The

_{ }galaxy NGC 2639 is an Sa galaxy with a measured maximum rotational_{ }velocity of 324 km/s and an apparent magnitude of B = 12.22 mag_{ }(after making corrections for any extinction)._{ }- (a)
- The
_{ }B-band Tully-Fisher relation for Sa spirals is

M_{B}= -9.95 log v_{max}+ 3.15

_{ }maximum velocity*v*in km/s. Use this relation to_{max}_{ }estimate the absolute B magnitude of NGC 2639._{ } - (b)
- Determine
_{ }the distance to NGC 2639 using its distance modulus_{ }. - (c)
- It
_{ }is customary to measure a radius R_{25}to where a spiral_{ }galaxy's surface brightness falls_{ }to a level of 25 B-mag/arcsec^2._{ }Spiral_{ }galaxies appear to follow a standard relation_{ }

log R_{25}= -0.249 M_{B}- 4.00

_{25}in kpc. What is R_{25}for NGC 2639 in kiloparsecs?_{ } - (d)
- In
_{ }the B band, the night sky has a brightness of_{ }22 B-mag/arcsec^2. If you want_{ }to measure a galaxy profile down to R_{25}(25 B-mag/arcsec^2), how many times*fainter*_{ }than the sky background is this?_{ }This is why its hard to_{ }do photometry from the ground._{ } - (e)
- To
_{ }measure the mass of a galaxy, we can use Newton's Law (or Kepler's Law)_{ }

v^{2}= GM / R

_{ }give the mass inside some radius R given_{ }a rotational velocity v._{ }Find the mass in Msun within R_{25}for NGC 2639,_{ }assuming that the rotation curve_{ }is flat outside a few kpc._{ } - (f)
- The
_{ }Sun has an absolute visual magnitude of M_{V}= 4.83 and_{ }a color index of B - V = 0.64._{ }Find the blue absolute magnitude M_B of the Sun, and use this to determine the luminosity_{ }of NGC 2639 in the B band, in Lsun._{ } - (g)
- Compute
_{ }the mass-to-light ratio of NGC 2639_{ }in the B band (in Msun/Lsun)_{ }interior to R_{25}, using the numbers found above._{ }Is this consistent with what we have_{ }learned about spiral galaxies?_{ }

The

_{ }Virgo cluster of galaxies is the nearest substantial cluster._{ }This cluster extends over nearly 10 degrees_{ }on the sky,_{ }and contains a number of bright galaxies._{ }Our first task is_{ }to find the distance to Virgo._{ }The following table_{ }gives a list of distance_{ }indicators and the results and uncertainties_{ }d_{i}± s_{i}_{ }(mean d, standard deviation s) for the_{ }Virgo cluster (adapted from Jacoby et al. 1992)._{ }More details on these specific_{ }indicators can be found here._{ }Method Virgo Distance (Mpc) 1. Cepheids 14.9 ± 1.2 2. Novae 21.1 ± 3.9 3. Planetary Nebula Luminosity Function 15.4 ± 1.1 4. Globular Cluster Luminosity Function 18.8 ± 3.8 5. Surface Brightness Fluctuations 15.9 ± 0.9 6. Tully-Fisher relation (spirals) 15.8 ± 1.5 7. Faber-Jackson/D-sigma relation (ellipticals) 16.8 ± 2.4 8. Type Ia Supernovae 19.4 ± 5.0 Derive

_{ }an average distance by computing the_{ }*weighted mean*_{ }d _{avg}= (d_{i}/_{i}^{2}) / (1/_{i}^{2})with

_{ }the sums running over the eight_{ }methods i = 1 to 8. Note that_{ }this is derived by minimizing chi-squared_{ }= ( d _{i}- D /_{i})^{2}and

_{ }setting d_{avg}= D at the minimum_{ }where d/dD = 0._{ }The

_{ }uncertainty in our estimator is given by_{ }finding where chi-squared_{ }increased by 1 above the minimum._{ }This is given by_{ }_{d}^{2}= 1 / (1/_{i}^{2})assuming

_{ }the uncertainties really reflect Gaussian_{ }errors. What is the_{ }uncertainty in our average_{ }distance estimate to Virgo, in Mpc?_{ }Spectra

_{ }of galaxies in the Virgo cluster indicate an average recession velocity of_{ }1136 km/s for the cluster._{ }When combined with the estimated_{ }distance from Problem 2,_{ }what is the value of the_{ }Hubble constant H_{0}, assuming_{ }v _{r}= H_{0}· rfor

_{ }the Hubble relation?Because

_{ }Virgo represents a significant mass_{ }in our corner of the Universe,_{ }we must correct the velocity_{ }of Virgo for our motion_{ }*towards*the cluster,_{ }estimated to be around 168 km/s._{ }What does this correction give us_{ }for the value of H_{0}?_{ }Recent

_{ }parallaxes from the Hipparcos_{ }satellite give some indication that our_{ }distance scale may be too short,_{ }and should be increased by 10%._{ }If we make this correction to our_{ }average distance to Virgo, what_{ }is the new value of_{ }the Hubble constant that we derive?_{ }What

_{ }is the*Hubble time*_{ }t _{H}= H_{0}^{-1}in

_{ }years, assuming this value_{ }of H_{0}?The

_{ }Virgo cluster contains hundreds of galaxies moving_{ }around under the influence of its_{ }gravitational potential._{ }The radial velocity dispersion_{ }of the Virgo cluster galaxies is_{ }_{r}= 666 km/s.The

_{ }virial theorem implies_{ }M = 5 _{r}^{2}R / Gwhere

_{ }the factor 5 comes from assuming Virgo is a_{ }uniform density sphere._{ }Using this relation, and_{ }assuming that the Virgo cluster is_{ }approximately a sphere of radius R = 1.5 Mpc,_{ }what is the total gravitational_{ }mass of the Virgo cluster, in Msun?_{ }How

_{ }long, in years, would it take a galaxy to cross_{ }the Virgo cluster?_{ }Assume the*rms*velocity_{ }_{r}. How does this compare to the Hubble time?_{ }What

_{ }is the Hubble velocity difference between_{ }opposite points on a_{ }diameter of the Virgo cluster?_{ }How does this compare to_{ }_{r}? How is this (the Hubble velocity over that distance)_{ }related to the timescales we just computed?Like

_{ }many large cluster such as Coma, the Virgo_{ }cluster contains a hot_{ }intergalactic medium (IGM),_{ }which is gas at an extreme_{ }temperature of 70 million K._{ }This means that Virgo_{ }is emitting X-rays via the*thermal bremsstrahlung*(free-free) process that we discussed in_{ }class for the much cooler H II regions._{ }- (a)
- The
_{ }luminosity comes from electron-ion interactions, and_{ }can be shown to have an volume emissivity_{ }

_{ff}= 1.42 × 10^{-40}( n_{e}/ 1 m^{-3})^{2}( T / 1K )^{1/2}W/m^{3}.

_{ }the X-ray luminosity of Virgo is L_{x}= 1.5 × 10^36 W, what_{ }is the average electron number density n_{e}(in m^-3) in the cluster IGM?_{ } - (b)
- Again
_{ }modeling Virgo as a uniform density sphere_{ }of radius 1.5 Mpc, what is the total_{ }mass (in Msun) of the IGM_{ }in the cluster, assuming that_{ }it is composed entirely of ionized hydrogen?_{ } - (c)
- What
_{ }fraction of the gravitational mass is_{ }accounted for by the hot IGM?_{ } - (d)
- If
_{ }the optical luminosity of the galaxies in the_{ }cluster give a total of L_{V}= 1.2 × 10^12 Lsun,_{ }what is the mass-to-light ratio_{ }in the V-band for the Virgo cluster?_{ }If an "average" galaxy has a mass-to-light_{ }ratio of 10 in V-band,_{ }what is the approximate mass_{ }in galaxies in the cluster?_{ }How does this compare to the_{ }hot gas mass of the IGM?_{ } - (e)
- For
_{ }the IGM gas, the average kinetic thermal_{ }energy per particle is_{ }

K = 3 k T / 2

_{ }both the electrons and protons. How long will it_{ }take to radiate this energy away in X-ray emission at the rate_{ }given above? How does this_{ }*cooling time*_{ }compare to the Hubble time?_{ }(Note: if the cooling time_{ }is less than the Hubble time,_{ }the cluster would cool off and stop emitting_{ }X-rays before it can grow in size._{ }If the cooling time is longer_{ }it can stay hot indefinitely_{ }as it gets heated by infalling galaxies.)_{ }

How

_{ }much energy can we get out of_{ }an accreting supermassive black hole?_{ }To answer this, we recycle_{ }the idea of the*Eddington luminosity*_{ }that we used to calculate_{ }the limiting luminosity of a big star:_{ }L _{Ed}= 4 G M c /while

_{ }for electron (Thomson) scattering_{ }= _{T}/ m_{H}0.04 m^{2}kg^{-1}which

_{ }is just another way of saying that if_{ }the luminosity is too high,_{ }then the radiation pressure_{ }overcomes the gravitational_{ }attraction of the massive_{ }object (see Homework Set 4,_{ }Problem 3)._{ }Calculate the_{ }Eddington luminosity_{ }for a 10^8 Msun black hole._{ }The

_{ }energy comes from the gravitational binding_{ }energy of the accreting matter._{ }Derive an expression_{ }for the luminosity L of matter_{ }accreting at a rate_{ }dM/dt (in Msun per year) from_{ }infinity to a radius equal to_{ }R _{acc}= q R_{sw}(a

_{ }multiple q of the Schwarschild radius_{ }) for our black hole of mass M._{ }Equating this to the Eddington luminosity_{ }for our black hole (scale to M / 10^8 Msun,_{ }find an expression for the_{ }*Eddington accretion rate*dM/dt(Ed) in_{ }terms of q and M / 10^8 Msun._{ }What is the Eddington accretion rate_{ }(in Msun per year) for our_{ }10^8 Msun black hole, assuming the gas gets to 3 R_{sw}(q=3)?_{ }The

_{ }Eddington accretion rate is the maxiumum rate_{ }you can feed a black hole as_{ }radiation pressure keeps_{ }you from feeding it faster._{ }Furthermore, observations show that active_{ }galactic nuclei (AGN) are emitting light_{ }at just this rate._{ }Therefore, this is how fast you must_{ }be feeding stars into a_{ }supermassive black hole in a_{ }quasar or radio galaxy_{ }to keep it luminous!_{ }

*smyers@nrao.edu*
*Steven T. Myers*