Pulsars. http://www.cv.nrao.edu/course/astr534/Pulsars.html High Energy Astrophysics email@example.com http://www.mssl.ucl.ac.uk/ Enrico Massaro notes. Pulsed emission; Rotation and energetics; Magnetic field; Neutron star structure; Magnetosphere and pulsar models;
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High Energy Astrophysics
Enrico Massaro notes
Astronomers were used to slowly varying or pulsating emission from stars, but the natural period of a radially pulsating star depends on its mean density and is typically days, not seconds.
There is a lower limit to the rotation period P of a gravitationally bound star, set by the requirement that the centrifugal acceleration at its equator not exceed the gravitational acceleration. If a star of mass M and radius R rotates with angular velocity =2/P
The canonical neutron star has M~ 1.4MSun and R~ 10 km,depending on the equation-of-state of extremely dense mattercomposed of neutrons, quarks, etc. The extreme density andpressure turns most of the star into a neutron superfluid that is asuperconductor up to temperatures T~109 K. Any star ofsignificantly higher mass (M~3MSun in standard models) mustcollapse and become a black hole. The masses of several neutronstars have been measured with varying degrees of accuracy, andall turn out to be very close to 1.4MSun.
From the Salpeter IMF (number of stars formed each year per cubic Mpc with mass between M and M+dM):
(M,M+dM)=2 10-12 M-2.35 dm
98% of the stars will end forming a white dwarf, i.e. the total number of WD in a typical galaxy is ~1010
Neutron stars will form if Mns>1.4MSun. They are thought to form form from the collapse of the core of stars with mass between 8 and 35 MSun. The total number of neutron stars in a typical galaxy is ~109.
Black holes will form is MBH>3MSun. They are thought to form from the collapse of the core of stars with M>35 MSun. The total number of black holes in a typical galaxy is ~106-107
The Sun and many other stars are known to possess roughlydipolar magnetic fields. Stellar interiors are mostly ionized gas andhence good electrical conductors. Charged particles areconstrained to move along magnetic field lines and, conversely,field lines are tied to the particle mass distribution. When the coreof a star collapses from a size 1011cm to 106 cm, its magnetic flux is conserved and the initial magneticfield strength is multiplied by 1010, the factor by which thecross-sectional area a falls. An initial magnetic field strength ofB~100 Gauss becomes B~1012 Gauss after collapse, so young neutronstars should have very strong dipolar fields.
Radius collapses from 7 x 10 m to 10 m
B ~ 5 x 10 Tesla = 5 x 10 Gauss
B ~ 3.3 x 1015 (P P)½ Tesla
or ~ 106 to 109Tesla for most pulsars
The traditionalmagnetic dipole model of a pulsar
Light cylinder (the cylinder centered on the pulsar andaligned with the rotation axis at whose radius the co-rotatingspeed equals the speed of light).
Where P is the pulsar period. This electromagnetic radiation will appear at the very low frequency=1/P<1kHz, so low that it cannot be observed, or even propagate through the ionized ISM. The huge power radiated is responsible for pulsar slowdown as it extracts rotational kinetic energy from the neutron star. The absorbed radiation can also light up a surrounding nebula, the Crab nebula for example.
The neutron star is surrounded by a magnetosphere with free charges that produce intense electric currents. The neutron star is a spinning magnetic dipole, it acts as a unipolar generator.There are two main regions:
The closed magnetosphere, defined as the region containing the closed field lines within the light cilinder
The open magnetosphere, the region where the field lines cannot close before RL and extend above this radius.
If gravity is negligible, The total force acting on a charged particle is:
This is a factor of 10 larger than the gravitational force and thus dominates the particle distribution.
The particles moves along the field lines and at the same time rotate with them. Charges in the magnetic equatorial region redistribute themselves by moving along closed field lines until they build up an electrostatic field large enough to cancel the magnetic force and give F=0 . The voltage induced is about 106 V in MKS units. However, the co-rotating field lines emerging from the polar caps cross the light cylinder (the cylinder centered on the pulsar and aligned with the rotation axis at whose radius the co-rotating speed equals the speed of light) and these field lines cannot close. Electrons in the polar cap are magnetically accelerated to very high energies along the open but curved field lines, where the acceleration resulting from the curvature causes them to emit curvature radiation that is strongly polarized in the plane of curvature. As the radio beam sweeps across the line-of-sight, the plane of polarization is observed to rotate by up to 180 degrees, a purely geometrical effect. High-energy photons produced by curvature radiation interact with the magnetic field and lower- energy photons to produce electron-positron pairs that radiate more high-energy photons. The final results of this cascade process are bunches of charged particles that emit at radio wavelengths.
Velocity- of -
EmissionA more realistic model...
The extremely high brightness temperatures are explained by coherent radiation. The electrons do not radiate as independent charges e; instead bunches of N electrons in volumes whose dimensions are less than a wavelength emit in phase as charges Ne. Since Larmor's formula indicates that the power radiated by a charge q is proportional to q2, the radiation intensity can be N times brighter than incoherent radiation from the same total number N of electrons. Because the coherent volume is smaller at shorter wavelengths, most pulsars have extremely steep radio spectra. Typical (negative) pulsar spectral indices are ~1.7 (S-1.7 ), although some can be much steeper (>3) and a handful are almost flat (~0.5).
Total radiation intensity
does not exceed
Summed intensity of spontaneous radiation of individual particles
For radiating particles in thermodynamic
equilibrium i.e. thermal emission.
Blackbody => max emissivity
So is pulsar emission thermal?
Consider radio: n~108 Hz or 100MHz; l~3m
Watts m Hz ster
Crab flux density at Earth, F~10 watts m Hz
Source radius, R~10km at distance D~1kpc
In = 10 watts m Hz ster
From equation (1):
this is much higher than a radio blackbody temperature!
high-B sets up large pd => high-E particles
B = 1.108Tesla
R = 104 m
cascades results in bunches
of particles which can radiate
coherently in sheets
Optical & X-ray emission in pulsars
High magnetic fields; electrons follow field lines very closely, pitch angle ~ 0o
- similar to synchrotron radiation,
intensity from curvature radiation << cyclotron or synchrotron
IR, optical, X-rays, g-rays
- radio coherent, X-rays and Optical incoherent
- location of radiation source depends on frequency
- radiation is directed along the magnetic field lines
- pulses only observed when beam points at Earth
- radio emission from magnetic poles
- X-ray and optical emission from light cylinder
TheP-Pdot Diagram is useful forfollowing the lives of pulsars, playing a role similar to theHertzsprung-Russell diagram for ordinary stars. It encodes atremendous amount of information about the pulsar populationand its properties, as determined and estimated from two of theprimary observables, P and Pdot Using those parameters, one canestimate the pulsar age, magnetic field strength B, and spin-downpower dE/dt. (From the Handbook of Pulsar Astronomy, by Lorimerand Kramer)
Pulsars are born in supernovae and appear in the upper left corner of the PP diagram. If B is conserved, they gradually move to the right and down, along lines of constant B and crossing lines of constant characteristic age.
Pulsars with characteristic ages <105yr are often found in SNRs. Older pulsars are not, either because their SNRs have faded to invisibility or because the supernova explosions expelled the pulsars with enough speed that they have since escaped from their parent SNRs. The bulk of the pulsar population is older than 105 yr but much younger than the Galaxy (1010 yr). The observed distribution of pulsars in the PPdot diagram indicates that something changes as pulsars age. One possibility is that the magnetic fields of old pulsars decays on time scales 107 yr, causing pulsars to move straight down in the diagram until they fall below into the graveyard below the death line.
The death line in the PPdot diagram corresponds toneutron stars with sufficiently low B and high P that the curvatureradiation near the polar surface is no longer capable of generatingparticle cascades (Prad scales with B2 and P-4).
Almost all short-period pulsars below the spin-up line near log[Pdot/P(sec)]~-16 are in binary systems, as evidenced by periodic (i.e. orbital) variations in their observed pulse periods. These recycled pulsars have been spun up by accreting mass and angular momentum from their companions, to the point that they emit radio pulses despite their relatively low magnetic field strengths B 108 G (the accretion causes a substantial reduction in the magnetic field strength). The magnetic fields of neutron stars funnel ionized accreting material onto the magnetic polar caps, which become so hot that they emit X-rays. As the neutron stars rotate, the polar caps appear and disappear from view, causing periodic fluctuations in X-ray flux; many are detectable as X-ray pulsars.
Millisecond pulsars (MSPs) with low-mass (M~0.1-1 MSun) white-dwarf companions typically have orbits with small eccentricities. Pulsars with extremely eccentric orbits usually have neutron-star companions, indicating that these companions also exploded as supernovae and nearly disrupted the binary system. Stellar interactions in globular clusters cause a much higher fraction of recycled pulsars per unit mass than in the Galactic disk. These interactions can result in very strange systems such as pulsar-main-sequence-star binaries and MSPs in highly eccentric orbits. In both cases, the original low-mass companion star that recycled the pulsar was ejected in an interaction and replaced by another star. (The eccentricitye of an elliptical orbit is defined as the ratio of the separation of the foci to the length of the major axis. It ranges between e for a circular orbit and e for a parabolic orbit.) A few millisecond pulsars are isolated. They were probably recycled via the standard scenario in binary systems, but the energetic millisecond pulsars eventually ablated their companions away.
- surface gravity, g = GM/R2 ~ 10 m s
Heavy nuclei (Fe) find a minimum energy when arranged in a crystalline lattice
Neutron star segment
Superfluid neutrons, superconducting p+ and e-
2x10 kg m
crystallization of neutron matter
4.3x10 kg m
10 kg m
10 kg m