Pulsars
and the spin-down

A radio pulsar is a neutron star whose beam sweeps past Earth. Two measured numbers carry the dipole story: the spin period P and how fast that period is growing, Ṗ. For magnetic-dipole braking the index is n = 3, so the characteristic age is τ = P / (2 Ṗ). The surface dipole field in the catalogue convention is B = 3.2×10¹⁹ √(P Ṗ) gauss, with a moment of inertia I = 10⁴⁵ g cm² (10³⁸ kg m²) and a radius of 10 km. The spin-down luminosity is Ė = 4π² I Ṗ / P³. The light cylinder, where a co-rotating field would move at the speed of light, is R_lc = c P / (2π). c = 299,792,458 m/s exactly. The four buttons are P0 and P1 printed by the ATNF Pulsar Catalogue version 2.9.0 (Manchester, Hobbs, Teoh, and Hobbs 2005). The Crab’s catalogue age is 1,257 yr. The supernova of 1054 is 972 years before 2026. A heavy star’s path to a neutron star is the stellar lifetime desk. A horizon, if the remnant is heavier, is the black holes desk.

Educational calculator. Idealized magnetic dipole. Braking index fixed at 3. Moment of inertia fixed at 10⁴⁵ g cm². Radius fixed at 10 km. True ages and true fields differ. The Crab’s true age is the supernova of 1054, 972 years by a count of calendar years to 2026, not the characteristic age. Catalogue periods are at the printed epoch, not a live ephemeris. Not a timing model.

Interactive · Period and derivative

Log period across · log period derivative up · short and fast-slowing is the upper left

This point
Cited preset
Constant age
Constant B
10¹² V sketch

Catalogue presets · P0 and P1, ATNF v2.9.0

Period

Period derivative · logarithmic · dimensionless

Pdot

Opens on the Crab, B0531+21, catalogue epoch MJD 48442.5. Lines of constant age slope up to the right. Lines of constant dipole field slope down. The dashed line is a 10¹² V polar-cap sketch, not a fitted death line. Only the four cited pulsars are marked.

Crab · B0531+21 · MJD 48442.5

1,257 yr

Characteristic age · dipole n = 3

Period33.3924123 ms
Pdot4.20972×10⁻¹³
Characteristic age1,257 yr
True age972 yr, from 1054
Dipole field3.794×10¹² G
Spin-down power4.463×10³¹ W
erg/s4.463×10³⁸ erg/s
Light cylinder1,593.3 km
Spin29.947 Hz

Period 33.3924123 ms (0.0333924123 s). Period derivative 4.20972×10⁻¹³, dimensionless. B0531+21 (J0534+2200) at catalogue epoch MJD 48442.5, the P0 and P1 printed by ATNF v2.9.0 from Lyne et al. 2015. Characteristic age is P / (2 Ṗ), which is 1,257 yr. The supernova of 1054 is 972 years before 2026, a count of calendar years, not a day inside 1054. The characteristic age is longer. The formula assumes the star was born spinning much faster than it spins now, and that the braking index stays at 3. Lyne et al. 2015 is 45 years of the Crab’s rotation. This number is not that history. Surface dipole field is 3.2×10¹⁹ times the square root of P Ṗ, which is 3.794×10¹² G. The 3.2 has two figures, so the field is about that order. Moment of inertia is 10⁴⁵ g cm² and the radius in the convention is 10 km. Spin-down luminosity is 4π² I Ṗ / P³, which is 4.463×10³¹ W (4.463×10³⁸ erg/s). Light-cylinder radius is c P / (2π), which is 1,593.3 km. Spin frequency, 1/P, is 29.947 Hz. Idealized dipole. Not a timing model.

ATNF v2.9.0 presets · dipole formulas on the printed P0 and P1
Pulsar	P	Ṗ	Epoch	τ	B	Ė	R_lc
Crab B0531+21	33.3924123 ms	4.20972×10⁻¹³	MJD 48442.5	1,257 yr	3.794×10¹² G	4.463×10³¹ W	1,593.3 km
Vela B0833−45	89.328385024 ms	1.25008×10⁻¹³	MJD 51559.319	11.32 kyr	3.382×10¹² G	6.924×10²⁹ W	4,262.2 km
J0437−4715	5.757451941593412 ms	5.729220×10⁻²⁰	MJD 55486	1.592 Gyr	5.812×10⁸ G	1.185×10²⁷ W	274.71 km
B1919+21	1.3373021601895 s	1.348207×10⁻¹⁵	MJD 48999.0	15.72 Myr	1.359×10¹² G	2.226×10²⁴ W	63,807 km

The buttons need JavaScript. The arithmetic is still τ = P / (2 Ṗ), B = 3.2×10¹⁹ √(P Ṗ) gauss, Ė = 4π² I Ṗ / P³ with I = 10³⁸ kg m², and R_lc = c P / (2π). Crab P0 = 0.0333924123 s and P1 = 4.20972×10⁻¹³ give 1,257 yr, 3.794×10¹² G, and 4.463×10³¹ W. The supernova of 1054 is 972 years before 2026. Vela gives 11.32 kyr. J0437−4715 gives 1.592 Gyr. B1919+21 gives 15.72 Myr. Not a timing model.

What the numbers are

Two numbers, and the beam. A pulsar is a neutron star, seen when the beam swings past Earth. The catalogue period is the time for one turn. The period derivative is how much that time grows each second. It is tiny, and it is dimensionless. The Crab’s catalogue period is 33.3924123 ms and its derivative is 4.20972×10⁻¹³. J0437−4715 turns in 5.757451941593412 ms. B1919+21 takes 1.3373021601895 s.

The age is an assumption. Characteristic age is the period divided by twice the derivative. Braking index 3 is the dipole. The Crab’s catalogue numbers give 1,257 yr. The guest star of 1054, the supernova tied to the Crab Nebula, is 972 years before 2026. The characteristic age is longer because the star was not born at a standstill, and because the braking index does not have to stay at 3. Vela’s catalogue age is 11.32 kyr. B1919+21’s is 15.72 Myr. J0437−4715’s is 1.592 Gyr, a recycling age, not the age of the star.

Field, power, and the light cylinder. The surface dipole field on this page is 3.2×10¹⁹ √(P Ṗ) gauss. That is the ATNF BSURF convention, with I = 10⁴⁵ g cm² and R = 10 km. The Crab computes 3.794×10¹² G, order 10¹² G. J0437−4715 computes 5.812×10⁸ G. The polar field in Harding’s equation (5) is twice this, 6.4×10¹⁹ √(P Ṗ), because a dipole is stronger at the pole. Spin-down luminosity is 4π² I Ṗ / P³. The Crab’s is 4.463×10³¹ W, which is 4.463×10³⁸ erg/s. The ATNF EDOT column prints that power in erg/s, rounded. The light cylinder is c P / (2π). Inside it, a co-rotating field would still be slower than light. The Crab’s is 1,593.3 km.

Literacy · the words on the readout

Period: P, the barycentric spin period, in seconds. The millisecond buttons are the same number times 1,000. One turn of the Crab’s catalogue period is 33.3924123 ms, which is 29.947 Hz.
Period derivative: Ṗ, how many seconds the period grows in one second. Dimensionless. The slider is logarithmic, from 10⁻²² to 10⁻⁹. A negative derivative, a glitch, or a measured Shklovskii term is outside this page.
Characteristic age: τ = P / (2 Ṗ). The 2 is the dipole, braking index n = 3, because τ = P / ((n − 1) Ṗ). It counts from a birth spin much faster than today’s. A year here is 365.25 × 86,400 seconds.
Dipole field: B = 3.2×10¹⁹ √(P Ṗ) gauss. Canonical I = 10⁴⁵ g cm² and R = 10 km. The 3.2 has two figures. This is the ATNF BSURF column, the equatorial convention. Harding’s polar field is twice it.
Spin-down luminosity: Ė = 4π² I Ṗ / P³. With I = 10³⁸ kg m² the result is watts. Multiply by 10⁷ for erg/s, which is how the catalogue prints EDOT, rounded. It is the rotational power, not a measured flux at Earth.
Light cylinder and the diagram: R_lc = c P / (2π), with c = 299,792,458 m/s. On the log-log diagram, constant age is a line of slope +1 and constant B is a line of slope −1. Millisecond pulsars sit at short P and tiny Ṗ. The dashed line is Harding’s equation (1) at a polar-cap potential of 10¹² V, the round value Xia Zhou holds fixed in IAU Symposium 337. Chen and Ruderman call the real radio boundary a death valley, not this one line. The shaded side is that sketch. No other pulsars are plotted.

Sources · the periods and the formulas

Cited, not invented

Manchester, Hobbs, Teoh, and Hobbs 2005, AJ 129, 1993 — the ATNF Pulsar Catalogue. This page uses version 2.9.0, queried for P0 and P1. The catalogue stores F0 and F1 and prints P0 = 1/F0 and P1 = −F1/F0². The web catalogue is atnf.csiro.au/research/pulsar/psrcat. If you use the catalogue, cite that paper and the web address, and the original timing papers where you can.
Lyne, Jordan, Graham-Smith, Espinoza, Stappers, and Weltevrede 2015, MNRAS 446, 857 — “45 years of rotation of the Crab pulsar.” ATNF v2.9.0 attributes the Crab’s F0 and F1 to this paper, at epoch MJD 48442.5. Printed P0 = 0.0333924123 s, P1 = 4.20972×10⁻¹³. The catalogue’s rounded AGE, BSURF, and EDOT are 1.26×10³ yr, 3.79×10¹² G, and 4.5×10³⁸ erg/s. This page recomputes them.
Dodson, McCulloch, and Lewis 2002, ApJL 564, L85 — the January 2000 Vela glitch. ATNF v2.9.0 attributes Vela’s F0 and F1 to this paper, at epoch MJD 51559.319, the glitch date in the paper (MJD 51559). Printed P0 = 0.089328385024 s, P1 = 1.25008×10⁻¹³.
Reardon and collaborators 2024, ApJL 971, L18 — precision timing of PSR J0437−4715. ATNF v2.9.0 attributes F0 and F1 to this paper, at epoch MJD 55486. Printed P0 = 0.005757451941593412 s, P1 = 5.729220×10⁻²⁰. Discovery of the pulsar: Johnston and collaborators 1993, Nature 361, 613.
Hobbs, Lyne, Kramer, Martin, and Jordan 2004, MNRAS 353, 1311 — long-term timing of 374 pulsars. ATNF v2.9.0 attributes B1919+21’s F0 and F1 to this paper, at epoch MJD 48999.0. Printed P0 = 1.3373021601895 s, P1 = 1.348207×10⁻¹⁵. Discovery: Hewish, Bell, Pilkington, Scott, and Collins 1968, Nature 217, 709.
Duyvendak 1942, PASP 54, 91 — further data bearing on the identification of the Crab Nebula with the supernova of 1054. This page counts 2026 − 1054 = 972 calendar years. It does not pick a day inside 1054. The pulsar discovery near the nebula: Staelin and Reifenstein 1968, Science 162, 1481. Vela’s supernova association: Large, Vaughan, and Mills 1968, Nature 220, 340.
NIST — speed of light in vacuum — c = 299,792,458 m/s exactly. The light cylinder divides the period by 2π and multiplies by that c.
Harding, astro-ph/0209160, “Radio Pulsar Death” — equation (1), Φ = 6.6×10¹² V × B_p,12 × P⁻² × R₆³. Equation (5), B_p = 6.4×10¹⁹ √(P Ṗ) G for I = 10⁴⁵ g cm² and R = 10 km. Harding writes that a constant-Φ line is the conventional sketch and that a real death line is not that one line. The polar field is twice this page’s 3.2×10¹⁹ dipole field.
Xia Zhou 2018, IAU Symposium 337 — sets the accelerator drop ΔV = 10¹² V so the death line’s move with the equation of state can be discussed on its own. The underlying paper is Zhou, Tong, Zhu, and Wang 2017, MNRAS 472, 2403. This page draws that 10¹² V line with Harding’s coefficient. It is a sketch.
Chen and Ruderman 1993, ApJ 402, 264 — “Pulsar death lines and death valley.” Turnoff is a valley in the field–period plane, not one line. The potential they build on is Ruderman and Sutherland 1975, ApJ 196, 51.

Classroom dipole. P and Ṗ for the four buttons are ATNF Pulsar Catalogue v2.9.0, not invented and not a later ephemeris. τ = P/(2 Ṗ). B = 3.2×10¹⁹ √(P Ṗ) G. Ė = 4π² I Ṗ / P³. R_lc = c P / (2π). The Crab’s characteristic age is 1,257 yr. The supernova of 1054 is 972 years before 2026. The dashed line is a 10¹² V sketch. Not a timing model. Earthrise preview: NASA / Bill Anders (public domain).

Keep going

The path

Presets	Crab B0531+21, Vela B0833−45, PSR J0437−4715, and B1919+21. Each sets the ATNF v2.9.0 P0 and P1. Nothing else is plotted as a pulsar.
Period	1 ms to 20 s. The unit buttons switch the number between milliseconds and seconds. The slider is logarithmic in seconds.
Period derivative	10⁻²² to 10⁻⁹, logarithmic. Dimensionless. Scientific notation is accepted, for example 4.20972e-13.
Diagram keys	With the plot focused, Left and Right step the period by a fortieth of a decade. Up and Down step the derivative the same way. Shift steps by a tenth of a decade.

Buttons	Enter or Space selects. The pressed button is aria-pressed. Targets are at least 44 pixels on a side.
Phone	Below 900 pixels the long section row becomes a single Sections control. Open it for the same links. The site menu is the Menu button in the top bar.
This dialog	The question mark, or ?, opens it. Escape closes it. Tab stays inside until you close it. Click the dim page behind the card to close it.

Pulsars
and the spin-down

Cited, not invented

Stellar lifetime

Black holes

Gravitational waves

HR diagram

Solar map

Pulsar desk

Controls

Readouts

Keyboard and touch

Characteristic age	P / (2 Ṗ). A year is 365.25 × 86,400 seconds. The headline is this age.
True age	Only the Crab has one here: 2026 − 1054 = 972. Other rows say the formula is not a birthday.
Dipole field	3.2×10¹⁹ √(P Ṗ) gauss. Two figures in the 3.2. I = 10⁴⁵ g cm². R = 10 km.
Spin-down power	4π² I Ṗ / P³ in watts, and the same power in erg/s. The catalogue rounds the erg/s figure.
Light cylinder	c P / (2π), with c = 299,792,458 m/s.
Diagram	Log P across, log Ṗ up. Age lines, field lines, and one dashed 10¹² V sketch. Millisecond pulsars are labeled at short period and small derivative. The shaded side of the sketch is an approximate death valley, not a catalogue.

Pulsarsand the spin-down

Cited, not invented

Stellar lifetime

Black holes

Gravitational waves

HR diagram

Solar map

Pulsars
and the spin-down