ASTR 200 : Lecture 20 Neutron stars 1
Equation of state: Degenerate matter We saw that electrons exert a `quantum mechanical' pressure. This is because they are 'fermions' and are not allowed to occupy the same quantum mechanical state. The number density ne of electrons is proportional to the number of protons (n e ρ ) so the (non-relativistic) equation of state for degenerate matter has : 5/3 P ρ BUT this gets modified if the electrons begin to move near the speed of light (that is, become relativistic). Then the equation of state will change... This was realized by the young Subrahmanyan Chandrasekhar, a student on his ocean voyage to start grad school at Cambridge 2
The Chandrasekhar limit Realized that there is an upper limit to the possible mass of a white dwarf Applied special relativity, which must become important when electron p~mec, where c is the speed of light As a WD becomes more massive, it shrinks. This means the available electrons must, in each volume cell, occupy and higher momentum states and fill them up to a Fermi momentum that is increasing: 3 n e 1/ 3 pf =( ) h 8π For pf~mec, 3 8 π 3 8 π me c n e crit = 3 p F = ( ) 3 h 3h 3
The failure of previous approach From the last lecture, for a white dwarf n e= 3M 0.1 3 8 π m H R WD G 3 m 3e m 4H ℏ6 M 2 As more mass is added, the electron density ne will reach ne crit, the value where electrons become relativistic, when the mass passes : 3 3 1/ 2 M start ℏ c ) 1.1 M solar relativistic 0.6( 3 4 G mh So for masses beyond this, we cannot use the equation of state previously derived. 4
Relativistic equation of state From the last lecture, the electron pressure was 8π p 3 P= 3 0 v p dp 3h f For relativistic electrons, v~c always, and so 1 /3 8 πc 2 π c 4 hc 3 3 4 /3 P= 3 0 p dp= 3 p F = ( π ) n e 8 3h 3h pf As before n e ρ, so now for relativistic degenerate matter P ρ 5 4 /3 instead of the 5/3 power. This means that after the relativistic state is entered, the pressure from degenerate electrons is rising less quickly as the mass density of the star increases.
The Chandrasekhar Mass Chandrasekhar's detailed calculation showed that for a carbon/oxygen white dwarf the electron degeneracy pressure could ONLY support the WD if : M WD < M Ch 1.4 M solar Note that this is the mass of the leftover core, not the original mass of the star. Mass loss after the main sequence ejects some mass. It is thought that stars: <0.5 Msolar will become He white dwarfs 0.5 < M < 5 become carbon/oxygen white dwarfs 5 < M < 7 will leave neon-magnesium white dwarfs 6
What about more massive stars? In the very final giant phase of a very massive star, the star consists of a set of fusion shells burning higher-mass elements. Each stage is capable of supporting hydrostatic equilibrium a shorter time The final stage is an inert iron core; inert because you cannot fuse iron together to get something heavier and release energy The core is supported by degenerate electrons. When the iron core passes the Chandrasekhar limit, it is unable to exist in equilibrium and free-fall collapses. Electron pressure has failed and cannot stop the collapse. As it collapses towards a point at the center of the star, the core reaches the density of an atomic nucleus and the electrons are forced into the protons via the reaction p+e n+ ν e 7 making the neutrons for the neutron star
Type II supernovae When the core free falls it releases : A massive amount of gravitational energy in ~0.1 sec ~1046 J (The sun radiates ~1044 J in its entire main sequence lifetime!). Blows the envelope of the star off into space. ~1057 electron neutrinos Recent example: SN 1987A 8 Neutrino detections from 1987A HST image of SN 1987A remnant
But what about the core? It is still free-falling neutrons... IF the core is less than ~3 Msolar, neutron degeneracy will stop the collapse at this point, with a radius ~3 km leaving a NEUTRON STAR 9
Radius of a neutron star: vs Mass We can use the same expression as last time to compute the radius of a neutron star, just replacing me with mn. 2 1/ 3 M R NS 0.95 ( ) 2 G mn mh mh ℏ 1 /3 Which gives: R NS M 3.2 ( ) M sun km (approximate!) So, a 2 solar mass neutron star (2.5 km radius) will have a mean density on the order of billions of tons per cubic cm! Above is only approximate because in fact the 'strong nuclear' force also comes into play, which makes neutron star equations of state uncertain; roughly, the constant out front above should be about 10 km rather than 3.2 10
Can we see a neutron star? They still have a LOT of leftover heat to radiate Surface T about 1 million Kelvin. Recently seen. Wein's law: peak blackbody energy ~0.5 kev 2.5 nm Below: X-ray spectrum of isolated NS, 11
Pulsars Even though neutron stars were predicted by Walter Baade and Fritz Zwicky in 1934, many astronomers doubted they could exist However, in 1967 graduate student Jocelyn Bell discovered a periodic radio source in a radio telescope signal near Cambridge, with a strong pulse of radiation every 1.33 seconds The pulse regularity rivaled the very best atomic clocks. Although aliens were briefly discussed, Thomas Gold and Franco Pacini realized the bursts must be coming from rapidly rotating neutron stars These `pulsars', as they are called, are often found in the core of supernova remnants 12
The Crab nebula Supernova remnant ~2000 pc away, ~3 pc across Remains of a supernova that happened in 1054 AD Expanding at ~0.5% c (!) Ancient historical records Chinese and japanese astronomers 'guest star' North American indians (at right, Chaco Canyon, NM) Remained visible for about 2 years Visible in daylight (!) initially 13
The Crab PULSAR At very center of the supernova remnant Radio pulses every 0.033 sec (!) This very young pulsar is even pulsing in the optical What produce the radiation? A magnetic field attached to the spinning neutron star IF the Earth lies along the cone the beam sweeps, we see the radio emission caused by electrons interacting with the powerful magnetic field 14
Binary Pulsars Two neutron stars in binary orbits Only the beam of one of the pulsars will intercept the line of sight to Earth The binary orbit involves so much gravitational energy that general relativity predicts the mutual semimajor axis will decay as the system emits gravitational waves 15
Binary Pulsar The orbital decay (and gravity waves) can be detected indirectly by monitoring the arrival times of the pulses, over decades Precise agreement with the prediction of general relativity. Earned Hulse and Taylor the Nobel prize and very strong confirmation of Einstein's theory of general relativity DIRECT detection of gravitational waves has come recently (see next lecture) 16