Astronomy 1 / Section 3 (S. Myers)

Problem Set #9 (due Mon 29 April 1996 5pm)

1. For a Hubble constant of H_0 = 82 km/s/Mpc as determined by the HST, the critical density in our universe is

   3 H_0² / 8 Pi G = 1.26 x 10^-26 kg/m^3 What density is this in the units of Msun/ Mpc^3? From what you read about the average mass of galaxies, and their average spacing in the Universe (Chapters 13 and 15), does it seem likely that the average density of the Universe is higher than, equal to, or lower than this critical density? What does this imply for the fate of the Universe?

2. The age of a Universe expanding at a constant rate is given by the inverse of the Hubble constant, t_0 = 1/H_0. Calculate this age (in years) for the HST determined Hubble constant value of H_0 = 82 km/s/Mpc. (Note: you will have to convert the Mpc in the Hubble constant into km so it will cancel out.)

   A Universe with a constant expansion rate is equivalent to a Universe without any matter in it! Since we know we exist, we had better consider models with matter. The presence of mass in the Universe slows the expansion with time, so the age is shorter than in an empty universe. For a ``flat'' universe with the critical density , the age is given by t_0 = 2/3H_0. Calculate this age (in years) for H_0 = 82 km/s/Mpc.

   The main-sequence turn-offs of the oldest globular clusters give an age of around 15 billion years (1.5 x 10^10 yrs). What value of H_0 for a flat cosmology would be needed to give this age? This is what is called the ``age problem'' in cosmology - the Universe cannot be younger than its oldest denizens!

3. Use the blackbody formula for the wavelength of the maximum emission (see Seeds Chapter 6, or Lecture 20 in the notes) to calculate the peak wavelength of the cosmic microwave background radiation with T = 2.7 K. Give this wavelenth in sensible units, like millimeters (ie. convert from nanometers).

4. When we talked about blackbody radiation, we showed that the temperature was related to the average energy of gas particles, and thus the average energy of the photons emitted by the gas. We can write this energy as E = k T where k is Boltzmann's constant (k = 1.38 x 10^-23 J/K if you want energy in Joules, or 8.63 x 10^-5 eV/K, if you want energy in electron volts). The mass of the electron is m_e = 9.11 x 10^-31 kg. What is the rest mass energy of the electron (using E = m c^2)? When the temperature of radiation is high enough, two photons can come together and actually create a particle-antiparticle pair converting the photon energy into mass! At what temperature T is the photon energy kT equal to the rest mass energy of the electron, so electrons and positrons can be created? If the temperature of the radiation in the universe is now 3 K, at what redshift z is the universe at this temperature? Use the relation for the redshift evolution of the temperature

   T = T_0 ( 1 + z ) (Note: this temperature occurs when the Universe is about 4 seconds old!)

   The mass of the proton is m_p = 1.67 x 10^-27 kg. What is the rest mass energy of the proton, and what temperature does this corresond to? At what redshift would the universe be hot enough to make protons and antiprotons? (Note: this temperature occurs when the Universe is less than 10^-4 seconds old!)

Be sure to visit the DRL rooftop Student's Observatory on a clear night before the end of the term, or the Flower & Cook observatory during on the last field trip this Wednesday 14 April 1996.

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