• Summary
• Reading List
• Problem Set
• Answers

Week 1 - The Discovery of Neutron Stars

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.

Answer for Problem 1

The outgoing flux is given by

This yields a outgoing force of

Meanwhile the force due to gravity is

Setting the sum of forces to zero and solving for L yields

LEdd =

4 π c G M mp σT

= 1.3 x 1038 erg s-1

M

Problem 2 - Minimum Masses

The observations of Sco X-1 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?

Answer for Problem 2

Sco X-1

1. Because the source is so close in the sky to the galactic center, a good guess for its distance is the distance to the center of the galaxy, 10 kpc.
2. The paper quotes a flux of about 5 photons cm^-2s^-1 at a wavelength of 3 Å. This gives an energy flux of 3.3 x 10^-8 erg cm^-2s^-1. Using 10 kpc as the distance yields a luminosity of 4 x 10³⁹ erg/s.
3. Using the Eddington luminosity calculated in Problem 1 yields a mass estimate of 3 .