Topics ASTR 3730: Fall 2003

Transcription

1 Topics Qualitative questions: might cover any of the main topics (since 2nd midterm: star formation, extrasolar planets, supernovae / neutron stars, black holes). Quantitative questions: worthwhile to review: relation between magnitudes / fluxes optical depth atomic processes: Boltzmann law, energy levels, bremsstrahlung dynamics of binaries (covered in lecture #13) hydrostatic equilibrium different time scales (free-fall time scale etc) virial theorem Eddington limit

2 You will not be asked to rederive basic relations from scratch (e.g. you might be asked to integrate the equation of hydrostatic equilibrium, but not `prove the virial theorem starting from the equation of hydrostatic equilibrium ). No questions: based on subtleties of definition of specific intensity vs flux on derivation of degeneracy pressure formulae on energy transport in stars (either radiation diffusion or conditions / stability for convection) requiring recitation of specific nuclear reactions (ideas about mass differences -> energy or binding energy curve might be asked) derivation of Chandrasekhar mass - just need to know bottom line for that

3 Two basic relations: F(r) = L 4pr 2 m = -2.5logF + constant x = log y Æ y =10 x x = ln y Æ y = e x log(ab) = log(a) + log(b) Ê logá aˆ = log(a) - log(b) Ë b log(a b ) = blog(a) Radiation Inverse square law: F(r) is the flux of energy at distance r from a source with luminosity L (usually bolometric, but could be in some band). relation between magnitude and flux Questions tend to involve calculating changes in F, and relating that to differences in m (or vice versa). Almost never need to know (i) the value of the `constant, or (ii) the absolute distance to the source.

4 Opacity and optical depth Consider radiation traveling through an absorbing medium. The intensity of the radiation obeys the equation: di ds = -rki k is the opacity, units cm 2 g -1 s is the distance along the path the radiation takes minus sign as the intensity is diminished by absorption I and k are normally functions of frequency, unless the absorption is due to scattering of the radiation by electrons (this is why the Eddington limit does not depend upon frequency). For constant density and opacity, integrate to get: I = I 0 e -rkds = I 0 e -t General definition of optical depth: dt = rkds

5 Optical depth: is dimensionless, just a number quantifies `how much of the radiation is absorbed by the gas or dust doing the absorbing In simple cases where the density and opacity are known, easy to work out: e.g. X-ray radiation from an X-ray binary passes through a cloud of fully ionized gas with r = g cm -3 and size 10 9 cm. What fraction of the flux is absorbed by the cloud? I 0 I Opacity for fully ionized gas will be the electron scattering opacity: k = 0.4 cm 2 g -1 Ds = 10 9 cm

6 Procedure: calculate t using the formula (in this case, no integration is needed, but in general you might need to integrate) get ratio I / I 0 from the solution to the radiative transfer equation In simple cases like this, don t need to distinguish between the intensity and the flux. So if you had to calculate the change in the magnitude due to the absorption, assume F / F 0 = I / I 0.

7 Radiative processes Bound-bound Bound-free Free-free E i E j Atomic transitions: hn = E i - E j nl = c Energies in ev or kev: 1 ev = erg Ionization or recombination Bremsstrahlung - important at high T when all the atoms are ionized and we have a plasma with ions + electrons. All three processes can involve either the emission or absorption of photons

8 Difficult to calculate the probability of particular atomic transitions. Questions normally require you to calculate the relative number of atoms with electrons in different levels: n 2 = g 2 e - ( E 2 -E 1 ) / kt n 1 g 1 Boltzmann law where g 1 and g 2 are the statistical weights or degeneracy of the two energy levels. For hydrogen g n = 2n 2 - you will need to be given this to do a question. Questions about bremsstrahlung often involve working out how much emission there is, using the formula (for hydrogen): e ff = T 1 2 n e 2 erg s -1 cm -3 rate per unit volume - need to multiply by volume or integrate over volume to get total bremsstrahlung L for hydrogen: n e = r / m H

9 Time scales for stars Dynamical or free-fall time scale: time scale on which star would collapse if pressure support lost (directly relavent to core collapse or star formation, but can be calculated for any situation). Calculate escape velocity: 1 2 mv 2 = GMm R M is the mass enclosed within radius R of the center of the star v esc = 2GM R Define the free-fall time scale as the time scale on which mass would fall to the center at the escape velocity: t free- fall = R v esc = R 3 2GM Free-fall or dynamical time scale, ~same as orbital time scale at that radius

10 Kelvin-Helmholtz time scale: t KH U L Time scale for the star to radiate all its thermal energy U at present luminosity L Use the virial theorem to get U, won t normally be able to calculate L ab initio so this is usually given. Nuclear time scale: t nuc E nuc L Sets main sequence lifetime. E nuc = mass of hydrogen in core x energy release per gram of H

11 Basic form: 0 = W + 3( g -1)U Virial theorem For an ideal gas (g = 5 / 3): 0 = W + 2U applies provided the star isn t presently involved in some dynamical time scale event (e.g. collapsing) W is the gravitational potential energy U is the thermal energy normally applied to the whole star In an application, you will normally need to: (i) combine with equation for total energy (which may or may not be conserved): E total = W + U (ii) calculate W: W = - Gm r 4pr2 rdr R Ú 0 W = - GM 2 R for an estimate if r(r) is specified

12 gradient of pressure Hydrostatic equilibrium dp dr = - Gm r r 2 gravitational force To apply this equation, need: gravitational acceleration g in spherical co-ordinates an equation of state: P = P(r) expression for mass m(r) enclosed within radius r, if gas we are considering has significant mass This will give a differential equation for the density r, which will need to be integrated with appropriate boundary conditions (normally P = 0 at r = R, dp / dr = 0 at r = 0).