Physics Made Easy

Astrophysics IV

Star forming regions
Massive stars
are born in warm molecular clouds; because of their size, they shape the environment around them through processes such as photoionisation, stellar winds and supernovae. It may also be the case that massive stars trigger or terminate the formation of less massive stars.
Low mass stars are born in cold, dark molecular clouds, Bok Globules and maybe also near massive stars. Recently, it seems that low-mass stars are born in cluster-like environments; however, it seems that these clusters do not survive as most low-mass stars are not found in clusters.

Stellar collapse (low mass): cool molecular H2 cores collapse when their mass exceeds the Jeans mass M_J\simeq 6M_O(\frac{T}{10K})^{\frac{7}{2}}(\frac{n_{H_2}}{10^{10}m^{-3}})^{-\frac{1}{2}}

The collapse can be triggered by loss of magnetic support, collision with other cores or compression due to nearby supernovae. The collapse is isothermal (and ‘inside out’) so there is efficient radiation of energy from ~106RO to ~5RO over a timescale of 105-106 yr. The collapse stops when the material becomes optically thick and can no longer remain isothermal. This is a protostar.

Conserving angular momentum: to ‘store’ the angular momentum once the star has formed, stars do not generally form on their own. Instead, most stars will form together with planetary systems or stellar companions.

Note that stars start off by rotating rapidly; stars like the Sun are slowed by magnetic braking.

Pre-main sequence phase: when Tc~106K, an accreting protostar will start burning deuterium. When deuterium burning is finished, the star contracts until Tc reaches ~107K, at which point hydrogen burning starts and the main sequence lifetime begins.

White dwarfs are the endpoints of stars with M≤1.4MO. Supported by electron degeneracy pressure, they are usually made of carbon and oxygen (although there are some He and O-Ne-Mg stars).

White dwarf mass-radius relations (see also degeneracy on the previous page)

Non-relativistic degeneracy P\sim\frac{GM^2}{R^4}\propto (\frac{\rho}{m_em_H})^{\frac{5}{3}}

R\propto \frac{1}{m_e}(\mu_em_H) ^{\frac{5}{3}}M^{\frac{-1}{3}}

R decreases with increasing mass; ρ increases with M.

Relativistic degeneracy- M independent of R.

There is a maximum mass beyond which collapse occurs. We will find out what it is below.

Chandrasekhar mass (limiting mass for white dwarfs): consider a star, radius R, containing N fermions of mass m_f=\mu_fm_H (as always, μ is mean molecular mass, in this case per fermion).

Number density n\sim\frac{N}{r^3}; volume per fermion~1/n

Heisenber uncertainty principle \Delta x\Delta p\sim \bar{h} \Rightarrow p\sim\bar{h}n^{\frac{1}{3}}

Relativistic Fermi energy E_F=pc\simeq c\bar{h}n^{\frac{1}{3}} \sim \frac{\bar{h}cN^{\frac{1}{3}}}{R}

GPE per fermion E_g\sim \frac{-GMm_f}{R}\simeq\frac{GM\mu_fm_H}{R} where M=N\mu_fm_H

Total energy per particle E=E_f+E_g=\frac{\bar{h}cN^{\frac{1}{3}}}{R}-\frac{GN(\mu_fm_H)^2}{R}

The stable configuration is where E is minimised.

If E<0, E can be decreased without bound by decreasing R– therefore there is no stable equilibrium point and the star undergoes gravitational collapse.

The point where E=0 gives the maximum value of N for which there is no collapse.

N_{max}\sim(\frac{\bar{h}c}{G(\mu_fm_H)^2)})^{\frac{3}{2}}

With rigorous calculation, we get N_{max}=1.457(\frac{2}{\mu_e})^2M_O– this is the Chandrasekhar mass.

Evolution of massive stars (M≥1.3MO): we already mentioned that more massive stars can continue to fuse heavier elements up to iron. The core undergoes alternate burning and contraction phases until an iron core is formed, with fusion of lighter elements occurring in onion-like shells.

Carbon burning (T~6×108K) C -> Ne, Na, Mg

Oxygen burning (T~109K) O -> Si, P, S, Mg

Silicon burning Si -> Fe

Core collapse supernovae are triggered after the exhaustion of nuclear fuel in the core of a massive star if the iron core mass > Chandrasekhar mass. Gravitational energy from the collapsing core provides energy for the explosion, most of which is emitted in the form of neutrinos. By an unknown mechanism, ~1% of the energy is deposited in the stellar envelope, which is blown away (ejected), leaving a compact remnant (neutron star or black hole).

Thermonuclear explosions: if an accreting CO white dwarf reaches the Chandrasekhar mass, the carbon is ignited under degenerate conditions. The nuclear burning raises T, but not P, and just as with the helium flash we get thermonuclear runaway. This results in incineration and complete destruction of the star; with 1044 J of nuclear energy release, no remnant is expected.

These explosions are the main producers of iron and also act as standard candles (which we discussed in the Cosmology section).

Supernova classification: there are many different types of supernova, and if truth be told, not even the experts can decide how to classify them. For that reason, only a general overview will be given here.

  1. Type I: no hydrogen lines in spectrum
  2. Type II: hydrogen lines in spectrum

Theoretical types: the thermonuclear and core collapse models discussed above.

The relationship between these types is no longer 1:1.

Neutron stars are the end products of core collapse of massive stars (between 8 and ~20MO). In the collapse, all nuclei are dissociated to produce a very compact remnant mainly composed of neutrons (with a few protons and electrons). The typical radius of the remnant is ~10km, with a density of ~1018 kgm-3.

Just like with white dwarfs, the fact that neutrons, like electrons, are fermions leads to a maximum possible mass which can be supported by neutron degeneracy- this mass is estimated to between 1.5-3MO.

The dissociation of the nuclei is endothermic, using some of the gravitational energy released in the collapse. These reactions undo all the previous nuclear fusion reactions.

Schwarzschild black holes: a black hole has a density such that the escape velocity is greater than the speed of light (and since not even light can escape, of course it’s going to look black).

Take photons to have ‘mass’ m=\frac{E}{c^2}

Escape velocity v_{esc}=\sqrt{\frac{2GM}{R}}

If vesc > c, R<Rs

Schwarzschild radius R_s=\frac{2GM}{c^2}=3km(\frac{M}{M_O}

End states of stars: to summarise, there are three main possibilities-

  1. Star develops a degenerate core, nuclear burning stops and the stellar envelope is lost -> degenerate (white) dwarf.
  2. Star develops a degenerate core and ignites nuclear fuel explosively -> complete disruption in a supernova.
  3. Star exhausts all of its nuclear fuel and the core exceeds the Chandrasekhar mass -> core collapse and a compact remnant (neutron star or black hole).

Summary- final fate as a function of initial mass (not the mass of the end state)

M≤0.08MO: No hydrogen burning, supported by degeneracy pressure and Coulomb forces -> planets, brown dwarfs.

0.08-0.48 MO: hydrogen burning but no helium burning -> degenerate helium dwarf.

0.48-8 MO: hydrogen and helium burning -> degenerate CO dwarf

8-13 MO: complicated burning sequences, no iron core -> neutron star

13-80 MO: iron core, core collapse -> neutron star or black hole

M≥80MO: pair instability, complete disruption -> no remnant

Binary stars: as explained earlier, to conserve angular momentum, most stars are members of binary (or multiple star) systems. The orbital period of such systems ranges from 11 minutes to 106 years.

Most binaries are far apart with little interaction- in close binaries (P<10yr), mass transfer from one star to another can occur.

Classification of binaries

  1. Visual binaries: we can see the periodic wobbling of two stars as they orbit their centre of mass. If the motion of only one star is seen, it is known as an astrometric binary.
  2. Spectroscopic binaries: we see the periodic Doppler shifts of spectral lines. If the lines of both stars are detected, it is known as double-lined; if the Doppler shifts of only one star are detected, it is known as single-lined.
  3. Photometric binaries: the periodic variations of fluxes, colours etc are observed. The trouble here is that the same sort of variations can be caused by single variable stars (Cepheids, RR Lyrae variables).
  4. Eclipsing binaries: if the inclination of the orbital plane is ~90o, one or both stars are eclipsed by the other one at some point during their orbit.

Binary mass function

Radial velocity \frac{v_r}{c}=\frac{\Delta \lambda}{lambda} from Doppler shift

M_1a_1=M_2a_2 from position of COM in COM frame.

Period P=\frac{2\pi}{\omega}=2\pi\frac{a_1\sin{i}}{v_1\sin{i}}=2\pi\frac{a_2\sin{i}}{v_2\sin{i}}

a=a1+a2

Gravitational force=centripetal force

\frac{GM_1M_2}{(a_1+a_2)^2}=\frac{M_1(v_1\sin{i})^2}{a_1\sin{i}}

\frac{GM_1M_2}{(a_1+a_2)^2}=\frac{M_2(v_2\sin{i})^2}{a_2\sin{i}}

 

(a_1+a_2)^2=a_1^2(1+\frac{M_1}{M_2})^2=(\frac{a_1}{M_2})^2(M_2+M_1)^2

Substituting for a1sin i, (a1+a2)2 etc (basically eliminating all the a terms), leads to

f_1(M_2)=\frac{M_2^3\sin^3i}{(M_1+M_2)^2}=\frac{P(v_1\sin{i})^3}{2\pi G}

f_2(M_1)=\frac{M_1^3\sin^3i}{(M_1+M_2)^2}=\frac{P(v_2\sin{i})^3}{2\pi G}$

These are the mass functions- they relate observables like the period and velocity to quantities of interest like masses M1, M2 and angle i.

For a double-lined spectroscopic binary, we can measure f1 and f2 and hence determine M1sin3i and M2sin3i

For visual and eclipsing binaries, i is known, so we can determine M1 and M2.

For M_1 \ll M_2, f_1(M_2)\simeq M_2\sin^3 i

Measuring v1sin i constrains M2

For eclipsing binaries, the radii of both stars can also be determined. This is the main source of accurate masses and radii of stars (plus luminosity if the distance to the binary is known).

Roche potential: here we have a restricted three-body problem where we determine the motion of a test particle in the field of two masses M1 and M2 in a circular orbit about each other.

Equation of motion of the particle in a frame rotating with the binary masses

\frac{d^2r}{dt^2}=-\nabla U_{eff}-2\underline{\Omega}\times\underline{v}

where Ω=2π/P and the last term accounts for the Coriolis force.

Ueff is an effective potential given by

U_{eff}=\frac{-GM_1}{|\underline{r}-\underline{r_1}|}-\frac{GM_2}{|\underline{r}-\underline{r_2}|}-\frac{1}{2}\Omega^2(x^2+y^2)

There are five stationary (Lagrangian) points of the Roche potential Ueff where the effective gravity \nabla U_{eff}=0. Three of these are saddle points, 1, L2, L3.

 

roche-lobe.jpg

Roche lobe: the equipotential surface passing through the inner Lagrangian point L1– this ‘connects’ the gravitational fields of the two stars (please forgive the apology of a diagram- the shaded areas are the Roche lobes of each star).

 

 

Classifications of close binaries

  1. Detached binaries: both stars underfill their Roche lobes (photospheres lie beneath respective Roche lobes). The two stars undergo gravitational interactions only.
  2. Semi-detached binaries: one star fills its Roche lobe. The Roche lobe filling component transfers matter to the detached component. These are known as mass-transferring binaries.
  3. Contact binaries: both stars overfill their Roche lobes. A common photosphere surrounding both components is formed.

Binary mass transfer: 30-50% of all stars experience mass transfer by Roche-lobe overflow during their lifetimes.

  1. (Quasi-)conservative mass transfer: one star accretes mass from another. The mass loser tends to lose most of its envelope, leading to the formation of a helium star. The accretor tends to be rejuvenated, so that it appears to be a younger, more massive star. The orbit generally widens.
  2. Dynamical mass transfer: the stars share a common envelope and eventually spiral into each other. The mass donor engulfs the secondary- either the envelope is ejected to leave a very close binary or the two stars merge to form a single, rapidly rotating star.

 

 

The Algol paradox: the less massive K star of the Algol binary appears to be more evolved- how can this be? In order to solve the paradox, it seems that initially the K star must have been the more massive, therefore evolving more rapidly than its companion B star. During the later stages of its evolution, however, mass is transferred from the K star to the B star- now the K star is less massive, whilst the added mass of the B star makes it more luminous.

Accretion and the Eddington limit

For photons, E=pc

Force F=\frac{dp}{dt}=\frac{1}{c}\frac{dE}{dt}=\frac{flux}{c}=\frac{L}{4\pi r^2c} per unit area for a perfect absorber

Actual fraction of radiation absorbed = κ per unit mass = κρ per unit volume

Actual force on a fluid element F=\frac{\kappa \rho L}{4\pi r^2c}

Gravitational force on a fluid element F_g=\frac{GM\rho}{r^2} per unit volume

In equilibrium \frac{\kappa \rho L}{4\pi r^2c}=\frac{GM\rho}{r^2}

Rearranging, we get the maximum luminosity for a star of mass M L_{Edd}=\frac{4\pi GMc}{\kappa}

In hydrosytatic equilibrium, there is a limit on the luminosity, but as L\simeq L_O(\frac{M}{M_O})^4, there is also a limit on the mass. If the limit is exceeded, internal radiation pressure drives the loss of excess mass.

The Eddington limit also affects the accretion rate. If the observed luminosity L=\frac{GM\dot{M}}{R}, then as latex \dot{M}$ (i.e. \latex \frac{dM}{dt}$) depends on L, it is also limited.

Maximum accretion rate \dot{M}=\frac{L_{Edd}R}{GM}=\frac{4\pi Rc}{\kappa}

Violating the Eddington limit: as can be seen from the mass of supermassive black holes, the Eddington limit can be violated. This can happen under the following conditions-

  1. If the pressure is high enough to allow radiation of energy via neutrinos.
  2. If accretion is in one direction and energy is radiated in another direction.

Limit to a star’s central pressure
Equation of hydrostatic equilibrium \frac{dP_r}{dM_r}=\frac{-GM_r}{4\pi r^4}

\int_{P_c}^{P_s}dP_r=-\int_0^M \frac{GM_rdM_r}{4\pi r^4}

For r<R, the radius of the star, $latex \frac{1}{r^4}>\frac{1}{R^4}

\therefore P_s-P_c>-\int_0^M \frac{GM_rdM_r}{4\pi R^4}

Of course, surface pressure Ps=0, so P_c>\frac{GM^2}{8\pi R^4}

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