Neutron stars and pulsars Neutron stars are formed in (some) core collapse supernova when an iron core becomes unstable to collapse. Basic properties: • neutron rich because many of the protons have been transformed into neutrons: - e + p Æ n + n e • degeneracy pressure of neutrons provides the pressure support against gravity, but matters are complicated † because nuclear forces are important. • density is comparable to nuclear values: r ~ 1015 g cm-3, radius is ~10 km
ASTR 3730: Fall 2003 Newly formed neutron stars are very hot, and embedded within young supernova remnants. Rapidly cool due to neutrino emission (in the very early stages), then photon emission: Galactic population of neutron stars must be enormous - a very small number have been detected directly from this cooling radiation.
ASTR 3730: Fall 2003 Neutron star history Idea that neutron stars might be formed in supernova explosions was suggested by Walter Baade & Fritz Zwicky in 1934. Detailed models for neutron stars were derived by Robert Oppenheimer and George Volkoff in 1939. Obvious that isolated neutron stars would be very faint and hard to detect. Neutron stars in binaries are extremely luminous, but only in X-rays which at the time were undetectable. Surprisingly, first discovered by accident via pulsed radio emission - pulsars, by Jocelyn Bell in 1967.
ASTR 3730: Fall 2003 Large number (~ thousand) of radio pulsars are now known, some of which also pulse in optical or X-ray: Crab pulsar
ASTR 3730: Fall 2003 Where neutron star masses can be measured (in binary systems), the mass is found to be very close to 1.4 Msun. Maximum possible mass for a neutron star is not known, but probably
2 - 3 Msun.
ASTR 3730: Fall 2003 Pulsar timing Basic observable for a radio pulsar is the time of arrival of each pulse at Earth. After correction for effects such as the Earth’s motion around the Sun, derive: P Pulse period (e.g. 1s, though a wide range) dP Period derivative (e.g. 10-12 seconds per ≡ P˙ dt second - so dimensionless) d 2P ≡ P˙˙ Second period derivative dt 2 Some (not all) pulsars make very good clocks - any trends in P can be predicted / extrapolated to extraordinary accuracy. † Frequency stability comparable to best atomic clocks.
ASTR 3730: Fall 2003 Isolated pulsars (ignoring some special cases we will discuss later) are powered by rotational kinetic energy. Spin down slowly as this energy is released: 1 E = IW2 rot 2 dE 1 d 1 dW rot = I W2 = I ¥ 2W = IWW˙ dt 2 dt ( ) 2 dt …where I is the moment of inertia of the neutron star and W is the angular velocity. Very roughly:
† 2 -1 I ~ MR take: M = 1.4 Msun, R = 10 km, W = 60 s
48 (enough for a Solar luminosity E = 5 ¥10 erg rot for ~40 million years)
† Eventually, pulsars slow down so much that the pulsed radiation ceases. † ASTR 3730: Fall 2003 What can we learn from measuring P and its time derivatives?
Period and period derivative Quantity: P P˙ has dimensions of time and is obviously related to the `age’ of the pulsar. To be more quantitative, write: † k is a positive constant, n is a constant dW called the braking index that describes = -kWn how the pulsar spins down. Simplest dt model for pulsar radiation predicts that the braking index n = 3. Integrate this equation, assuming that:
† • at time t = 0, W = W0 • at present time t = t, angular velocity is W
ASTR 3730: Fall 2003 W t Ú W-n dW = -k Ú dt W0 0 W È W-n +1 ˘ Í ˙ = -kt Î -n +1˚ W0 È W-n +1 W-n +1 ˘ Í - 0 ˙ = -kt Î 1- n 1- n ˚
Second† term is negligible provided that:
• W0 >> W i.e. pulsar was initially spinning much faster than it is today • n > 1 so the powers are negative (actually n = 3 or thereabouts so this is OK)
ASTR 3730: Fall 2003 W1-n Result is then: -kt = 1- n Eliminate k using the first equation for spin down: W˙ = -kWn 1 W t†= 1- n W˙ Since W = 2p / P can alternatively write th†is in terms of the period and period derivative with respect to time: † 1 P t = n -1 P˙ e.g. for the Crab pulsar: P = 33.4 ms, P˙ = 4.21¥10-13 ss-1 Taking n = 3, this gives an age of t = 1255 years for the pulsar. † Good estimate to the true age, since the Crab nebula was formed from a †supernova in 1054 AD.
ASTR 3730: Fall 2003 Period plus first two period derivatives With a measure of all three quantities, we can determine the braking index without recourse to theory: W˙ W˙ = -kWn Æ k = - Wn W˙˙ = -k ¥ nWn-1 ¥ W˙ W˙ Substitute for k and tidy up: W˙˙ = nWn-1W˙ Wn WW˙˙ † n = W˙ 2 Can write this in terms of P and its derivatives as before. Typically n is around (but not exactly equal) to 3. †
ASTR 3730: Fall 2003 The pulsar population
lines of constant age
period derivative
pulsar death line
period ASTR 3730: Fall 2003 Emission from pulsars The optical emission from e.g. the Crab pulsar appears to be synchrotron emission. How does the neutron star produce energetic particles and magnetic fields?
Speed of light Rotation Open magnetic cylinder - where axis Magnetic field lines particle corotating axis with the neutron star have v = c v = Wr = c c Closed r = magnetic W field lines
†
ASTR 3730: Fall 2003 Concept leads to a lighthouse model for pulsars:
Charged particles are accelerated off the surface of See pulses of the neutron star radiation when by strong electric the beam sweeps fields past our direction Radiation is beamed along directions of open field lines
ASTR 3730: Fall 2003