Selection of Homework Questions

   

Topic 1: History & Preliminaries

1. A globular cluster moves at 150 km/s on a circular orbit of radius 25 kpc.

   (i) What's the period of the cluster?
   (ii) What's the mass and mean density interior to its orbit?
   (iii) If the cluster has size ~10pc and internal velocity dispersion ~10 km/s, what's its mass and mean density?

2. What, roughly, are the speeds and periods of :

   (i) comets in the Oort cloud at a radius of ~¼ pc.
   (ii) gas in an accretion disk 1000 AU from a 10⁸M black hole.
   (iii) stars orbiting in the galactic center cluster of size ~30 arcsec and density ~10⁶ M pc^-3

3. Cosmoligical estimates are often easy using psm units:

   (i) Hubble's original estimate was H_o = 530 km s^-1Mpc^-1. What cosmic age does this imply?

   (ii) For H_o = 72 km s^-1 Mpc^-1, what's the critical density in M pc^-3   (use _c = 3H_o²/8G).

   (iii) Compare this to the density of typical clusters of size ~1 Mpc and dispersion ~500 km s^-1.

   (iv) If such clusters virialize by dissipationless collapse (i.e. R_final ~½ R_initial), what was their density just before collapse? If collapse starts when / ~ 1 (ie they have twice the mean density), what redshift did the collapse begin?

4. Consider the formation of ellipticals:

   (i) What's the total kinetic energy (KE) of a giant E galaxy with L ~ 10¹¹ L, ~ 300 km s^-1 and M/L ~ 10?

   (ii) From the virial theorem, the galaxy's binding energy (BE) is equal to its KE, which was liberated when it formed. If it collapsed on its current dynamical timescale, what was the "collapse luminosity", L_BE, in PLU and L?

   (iii) Is this significant compared to the typical luminosity of the associated starburst?

(2) Magnitudes and Mass to Light Ratios :

1. If I_B is the B band surface luminosity density measured in L_B, pc^-2 and µ_B is the corresponding B band surface brightness in B mag arcsec^-2 (mag/ss), show that
   

   µ_B = 27.05 - 2.5 log(I_B)

   Derive similar relations for V, I, and bolometric (you will need to use the solar absolute magnitudes given in the notes : ). These are potentially very useful relations. (B&M Q2.2)

2. NGC 1399 has a central surface brightness of µ_V ~ 16.0 mag/ss. What is its projected surface luminosity density, I_B, in L_,V pc^-2 ?

3. If it has a core radius of ~1.0 arcsec, a redshift of 1350 km/s, and a central velocity dispersion of ~150 km/s, estimate the core's luminosity density and mass density, and hence its M/L_V ratio. Include "h" in your answers.

(3) You outshine the stars !

Putting mass-to-light ratios into physical units can be surprising :

1. What's M/L_bol for the sun, in units of kg/Watt? About 50% of the sun's mass is in its nuclear burning core. What's the M/L_bol of this "nuclear furnace"? Think about this value: does it jive with the cliche of the sun as a "roaring fusion furnace"? You should conclude that stars are not luminous because they're intrinsically luminous per kg -- indeed they are quite feeble in that sense. They are luminous only because their furnaces are so massive.

2. To emphasise this, estimate your own M/L_bol ratio, using the same units. Assume you weigh 100 kg and radiate like a black body of area 2 m² at 300K   -- not unreasonable.

3. Imagine a big ball of 2×10²⁸ people -- a bizarre and fairly unpleasant notion -- it has about the same mass and size as the sun. Assume that people survive until they starve (they don't eat eachother!) so that the "lifetime" of this human star is about 1 week.

   (i) What would its luminosity be?

   (ii) Compare the total energy liberated by the sun and the human star integrated over their respective lifetimes.

   (iii) Compare this ratio to the ratio of typical electron binding energies (chemistry) with typical nuclear binding energies (fusion).

   Notice: solar type stars live so long for two reasons: (a) their fuel is indeed very energy rich; but (b) they burn it at an extremely frugal rate -- their furnaces are surprisingly feeble, per kg, well below even to your own metabolic rate.

4. Finally, use the stellar mass-luminosity relation (L M^3.5) to find out what star mass and spectral type has roughly the same M/L_bol as you.

   When your supervisor nexts asks you whether you have "fire in the belly" for your work, you can honestly reply, "more, even, than the sun and stars!"

(4) Alien Astronomers in Virgo study the Milky Way

In Topic 1.3b we learned that the Milky Way had an exponential disk scale length of 3.5 kpc with surface brightness near the sun (8.5 kpc) of 15 L_V, pc^-2, and a total luminosity of L_V = 1.4 x 10¹⁰L_V,. Estimate m_V & M_V for the MW as viewed from the Virgo cluster (distance 15 Mpc). What is the surface brightness of the disk (µ_V in magnitudes per square arcsec) at the radius of the sun and at the center; and what is the diameter, in arcsec, as defined by the 25^th V mag/ss isophote. Assuming the rotation curve stays flat at ~220 km/s outside the solar circle, what is the value of M/L_V measured inside the D₂₅ isophote?

(5) Concordance Cosmology :

Use the parameter diagrams given in the notes, 1.3m () to gain familiarity with some of the basic properties of the concordence model.
1. "More distant objects look smaller" Not true. Out to what redshift does this statement hold true ? Over what range of redshift does a 10 kpc galaxy appear to be between 1 and 1.5 arcsec in size ? What is the % error in using the small angle formula and linear Hubble law in estimating angular sizes at z ~ 0.1; 0.5; 1 ?

2. At what redshift was the expansion of the universe "coasting" (ie neither decelerating nor accelerating) ? At what fraction of the current age did this occur ?

3. What is the look-back time to a z ~ 7 quasar ? How long was it since the big bang, in Gyr and as a fraction of the current age ? How much denser was the universe at that time, and what was the CMB temperature ?

4. JWST will see galaxies at z ~ 10. If their diameters are ~20 kpc, how big will they appear to be ? Pushing further, LOFAR expects to see HI (21 cm) emission at z ~ 30 during the "Dark Age" (after recombination but before reionization). How big will 100 kpc structures appear at that time ? Yet further, at recombination (z ~ 1000) what physical scale corresponds to 1 degree (the first accoustic peak in the CMB power spectrum).

5. Recombination occurs at z ~ 1000. Roughly, what's the hydrogen density at this time (in cm^-3; allowing for the fact that baryons are only ~10% of the total matter density). What's the ratio of radiation density to hydrogen density at this time ? At what redshift and corresponding cosmic age is the total matter density equal to the radiation density ? What's the temperature at that time ?

(6) Scaled H_o :

A few examples of dealing with different values of h = H_o / 100.
1. What are the h dependencies of : diameter (from angular size); mass density (from dynamics); luminosity (from flux); mass (from dynamics); and M/L.
2. An old paper by Sandage uses H_o=55 km/s/Mpc to give the central luminosity density and mass density in a galaxy to be 10³ L_B,/pc³ and 10⁴ M/pc³. Express these values using "h" and evaluate them for H_o=72 km/s/Mpc.