Last update: Friday, May 7, 1999

This problem set is meant as a sample for the Final Exam, mostly covering the material that was not on the first two midterms. You may also complete it by the due date, and hand it in for extra credit. Note that you should also review the problems from the previous homeworks, as well as the reading assignments and/or other review materials that will be posted on the course page.

Observations

_{ }of star forming regions reveal that_{ }there are small dense nearly_{ }spherical clouds known as**Bok globules**._{ }A particular Bok globule is observed_{ }to have a radius of 0.5 pc and_{ }has a total mass (in H_{2}) of 250 M_{sun}. What is the average number_{ }density of molecular hydrogen n(H_{2}) in the globule? How many molecules are there_{ }total in the globule?If

_{ }the thermal energy per molecule of gas_{ }is given by 3kT/2, what is the total_{ }thermal energy (kinetic energy)*K*in the_{ }globule (in Joules), assuming a_{ }cloud temperature of 10 K?The

_{ }gravitational binding energy of_{ }a uniform density cloud_{ }is*U*= -3/5 · G M^{2}/ RCalculate the binding energy

*U*(in Joules) of the Bok globule.The

_{ }virial theorem states that for a stable,_{ }gravitationally bound system_{ }2 *K*+*U*= 0and

_{ }therfore if*U*< 2*K*then the_{ }system has enough gravitational_{ }binding energy to collapse (remember*U*is negative)._{ }Would you expect the Bok globule to collapse to_{ }form a cluster of stars, or to support_{ }itself with gas pressure?_{ }If it does collapse, calculate the collapse_{ }time (free-fall) in years._{ }If it does not collapse, how_{ }massive would it have to be in order_{ }to collapse?The

_{ }Sun's circular velocity around the center_{ }of the Milky Way_{ }galaxy is measured to be v_{c}= 220 km/s at its radius of R = 8 kpc._{ }What is the implied mass interior to this radius_{ }(in Msun)? How long does it take the Sun to_{ }make one revolution about_{ }the center of the galaxy?_{ }How many revolutions will it have_{ }made in the 4.5 Gyr of its life?_{ }(This can be thought of as_{ }its age in "galactic years".)_{ }Observations

_{ }of the 21-cm line of hydrogen in a_{ }direction 30 degrees from the galactic_{ }center (in the same side that the_{ }Sun is orbiting toward) show a maximum_{ }radial velocity (relative_{ }to the Sun) of v_{r}= 90 km/s. This is interpreted_{ }to come from an atomic hydrogen_{ }cloud moving with_{ }its circular velocity away_{ }from the Sun at the*tangent point*_{ }(the closest approach along_{ }that line of sight to the galactic center)._{ }Sketch the geometry_{ }of this observation as viewed_{ }from far above the_{ }disk of the galaxy. Be sure to_{ }indicate the position of the Sun,_{ }the galactic center, the_{ }angle corresponding to the galactic longitude of 30°,_{ }and the location of the cloud_{ }on the tangent line from us_{ }to its circle around the galactic center._{ }At

_{ }what radius r from_{ }the center of the_{ }galaxy is this point?_{ }The

_{ }measured radial velocity is the_{ }difference between the true circular_{ }velocity v_c(r) and the_{ }component of the Sun's_{ }motion in the same direction._{ }Compute the component_{ }of the Sun's velocity in this_{ }direction, and use it_{ }to correct the radial velocity_{ }v_r of the cloud to find_{ }the true circular velocity v_c(r)._{ }What is the implied mass of the_{ }galaxy interior to radius r?This

_{ }is how the rotation curve of_{ }the interior of the galaxy_{ }is built up.During

_{ }the course of its lifetime_{ }on and off the main sequence,_{ }a massive star will fuse its_{ }hydrogen all the way to iron._{ }If an iron-56 nucleus_{ }has a mass of 55.847 amu,_{ }what fraction of the mass_{ }of the original hydrogen_{ }nuclei is turned into energy_{ }during the course of this process?_{ }For

_{ }gas at solar abundance, approximately_{ }2% of it is_{ }in heavy elements like iron._{ }Calculate the luminosity in stars_{ }corresponding to turning 1 M_{sun}of hydrogen into 1 M_{sun}of gas 2% iron,_{ }assuming that the energy liberated_{ }is produced over the lifetime_{ }of a massive star ~ 10^{8}years._{ }The

_{ }**Great Attractor**is a large cluster_{ }or supercluster of galaxies_{ }located in the direction of_{ }the constellations_{ }of Hydra and Centaurus._{ }This region of the Universe is moving_{ }away from us with a recession velocity of 12000 km/s._{ }Assuming a Hubble constant of H_{0}= 60 km/s/Mpc, what is_{ }the distance to the Great Attractor_{ }in Mpc, and its redshift*z*?_{ }Our

_{ }galaxy, and the Local Group of galaxies_{ }of which we are part,_{ }has been shown to be_{ }moving*toward*the Great Attractor_{ }with a peculiar velocity of 570 km/s_{ }(measured against the Cosmic Microwave_{ }Background). Assuming that_{ }we are marginally bound to_{ }the Great Attractor_{ }(that is, we are moving with_{ }the escape velocity),_{ }what is the implied mass of the_{ }Great Attractor,_{ }in Msun.?The

_{ }absolute visual magnitude of a bright_{ }globular cluster in the Milky_{ }Way is M_{v}~ -10. What is the approximate_{ }luminosity of a globular cluster of_{ }this magnitue in Lsun?_{ }What

_{ }would the apparent_{ }visual magnitude of this globular cluster_{ }be if it were at a redshift_{ }of*z*= 0.1?_{ }(Assume H_{0}= 60 km/s/Mpc and a flat k=0 universe with =1.)_{ }If

_{ }our galaxy has a rotation curve_{ }that stays flat at a_{ }circular velocity of 220 km/s_{ }out to R = 20 kpc,_{ }what is the mass in M_{sun}enclosed within this radius?_{ }If

_{ }this enclosed mass collapsed from a sphere_{ }that originally had a_{ }radius of 100 kpc,_{ }how long would it have taken_{ }the galaxy to have formed if this collapse_{ }occured as free-fall?_{ }If

_{ }two black holes form a binary system,_{ }then they will spiral into each other as the orbital_{ }energy is carried away by gravitational waves_{ }- the*Laser Interferometric Gravity-wave Observatory*_{ }(LIGO) now being built is designed to search_{ }for these systems. What is the Schwarzschild_{ }radius of a 6 M_{sun}black hole, in km?_{ }In

_{ }general, when the black holes orbit decays_{ }to around 10 times_{ }the Schwarzschild radius,_{ }then the black holes_{ }will merge together in one_{ }orbit or less. What is the orbital_{ }period for two 6 M_{sun}black holes orbiting each_{ }other at a distance of 10_{ }Schwarzschild radii?The

_{ }COBE satellite has determined the_{ }temperature of the Cosmic_{ }Microwave Background_{ }to beT _{0}= 2.728 ± 0.004 K.Use

_{ }the Wien Law for the wavelength_{ }of the maximum emission to_{ }calculate the peak wavelength of the_{ }CMB radiation.At

^{ }a wavelength of 1 cm,^{ }what is the blackbody intensity in^{ }units of W m^{-2}Hz^{-1}sr^{-1}?^{ }Since the CMB is isotropic, what is the^{ }total flux density (in W m^{-2}Hz^{-1}) over the whole sky?^{ }Also, express this in terms^{ }of the normal radio astronomy^{ }flux density units of Janskys^{ }( 1 Jy = 10^{-26}W m^{-2}Hz^{-1}).Compare

_{ }this flux density to_{ }that received from the Sun and_{ }Jupiter (T=155 K). (Look_{ }up the radius and distance of_{ }Jupiter at closest approach_{ }to Earth.)Since

_{ }the microwave background_{ }bathes the Earth in its_{ }blackbody radiation (and as a consequence_{ }raises the Earth's temperature by 3 K!),_{ }this is the*one*case you can_{ }actually use the Stefan-Boltzmann formula_{ }to compute the total_{ }flux (in W m^{-2})._{ }Do so.The

_{ }age of the Universe_{ }at any given redshift, or_{ }the time interval between two_{ }different redshifts, depends_{ }upon the integral of the expansion_{ }rate of the Universe over this span._{ }In particular, for a critical universe with_{ }*k*=0 and_{0}=1dR/dt = [ (8G/3c ^{2}) u R^{2}]^{1/2}where

_{ }we have used the energy_{ }density u instead of the_{ }matter density. In the matter-dominated_{ }erau = c ^{2}and

_{ }in the radiation-dominated_{ }erau = a T ^{4}( a = 4/c = 7.56 x 10^{-16}J m^{-3}K^{-4})which

_{ }can be substitued in the dynamical equation_{ }above to obtain the differential_{ }equation dt = f(R) dR which in turn can_{ }be integrated to obtain t(R)._{ }Use the evolution of T(R) and_{ }(R) in terms of the_{ }current density_{0}and T_{0}, and the definition of scale factor_{ }with redshiftR _{0}/ R = ( 1 +*z*)to

_{ }obtain an numerical expression for_{ }t as a funtion of*z*in these_{ }two cases.The

_{ }cosmic microwave background was produced_{ }at a temperature of_{ }T _{rec}3600 Kwhen

_{ }the free electrons finally_{ }recombined with the protons_{ }and the universe became neutral_{ }instead of ionized._{ }Assume that the Universe_{ }was matter-dominated and calculate_{ }the redshift*z*and the look-back_{rec}_{ }time of recombination ( t_{0}- t_{rec}). Also estimate the_{ }time of recombination after the "big bang" t_{rec}under the assumption_{ }of matter domination._{ }At

^{ }a redshift only a factor of^{ }ten higher than that of recombination,^{ }the universe was still^{ }radiation dominated, as it was^{ }at all earlier times.^{ }Physicists predict that there^{ }is an as-yet undiscovered particle, called the^{ }Higgs boson, which is extremely important^{ }in setting the masses of the lighter particles.^{ }It is expected to have a mass^{ }of 500 GeV c^{2}( 5 x 10^{11}eV c^{2}, in energy units ). At what temperature^{ }would you expect the Higgs to be^{ }created and destroyed regularly in particle-antiparticle^{ }pairs (ie. when is kT ~ mc^{2})?Compute

_{ }the corresponding redshift*z*and time t_{Higgs}_{Higgs}after the Big Bang.Fo

_{ }r a summary of particle physics_{ }and new particle searches, see_{ }*http://pdg.lbl.gov/*.Is

_{ }there a particle that is its_{ }own black hole? The Schwarzschild_{ }radius for an object of mass m is_{ }R _{s}= 2 G m / c^{2}and

_{ }the de Broglie wavelength for asub> relativistic particle of mass m issub>= h / m c . Calculate

_{ }the mass m_{pl}for which= R _{s}(the

_{ }factor of pi is just convention) --- this_{ }is known as the*Planck mass*._{ }If

_{ }you have an energy of E_{pl}= m_{pl}c^{2}, you can create a Planck_{ }mass particle._{ }At what temperature_{ }will the radiation energy_{ }of the universe kT be_{ }equal to E_{pl}?_{ }At

_{ }what redshift*z*_{pl}will_{ }the Universe be_{ }able to make Planck_{ }mass_{ }particles at will?_{ }This

_{ }is a time called the "Planck epoch"_{ }and at this earliest_{ }stage of the Universe the quantum_{ }nature of gravity is_{ }important and the notions of space and_{ }time as we know_{ }them break down. This_{ }is effectively the earliest_{ }moment of the Big Bang_{ }that we can understand._{ }

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*Steven T. Myers*