• Summary
• Reading List
• Problem Set
• Answers

Week 1 - The Discovery of Neutron Stars, Black Holes and Gamma-Ray Bursts [2]

Summary

The first evidence for neutron stars, black holes and gamma-ray bursts was uncovered during the sixties. The stability and frequency of radio pulsars alone was sufficient to make a convincing argument that they were neutron stars. The energetics alone of gamma-ray bursts took nearly three decades to determine, and the nature of a fraction of them is still completely unconstrained.

• Monday 5 January: Organizational details
• Wednesday 7 January,
  Friday 9 January: LMXBs, HMXBs, GRBs, QSOs, PSRs

I encourage you to follow check out this interesting take on the discovery of pulsars: A Science Odyssey: On The Edge: Little Green Men.

Reading List

• ``Evidence for X Rays from Sources outside the Solar System''
  [ ADS, APS, PDF ]
  REF: Giacconi, R., Gursky, H., Paolini, F., Rossi, B. 1962, Physical Review Letters, 9, 439 .

• ``3C 273 : A Star-Like Object with Large Redshift''
  [ PDF ]
  REF: Schmidt, M. 1963, Nature, 197, 1040 .

• ``Red-Shift of the Unusual Radio Source : 3C 48''
  [ PDF ]
  REF: Greenstein, J. L., Matthews, T. A. 1963, Nature, 197, 1041 .

• ``Observations of a Rapidly Pulsating Radio Source''
  [ PDF ]
  REF: Hewish, A., Bell, S. J., Pilkington, J., Scott, P. F., Collins, R. A. 1968, Nature, 217, 709 .

• ``Observations of Gamma-Ray Bursts of Cosmic Origin''
  [ ADS, PDF ]
  REF: Klebesadel, R., Strong, I. B., Olson, R. A. 1973, Astrophys. J. Lett., 182, L85 .

Problem Set

Problem 1 - The Eddington Luminosity

There is a natural limit to the luminosity a gravitationally bound object can emit. At this limit the inward gravitational force on a piece of material is balanced by the outgoing radiation pressure. Although this limiting luminosity, the Eddington luminosity, can be evaded in various ways, it can provide a useful (if not truly firm) estimate of the minimum mass of a particular source of radiation.

1. Consider ionized hydrogen gas. Each electron-proton pair has a mass more or less equal to the mass of the proton (m_p) and a cross section to radiation equal to the Thompson cross-section (σ_T).
2. The radiation pressure is given by outgoing radiation flux over the speed of light.
3. Equate the outgoing force due to radiation on the pair with the inward force of gravity on the pair.
4. Solve for the luminosity as a function of mass.

Problem 2 - Minimum Masses

The observations of Sco X-1 and the quasars 3C 48 and 3C 273 can give a lower limit on the mass of the sources if they are gravitationally bound.

The source discovered by Giacconi et al. is now known as Sco X-1.

1. What is the most likely distance to Sco X-1 given its location on the sky?
2. At this distance given the flux estimate in the Giacconi et al., what is the luminosity of Sco X-1?
3. What is the minimum mass of Sco X-1?

You can estimate distances to 3C 48 and 3C 273 using the redshift of the objects. At the time, the Hubble constant was thought to be around 100 km/s/Mpc. The conventional wisdom is that it is 72 km/s/Mpc. Feel free to use either value.

1. What are the distances to 3C 48 and 3C 273? Feel free to neglect cosmological effects (for bonus points use Ω_M = 0.3 and Ω_Λ=0.7).
2. Do your distances agree with those in the papers?
3. The paper gives estimates of the luminosity in the optical and radio of these sources. What are the minimum masses of the objects using the optical and radio luminosities?

Problem 3 - Gamma-ray Burst Energetics

About how much energy is released in a gamma-ray burst if they are

1. In low Earth orbit?
2. At the distance of the moon?
3. At the disance of Pluto?
4. Near the center of the galaxy?
5. At cosmological distances?

What are the Eddington limiting masses for each of these scenarios? In which scenarios, would you be surprised if a gamma-ray burst repeated?

Estimate the amount of energy released by the following events. Use the formula GMm/R.

1. A nuclear explosion - 5 kg of fissile plutonium. Each plutonium atom releases about 200 MeV as it splits.
2. The collision of two asteroids. Assume each is 10 km in radius with a density of 3 g cm^-3.
3. The collision of two neutron stars. Assume each is 10 km in radius with a density of 10¹⁵ g cm^-3.
4. The collision of an asteroid with a neutron star.