1. Place an upper limit on the radius of the Celestial Sphere using the speed of light.
   [Hint: The dome of the sky must rotate once every sidereal day.]
   
2. The most distant planet known to the ancients was Saturn, about 10 AU distant.
   The stars were beyond Saturn, so use this to place a lower limit on the rotation speed of the Celestial Sphere.
3. Could you measure the distance to the Celestial Sphere using parallax?
   Modern ground-based measurements can easily measure a parallax of 0.1 arcsecond. But what is your baseline?

1. What is the number density of baseballs?
2. How far could you see before your line-of-sight is blocked by a baseball?
3. Is this universe transparent to baseballs?
   In practice, we can detect galaxies over 4 Gpc away, so that makes a good working definition of transparent.
4. Now imagine replacing baseballs with balloons of the same mass and number density. Assume that the air in the balloons has the density of air at sea level (0.001225 g cm⁻³) to work out the radius of these balloons.
   Is this universe filled with balloons transparent?
5. The pressure in space is somewhat lower than at sea level, by about a factor of 10¹⁶. Assuming the balloons can expand to an equilibrium size without popping, how big do they become?
   Assuming that the stretched-out balloons remain opaque, is this universe transparent?

1. The absolute visual (V-band) magnitude of the sun is M_⊙,V = 4.83. What is its apparent magnitude at a distance of
   1. d = 1 AU.**
   2. d = 1.34 pc (the distance to αCen, a solar-type star).
   3. d = 10 pc.
   4. d = 1 kpc (far yet still well within the Milky Way whose stellar disk has a radial extent of roughly 20 kpc).
   5. m−M = 31 (the distance modulus of the Virgo cluster of galaxies).
   6. Comment on the visibility of the sun at each of these distances.
      A bright star has m ≈ 1 (a star of the first magnitude) while the faintest stars visible to the unaided human eye have m ≈ 6.
2. The luminosity of the Milky Way Galaxy in V is L ≈ 3 x 10¹⁰ L_⊙.
   1. What is the absolute magnitude of the Milky Way?
   2. What would its apparent magnitude be if it were part of the Virgo cluster?
   3. Could you see it in Virgo with the naked eye?
   4. Could you image it with a modern detector on a modest telescope? (with a limiting sensitivity of m=20).
3. The radio source 3C 273 was one of the first quasars discovered. It has a visual magnitude m = 12.9 at a redshift z = 0.158.
   1. What is the proper distance to 3C 273 if H₀ = 70 km/s/Mpc?
      (You may use the approximation V ≈ cz).
   2. What is the corresponding luminosity distance?
   3. What is the absolute magnitude of 3C 273?
   4. How does the luminosity of 3C 273 compare to that of the Milky Way?
      [Quasars are quasi-stellar sources - radio sources that look like stars (point sources) optically. Their luminosity is produced in a tiny volume, typically smaller than the solar system.]

   ASTR/PHYS 428 only
   

4. Re-do the questions about 3C 273, but now use the full relativistic formula instead of the approximation V ≈ cz.
   How much difference does this make?

*All logarithms are base 10.
ξ is the "zero point" of filter-specific photometric systems defined in arbitrary and capricious ways that you needn't worry about here.

**Recall that the definition of the pc is the distance corresponding to a parallax angle of one arcescond with Earth's baseline of 1 AU.
So - how many arcseconds are there in one radian? That's how many AU there are in a pc.

1. What slope would you observe in the number count-magnitude [log(N)-m] diagram for such objects?
   [Hint: remembering the relation between luminosity, flux, and magnitude, integrate in spherical shells to find the number N of objects observed at each flux/apparent magnitude m.]
2. Surface brightness is the luminosity per unit area. Considering shells of width Δ at distance d, derive an expression for the surface brightness of shells on the sky in terms of n, L, d, and Δ.
   How does the surface brightness depend on distance?
   [Hint: what is the total luminosity of sources with density n and individual luminosity L?]
3. Integrate the light received from all shells, from d = 0 to infinity
   Would the sky be dark?
4. In what ways does modern cosmology differ from the assumptions above?
   [Hint: there are three distinct effects that mitigate Olber's paradox.]

1. Approximately how many years would it take to accelerate to half the speed of light?
   You may ignore relativitic corrections; just use v ≈ a*t.
2. How much of the initial rest mass M_i must be converted to kinetic energy to achieve this speed?
   Assume perfect efficiency.

   The rocket equation is far from perfectly efficient: the speed achieved depends on the exhaust velocity of the fuel v_ex and the ratio of the initial to final mass
   v = v_ex ln(M_i/M_f)
   Rockets are an expensive means of propulsion because you have to use fuel to move fuel and carry enough fuel to get where you're going — never mind getting back.

3. For a typical chemical rocket, v_ex = 3 km/s. What would the inital to final mass ratio need to be to reach half the speed of light?
4. It is conceivable that a nuclear rocket might achieve v_ex = 50 km/s. What is the ratio now?
5. How does this compare to the ratio of the Earth's mass to yours?

If you want a more optimistic answer, set v_ex = c, but don't look into the light!