Pulsars and neutron stars/Pulsar properties

Naming pulsars
Every pulsar has a unique name that defines its position in the sky (in J2000 coordinates). For instance PSR J0437-4715 is a pulsar with right ascension 04:37 and declination -47:15. In the past astronomers used B1950 coordinates and so some pulsars also has a "B" name. For instance, PSR J1939+2134 is also known as B1937+21. As the number of pulsars grows it is becoming more challenging to give each pulsar a unique name. Letters are used to distinguish pulsars that are close together (such as in the double pulsar system, or in globular clusters). For instance, PSRs J0024-7204C and J0024-7204D are two different pulsars in the globular cluster 47 Tucanae. Some, well studied pulsars are also known by common names. For instance, the pulsar in the Vela supernova remnant is known as the "Vela pulsar", J0835-4510 or B0833-45.

Pulse period
The fundamental property of a pulsar is its pulse period (P) - the time between adjacent pulses. This is usually understood as the time of rotation of the neutron star and so is sometimes also called the pulsar's "rotational period" (although note that the unknown pulsar radial velocity and other effects will lead to a slight variation in the measured period). It is also common to consider the pulsar pulse frequency $$\nu$$:

$$P = 1/\nu$$.

Pulsars slow down over time. The slow-down rate can be described by the time derivatives of the pulse period or frequency as:

$$\dot{P} = -\frac{1}{\nu^2}\dot{\nu}$$

$$\ddot{P} = -\nu^{-2}\ddot{\nu} + 2\nu^{-3}\dot{\nu}^2$$

In the Figure below (the various symbols and lines are described later) we show the measured pulse periods plotted versus their period derivatives for the majority radio pulsars in the ATNF pulsar catalogue v1.53. Note that a few pulsars have measured period derivatives that are less than zero and so indicate that the pulsar rotation is increasing, instead of decreasing. Such pulsars are associated with globular clusters and the measured negative spin-down rates are likely caused by the acceleration of the pulsar in the cluster's gravitational field. Such pulsars are not included in the diagram.

Script for making the figure.

The majority of pulsars have spin periods between ~0.1 and ~4 seconds. The longest period radio pulsar, PSR J2144-3933, has a period of 8.5 seconds. Such pulsars typically have slow down rates of ~10-14s/s. Another group of pulsars, known as the "millisecond pulsars", appear in the bottom left-hand part of the diagram. Such pulsars have millisecond periods (with the fastest being PSR J1748-2446ad with a period of 1.4ms) and a slow down rate of ~10-20s/s. The millisecond pulsar region and the pulsars joining their region to the normal pulsars are known as the "recycled pulsars".

The distribution of pulsar periods shown in the diagram is not the true, intrinsic, period distribution. Pulsar surveys are most sensitive to the normal pulsar population and have reduced sensitivity to very long or very short periods.

As will be described later in this section, pulsars are moving - they are observed to have a proper motion across the sky. As shown by Shklovskii (1970), and known as the "Shklovskii effect", this means that observed period derivatives (either the pulse period derivative or the orbital period derivative) will be higher than the intrinsic period derivative.

$$\dot{P}_{\rm observed} = \dot{P_{\rm intrinsic}} + \frac{P \mu^2 d}{c} $$

Where $$\mu$$ is the proper motion.

Estimating pulsar ages
The two black, dashed lines In the period-period derivative diagram above indicate pulsars with ages of 1 kyr (the top line) and 1 Myr (the lower line). Of course, nobody really knows the age of any pulsar and so these lines are simply indicative. However, it is possible to know the age of some supernovae that have associated pulsars and it is reasonable to assume that the neutron star itself was made during the supernova and the pulsar switched on at soon after. 59 pulsars in the pulsar catalogue are associated with supernova remnants (although it is likely that not all of these actually are true associations). The Crab supernova occurred on July 4 1054 and was recorded by the Chinese, Japanese, Korean, Arab and possibly Native Americans and Europeans. The Chinese record is the most often quoted and yet arguments abound in the literature about seemingly contradictory statements (wikipedia certainly has plenty to say on the subject).

For most pulsars more indirect methods are required for estimating their ages. Typically, pulsar astronomers calculate the pulsar's characteristic age:

$$\tau_c = P/(2\dot{P})$$

The two black, dashed lines in the period-period derivative diagram are characteristic ages.

A more detailed analysis that contains the decay of the pulsar's magnetic field is often called the true age (Helfand & Tademaru 1977):

$$\tau_{true} = \frac{\tau_D}{2}\ln\left[1+\frac{2\tau_c}{\tau_D} \right]$$

where $$\tau_D$$ is the decay time of the magnetic field.

It is also possible to consider the age of a pulsar from the time that it has taken to arrive at its current position assuming that it was born in the Galactic plan and travelled at a constant velocity. Lyne, Anderson & Salter (1982) showed that his kinetic age, $$\tau_k$$ is given by:

$$\tau_{tk} = \frac{D\tan b}{v_{tz}}\left[\frac{\cos^2 b}{1+\left(v_p/v_{tz}\right)sin b}\right]$$

where $$D$$ is the distance to the pulsar, $$b$$ the pulsar's galactic latitude, $$v_{tz}$$ the pulsar's transverse velocity perpendicular to the Galactic plane and $$v_p$$ the pulsar's velocity parallel to the Galactic plane.

Estimating magnetic fields
For a purely dipolar magnetic field, a representative magnetic field on the pulsar's surface will equal:

$$B_s [{\rm Gauss}] = 3.2 \times 10^{19} (P\dot{P})^{1/2}$$ (this assumes a neutron star radius of 10km and a moment of inertia of $$10^{45}{\rm g cm}^2$$.)

Note that Shapiro & Teukolsky (1983) derive

$$B_s [{\rm Gauss}] = 6.4 \times 10^{19} (P\dot{P})^{1/2}$$

which assumes a braking index of 3 and a uniformly magnetized stellar interior. The common practice of using the factor of $$3.2 \times 10^{19}$$ is a representative surface field. Of course, the actual magnetic field is determined by many factors and cannot be known to within a factor of two.

Spin down energy loss rate
The spin down energy loss rate (in ergs/s) can be estimated from

$$\dot{E} = 4\pi^2 I \frac{\dot{P}}{P^3}$$

where $$I$$ is the pulsar's moment of inertia (usually taken as $$10^{45} {\rm g cm}^2$$).

Proper motions and velocities
Pulsar proper motions have been measured using interferometric observations (VLBI) and also using the timing method. Proper motions can be measured in equatorial or ecliptic coordinates. For pulsar timing measurements it is usual that the error ellipse for proper motions measured in ecliptic coordinates is significantly smaller than that for equatorial coordinates, but for ease of comparison, it is common to publish results in equatorial coordinates.

The proper motion in right ascension and declination are normally written as $$\mu_\alpha \cos \delta$$ and $$\mu_\delta$$ respectively where $$\delta$$ is the pulsar's declination. Note that the $$\cos \delta$$ is often not explicitly written. For instance, the TEMPO2 software package labels $$\mu_\alpha \cos \delta$$ simply as PMRA.

The total proper motion can be calculated from:

$$\mu_{\rm tot} = \left((\mu_\alpha \cos \delta)^2 + \mu_\delta^2 \right)^{1/2}$$

Transverse velocities can be estimated from the total proper motion:

$$V [{\rm km s}^{-1}] = 4.74 D [{\rm kpc}] \mu_{\rm tot} [{\rm mas yr}^{-1}]$$

Proper motions can also be determined in Galactic coordinates (see, for instance, Harrison, Lyne & Anderson 1993).

Pulsar distances
Pulsar distances are difficult to measure. In a few cases interferometric observations or pulsar timing has led to a measurement of the pulsar's annual parallax, $$\Pi$$. In this case the pulsar's distance is simply:

$$D = \gamma/\Pi$$

$$\gamma = 1$$ is used to convert from the parallax measurement (usually in mas) to the distance (usually in kpc).

If the parallax is not known then the pulsar distance must be estimated by other means. A pulsar may be associated with a source (such as a globular cluster, supernova remnant or galaxy) which has a known distance. It is also sometimes possible to get a bound on the distance using HI absorption measurements. However, for most pulsars the distance is usually estimated from the pulsar's distance and an electron density model.

Flux densities, luminosities and spectral indices
The luminosity of a pulsar at a particular observing frequency ($$L(f)$$) can be derived from

$$L(f) = S(f) D^2$$

where $$S$$ is the flux density at a observing frequency $$f$$ and $$D$$ is the distance. Radio luminosities are generally published in units of $${\rm mJy} {\rm kpc}^2$$.

The spectral index describes how a pulsar's flux density changes with frequency. If two measurements have been made $$(S_1,f_1)$$ and $$(S_2,f_2)$$ then the spectral index is:

$${\rm SI} = -\frac{\log_{10}(S_1/S_2)}{\log_{10}(f_1/f_2)}$$