ESCI 330 Homework

Due one week after completion of the chapter. Check back for additions, corrections, hints, etc.

See due dates posted next to each assignment

Correction to formula 9.14 (gravitational redshift):

The lefthand side was wrong - it's λ_f / λ_i = (1-2GM/Rc² )^-1/2

By popular demand, pictures of the three-legged cat

Sun and Stars:

Chap 1:

1)     Pick any hydrogen atomic transition with n_i between 5 & 10 and n_f between 15 and 20 and calculate E and λ.

2)     An atom in the second excited state (n=3) of hydrogen is just barely ionized when a photon strikes it. What is λ if all the photon’s energy is transferred to the atom?

3)     The photosphere has a temperature of 6000K. From Wien’s law, what is the peak λ? What is the average energy of these photons? From (1.5) estimate the average kinetic energy of electrons in the photosphere and compare to the energy of the photons.

4)     Using the small angle formula θ = l/d (θ is angular size in radians, l is linear diameter, d is the distance) calculate the apparent angular size of the sun as seen from Pluto.

5)     From 1.2 derive approximate expressions for Planck’s radiation law
a) at high f (hf/kT >> 1), called the Wien distribution
b) at low f (hf/kT << 1), called the Rayleigh-Jeans approximation
Use e^(hf/kT) ~ 1 + hf/kT for hf/kT << 1

6)     What is the ionization energy of sodium (Z=11)? What kinetic energy would a colliding electron have to have to ionize sodium (assuming all its kinetic energy is given to the atom in the collision)?

Chap 2:

1)     People are usually surprised to learn that the sun is slightly dimmer at sunspot minimum. Given L α T⁴ and that the sunspot cycle correlates with other types of solar activity, how would you explain this fact?

2)    Calculate the energy released in the final steps of the PP I and PP II cycles. Comment on the relative efficiency of these processes as energy sources.

PP 1: ³He + ³He à ⁴He + ¹H + ¹H

PP II:  ⁷Li + ¹H à 2 ⁴He

3)     For a granule in the photosphere of 700 km diameter, calculate its angular diameter as seen from earth.

4)     There are sometimes considered to be three populations of stars (roughly “generations”):

Pop III (1^st gen) Z ~ 10^-9

Pop II (2^nd gen) Z ~ 0.002

Pop I (3^rd gen) Z ~ 0.02

Also recall that X=0.735 at the surface of the sun (presumed to reflect its initial value from the sun’s birth) and 0.341 in the core, and Y=0.248 at the sun’s surface (also presumed to reflect its initial value from the sun’s birth).  Estimate Y at the core. Using this information and the concept of the three “generations” to conjecture how the chemistry of the universe has changed since the Big Bang.

5) Only 10% of the sun’s hydrogen is available for fusion (that in its core). If X=0.735 overall initially, what will be the sun’s final overall X when it dies? What will X be in the core when the sun dies? What will the overall Y be when the sun dies? What will Y be in the core when the sun dies?

6)  Enclosed are flux measurements for the H and K lines of calcium for a number of sunlike stars. At first glance, what do these diagrams look like diagrams of for the sun? Are they technically those kinds of diagrams? If not, why do they look so much like them? Hint: think about how/where the H and K lines arise in the sun and what kinds of activity might change their flux.

HD 136202 (F8IV-V) 23 yrs	The Sun (G2V) 10.0 yrs	HD 103095 (G8VI) 7.3 yrs
HD 81809 (K0V?) 8.2 yrs	HD 3651 (K2V) 13.8 yrs	HD 10476 (K1V) 9.6 yrs
HD 166620 (K2V) 15.8 yrs	HD 160346 (K3V) 7.0 yrs	HD 16160 (K3V) 13.2 yrs
HD 4628 (K4V) 8.4 yrs	HD 201091 (K5V) 7.3 yrs	HD 32147 (K5V) 11.1 yrs

Chap 3: DUE WEDNESDAY FEB 20

1) A variable star changes in brightness (L or F) by a factor of 4X. What is the corresponding change in magnitude?

2) If a star has an apparent mag of -0.4 and parallax of 0.31" what is its distance and absolute mag?

3) The V mag of two stars are both observed to be 7.5 but their B mags are B₁ = 7.2 and B₂ = 8.7. What are the color indexes of the stars? Which star is "bluer"? Explain.

4) Mars orbits the sun at an average distance of 228 million km in 686.98 days. Assuming the orbit is a circle, calculate its orbital velocity. From this calculate the aberration of starlight which would be measured from a Mars-bound telescope.

5) If you haven't noticed, astronomers use strange units. Show that V_t = 4.74 μd if μ is measured in arcsec/year, d is measured in pc and V_t in km/sec. In other words, take eqn 3.2 and on the righthand side convert km to pc, sec to years, and radians to arcsec.

6) A binary system has an orbital period of 10 yrs. The stars have max radial velocities of 10 and 20 km/sec. What are the individual masses of the stars if the orbital inclination is 0º? 45º?

7) An eclipsing binary has an orbital period of 2d22h. Each eclipse lasts 18h and totality for 4h. Find the stellar radii in terms of the orbital radius a. If spectroscopic observations show the orbital velocity of one star relative to the other to be 200 km/sec, what are the stellar radii in km?

Chap 4: DUE MONDAY FEB 25

1) Using the data for the following stars, calculate the extinction of starlight in each case. What conclusion(s) can you draw (even tentatively) about the distribution of dust in our galaxy?

Deneb: M = -6.9  m = 1.3  D = 430 pc

Bellatrix: M = -3.6  m = 1.6  D = 210 pc

Antares: M = -4.5  m = 1 D = 120 pc

2) Use the values of L and T plotted on figure 4.5 to calculate the radii of Betelgeuse, Barnard's Star, and SiriusB. Compare your calculated values to those plotted on the graph.

3) The surface temperature of Barnard's Star is ~2800K. Is this hot enough for the spectral line due to the dissociation of CO to appear in its spectrum? Hint -  see fig 4.16.

4) A star has m = 12.5 and M = 3.3. Calculate its distance. Repeat the calculation with an extinction of 2.1 due to intervening dust. Calculate the percentage error or not including extinction:

% error = |difference in calculated distances| / distance with dust included (X 100)

5) Show that both versions of eqn 4.1 gives the same result for a 0.43 solar mass star.

6) This question asks you to sit back and reflect on what you've learned so far rather than do a calculation: Based on Chapters 1-4, do you think astronomers have mapped our stellar neighborhood as well as geologists have mapped the rock types and structures of Connecticut? Explain your reasoning (and include any limitations, approximations, problems, etc. you think especially relevant).

Chap 5: DUE WEDNESDAY FEB 27

1) Calculate the Jeans mass for cloud made of silicon (dust) if there are initially 30,000 atoms per cubic meter. T = 10K.

2) A starforming region has an initial mass of 2 solar masses, rotational velocity of 0.5 km/s, radius of 0.2 pc, and magnetic field strength 10^-11 T. Calculate the final values of v and B assuming it collapses to 1.5X the radius of the sun. Comment on the realism of your final answers.

3) Calculate the radius of a forming protostar of temperature 2000K and luminosity 0.1 that of the sun.

4) Consider an HI region of radius 1 pc, temperature 100K, and number density 10¹⁰ /m³.  Is it likely to form a single solar type star? Why or why not?

5) Is it observationally likely that a solar system sized disk (~50 AU) could be directly observed by a ground-based telescope if it were 150 pc away? Why or why not?

6) What is the critical density of dust that could be held onto by a white dwarf (~ size of earth, T ~ 10,000K, ~ 1 solar mass). Hint: use eqn 5.9

Chap 6: DUE MONDAY MARCH 3

1) a) calculate the central pressure of a typical red giant (M = 2 solar masses, R = 1au)

    b) Calculate the central pressure of a typical white dwarf (assume hydrostatic equilibrium applies) (M = 1.2 solar masses, R = R _earth)

    c) Comment on your answers to parts a and b in comparison to the sun.

2) Using (6.1) calculate the real molecular weights at the surface and core of the sun (using currently estimated values of X,Y,Z). Is μ = 1/2 a good approximation to use?

3) Using μ = 1/2 , calculate the core temperature of both stars you explored in question 1. Compare these answers to the estimated core temperature of the sun.

4) Using the values for the red giant in question 1 & 3, calculate dT/dr and L (in terms of K). Assuming this red giant has a luminosity 500X the sun, calculate K.

5) Using eqn (6.3) and values of K,Z,X, ρ, T for the sun (using core values of Z,X, ρ, T), find the constant.

6) Calculate the Gamow energy for fusion of 2 helium nuclei. Calculate the probability of such a reaction occurring in the sun and comment on why the sun does not currently fuse helium in its core. Assume E_p ~ 1/4 value for hydrogen collisions (because He is 4X more massive than a proton)). What would have to happen to the sun to make such a reaction more likely?

7) For a star of the sun's surface temperature, what is the number density n for which the radiation pressure = gas pressure?

Chap 7 DUE WEDNESDAY MARCH 5

1) A Cepheid variable varies from spectral class A0 to F0 and back in one cycle. If its change in apparent brightness is 2 magnitudes, how many times brighter is it at max as compared to min? Estimate its surface temperature at max and min.

2) An RR Lyrae star has an apparent magnitude of 11.2. How far away is it (neglecting the effects of extinction)?

3) A binary system consists of a 20 solar mass star and a 1 solar mass star. Calculate the initial ratio a³/P². If the heavier star loses 50% of its mass over its lifetime, what will be the final value of  a³/P² ? Hypothesize what changes an observer might see in the orbits over time.

4) Use the concept of the Gamow energy (chap 6) to explain why succeeding fusion cycles require higher temperatures.

Chap 8 DUE MONDAY MARCH 10

1) Using the data in figure 8.1, estimate the radius of the sun when it achieves helium flash.

2) Using figure 8.1, for a 5 solar mass star, at what stage in its life is it hottest? Brightest?

3) From figure 8.6, how many times brighter is a type II supernova at peak compared to when its outburst is first noticed? How much dimmer is it than max a year from its peak?

4) Referring to Fig 8.19, explain the abundance peaks at mass numbers 4,12,16, 56. What process creates these high abundances?

5) Referring to figure 8.19, why are water and carbon dioxide fairly common in our solar system?

6) Referring to figure 8.19, how much more abundant is hydrogen than carbon in the solar neighborhood?

7) Using the atomic mass numbers, explain the dip in abundance in figure 8.19 for elements between calcium and chromium.

Chap 9 DUE MONDAY MARCH 24: You should be considering your final paper topic

1) From (9.1) calculate the maximum mass of a white dwarf made primarily of carbon, and primarily of oxygen.

2) From (9.4), given R ~ R earth, M ~ M sun, estimate K. Using this value, estimate the radius of a white dwarf at the Chandrasekhar limit (1.4 M sun).

3) For a neutron star of radius 10 km, period = 10^-6 sec, angle between the two axes 15 degrees, and luminosity of 10³⁰  W, calculate the magnetic field strength.

4) Using (9.12), calculate the precession of the perihelion of Mercury in radians/revolution. Convert this to "/century and show it is ~ 43"/century.

5) Calculate the gravitational redshift for light trying to escape from the sun.

6) A white dwarf orbits an invisible companion in an orbit apparently seen face on. If the period is 30 yrs and average separation between the 2 objects is 20 au, what is the mass of the unseen companion? Is it likely to be a black hole? Explain.

Galaxies and Cosmology: Chap 1 DUE WEDNESDAY APRIL 2

1) Plot a rough rotation curve for a mass distributions where V α R^-0.15 . What assumptions can you make about this mass distribution?

2) Assuming that the number density of OB stars at midplane is 80 per 100,000 cubic parsecs, estimate the number density at the edge of the thin disk.

3) Calculate the distance from midplane at which the number density of G stars has dropped to half its midplane value.

4) Assuming dust is not an issue, what is the apparent magnitude of an RR Lyrae star located in a globular cluster 45,000 pc away? What would A (the dust extinction) have to be in order for this star to not be visible in a telescope that can see down to mag 28? Comment on what parts of the Milky Way we can best see RR Lyrae stars in and which we cannot.

5) A new stars is discovered in he Milky Way. Unfortunately, its location is right on the boundary between the bulge, thick disk, and halo. What populations might it belong to? What information would you want to know to make a possible determination of its actual population?

6) Using Figure 1.13, estimate the velocity of a star orbiting at the visible edge of the Milky Way. Use this value to estimate the mass of the Milky Way.

7) Notice that in Fig 1.13 the rotation curve is roughly Keplerian from 0.1-3 Kpc from the center. Explain why this is so.

Chap 2 DUE WEDNESDAY APRIL 9

1) An elliptical galaxy has a velocity dispersion of v = 1.7 X 10⁶ m/s. If r_h = 60 kpc, what is the total mass of the galaxy? If r_h is uncertain by +/- 10%, what is the possible range in mass? Hint: square v.

2) A small satellite galaxy orbits the Milky Way at an average distance of 50 kpc in 3 billion years. What is the suggested mass of the Milky Way?

3) A galaxy has a redshift of 0.05. Using figure 2.30, what is the range of most probable distances?

4) A Cepheid in another galaxy has a period of 20 d. What is its average absolute magnitude (Fig 2.25)? If its apparent mag is +17, how far is the galaxy?

5) A type Ia supernova is seen in a distant galaxy with a peak blue apparent magnitude of +11. What is the absolute magnitude and distance if H = 70, 72, and 75?

6) A nova is seen in a neighboring galaxy which has a decline of 1.8 mag per day. What is the peak absolute mag? If the peak apparent mag is 8, what is the distance?

7) A spiral galaxy has a rotation width of 200 for its 21 cm line. What is the range of possible absolute magnitudes (in IR)?

Chap 3 DUE MONDAY APRIL 14

1) The radio lobes of a galaxy span 2 Mpc. If the galaxy is 10 Mpc away, what is the angular size of the lobes?

2) The emissions of a quasar vary on scales of 5 hours. How large is the central region in km?

3) An AGN has a luminosity of 10³⁸ W and black hole of mass 10⁷  solar masses. It accretes 1 solar mass of material per year. What is the inner radius of the accretion disk? Compare this to the Schwarzschild radius of the black hole.

4) What is the Eddington luminosity for the black hole in problem 3?

5) For the black hole in problem 3, calculate the temperature of the accretion disk at 4 AU.

Chap 4 Due Wednesday April 16

1) If a galaxy cluster has a mass of 10¹⁴ solar masses, what is the predicted doppler velocity dispersion?

2) For the above cluster, assuming its gas halo has a radius of 3 Mpc, what is the predicted temperature of its ionized hydrogen?

3) Calculate the Einstein angle for an Einstein ring created by the above cluster if the cluster is 40 Mpc away and the object it is lensing is 70 Mpc away.

4) Calculate the observed angles of 2 lensed images if the einstein angle is 7" and the lensed object is offset by 20".

5) Consider the data in figure 4.20. For what Z do the two models give distances which diverge by ~20%? Discuss how we might use this to observationally determine which cosmological model best reflects our universe.

Chap 5 Due Wednesday, April 23

1) Referring to (5.17)
 

a) for what value of q_o is v=Hd=cz strictly true?

b) What does q_o = 0 mean for the expansion of the universe?

c) Calculate the distance to a galaxy of Z = 0.3 for q₀ = +/- 0.5. Calculate the error using V=HD=cZ would have given compared to these calculations.

2) Calculate the age of a Λ=0 k=0 flat universe and Λ=0 k=-1 empty open universe if H_o = 55 and 90. Are any of these ruled out by observations that the oldest globular clusters are about 13 billion years old?

3) How much smaller was the universe at z=0.2? 0.8? 1.4?

4) Philosophers and historians of science LOVE talking about the development of the FRW model and will use it as a classic example of the nature of science (how models are developed and tested). Why? Is it a good example? Explain your logic.

5) Consider the k=+1 Λ=0 >Λ_E Lemaitre model. Look at the diagram in the book (R vs t). Describe how H changes during the three "epochs". Hint - refer to eqn 5.15.

Chap 6: Due Monday May 5

1) How might the formation of structure in the universe have been different if Δρ/ρ ~ 10^-3 ?

2) If the ratio of the number density of neutrons to protons had been 0.2 at the beginning of nucleosynthesis, what would have been the resulting value for Y?

3) Are the following combinations of quarks possible? Explain your reasoning.

(uus)
(ccs)
(ddt)
(bsc)
(ttt)

4) Use 6.12 to verify the temperatures at the end of the Planck Era (10^-43 s), GUT era (10^-36s) and Electroweak era (10^-12 s).

5) How much energy is needed to produce an electron-positron pair? What temperature does this correspond to (assume ideal gas law)? How old was the universe when it fell below this threshold?

Chap 7: TBA Due Wednesday May 7

1) You are the peer reviewer for a prestigious astrophysics journal and you receive a new paper to review for possible publication. The authors claim that they have observational evidence for Ω_m,0 = 0.7. Based on Figure 7.12, what questions would you have about the results?

2) Take a look at Figure 7.27. Comment on the "precision" of the so-called precision cosmological model this data describes. For which parameters has true precision been achieved? For which parameters do we have a long way to go?

3) From 7.2, what has to be true about the values of the densities of matter and dark matter for an

a) decelerating universe
b) accelerating universe
c) constant velocity universe?

4) Compare figures 7.25 and 7.26. What constraints can you put on the most probably value for the density of baryonic matter in the universe?

Chap 8: no homework