint for problem 3:
Problem 20.3 assumes that you have solved problem 20.2 which we didn't assign!
Here is what you need.
Assume Jupiter is spherical and composed solely of pure molecular hydrogen (not a bad
assumption). For such a composition at high density, a good relation
between the pressure and density in Jupiter's interior is:
P(r) = K rho(r)^2
where K is a constant. The pressure increases as the square of the density,
with no dependence on temperature. This type of model is called a "polytrope" (see
page 369 in the text).
By combining the above equation of state with the equation of hydrostatic equilibrium,
you can obtain a 2nd-order differential equation for Jupiter's density as a function
of radius (dist from center). Problem 20.2 carries you through the process, which yields:
You will need to solve for the value of k in terms of Jupiter's radius, R_J.
For part a), you should start with the density distribution, rho(r), given above.
Hint: Since this analytic model does not assume constant density, you will need to
integrate over concentric rings to find the moment of inertia about Jupiter's
rotation axis. I = integral a^2 dm
where a = r sin(theta) and dm=rho(r) 2 pi a r d(theta) dr (this dm is rho * vol of shell).
You will need to integrate by substitution.
From problem 20.2, you should adopt a
central density of Jupiter of rho_c = 4 g/cm^3 (don't use 20 g /cm^3).
Find I/MR^2. For comparison, the measured value
is: I/MR^2 = 0.253.
Additional Problem:
a) Derive the gravitational potential, Phi, as a function of theta and r
from a pair of masses, each of mass m, located on the z axis at
+s/2 and -s/2. Use spherical coordinates, and the law of cosines as follows.
8. From CH. 22 pgs 907-935: Problems 1,3, 9, 23, 24, 29
For problem #1, you need to know the orbital velocity of the Local Standard of Rest (220 km/s), which is given as equation 22.23 on page 947 after a discussion of the Local Standard of Rest. Alternatively you would need to know the mass interior to 8 kpc and use Kepler's 3rd law. But that's somewhat circular since how we know the mass interior to 8 kpc is by knowing the orbital velocity of the LSR.
For problem #9 part (a) you should either just use 25 kpc (the radius that the values in table 22.1, page 918, are valid for) or they should use the mass given in table 22.1 and scale it down for a size of 8 kpc. Since they'd be assuming constant density for the gas they should get the same answer for both calculations.
Due April 6.
10. From CH.23: Problems 2, 8, 9, 15, 18.
For Problem 2 eliminate parts c and d, and replace them with
three parts:
Hint:
For problem 23.15, consider a spherical elliptical galaxy, and
keep track of the brightness from each annulus of area 2 pi r dr.
__ Due April 20.
11. Structure of the Universe: Chapter 25 (and a little of Chap.14)
CH. 14: Read the first 5 pages of Chapt. 14, and do Problem 14.3 (about Cepheid Variables).
Hint: The V-band extinction to M100 is a = 0.15 mag. Use the period-luminosity relation (Eq. 14.2) to
get the absolute magnitudes as a function of pulsation period, P.
CH. 25: Problems 25.2,, 25.14, 25.16 .
"WEB" Problem:
Hints:
25.2 : Major and minor axes are 19 and 14 mm, respectively, implying theta = 43 deg.
25.14: See Eqn. 23.8
25.16: Use Eqn 25.5; Your answer should be between 5-20 Mpc
c) Determine the distance to M101 from M_B=-21.5 and
Bmag = +8.2 (you should get 5-10 Mpc)
d) From the velocity in part b, determine the proper motion
(angular motion perpendicular to line of sight)
of the stars in arcsec/yr, as seen from Earth.
e) Could van Maanen have detected the rotation of M101?
a) Find an expression for the apparent magnitude, m, of a galaxy as a function of
its absolute magnitude, M, and its redshift, z. Assume H = 72 km/s per Mpc. Assume z < 0.1 .
b) Derive an expression for the angular diameter, theta (in arcsec), of a galaxy as a function of
its physical diameter, D, (in units of kpc) and its redshift, z.
Due April 27.
12. From CH.27: Read pgs 1221-1248; Problems 3, 11, 12, 15, 16, 21, 24
Due May 4.
13. From CH.27: Read pgs 1248 - End of Chapter;
Problems 25, 26, 36, 37
For all problems, assume a flat, matter-dominated universe with lambda = 0
(not needed for 26).
Due May 11
Final Exam: Friday, May 13 at 12:30 in 102 Moffitt.
End of Semester 14. From CH.28:
15. From CH.26: Problems 2, 7, 8, 11, 13
9. From CH.22: Problems 21 and 30
A new map of the entire sky was obtained in near-IR wavelengths,
with filters, "J" (1.2 micron), "H" (1.6 micron), and "K" (2.2 micron).
----------------------------------------------------------------
Additional Problem:
Go to the 2-Micron
All Sky Survey (2MASS) website. Choose three objects within our
Milky Way Galaxy and examine the IR image of your objects. Find, on
the web, an image of each of your objects taken in visible light.
Write two or three sentences describing the differences, if any,
between the IR and visible images.