About This Page
A Simulation, Not an Animation
A pulsar is a neutron star that spins and has a strong magnetic field: about one and a half times the mass of the Sun packed into a ball some 24 km across. Its magnetic poles are not aligned with its spin axis, so beams of radiation from near the poles sweep around the sky. If a beam crosses Earth, we see a pulse once per turn. The fastest known pulsar turns 716 times a second.
Everything above is calculated, in three parts:
- The star's surface is ray traced. For each pixel, your graphics card follows a ray of light back through the curved spacetime around the star, using general relativity, to find the point on the surface it came from. The pixel's color and brightness come from that point's temperature, its speed and the gravity the light climbs out of.
- The magnetic field lines are integrated from an exact solution of Maxwell's equations for a spinning magnet in empty space. Each point on each line is then placed on screen where its light would appear, bent by the star's gravity.
- The pulse plots are what a distant observer would record: X-rays from the hot spots, summed over the whole visible surface at every moment of the turn, and radio waves from a beam above the poles, including its polarization.
Change a setting and all three are recalculated. The equations, their simplifications and the tests they pass are on the method page. It runs best on a laptop or desktop computer.
In Everyday Terms
How Extreme Is a Neutron Star?
For a star of 1.4 suns and 12 km. How each number was worked out is on the method page.
At a Glance
The Numbers Behind the Image
What You’re Seeing
Reading the Image
The hot spots
Near the magnetic poles, charged particles accelerated in the magnetosphere strike the surface and heat it to around a million kelvin. The spots shine mainly in X-rays. Here the temperature is scaled down into visible light so the spots appear white and the rest of the star a dull orange; the shapes and brightness ratios are unchanged.
The tilted magnetic axis
A pulsar's magnetic poles are not at its spin poles. Here the magnetic axis (red) is tilted 60° from the spin axis (white), so the hot spots and the field lines sit on the star's sides and are carried around as it turns. The tilt is what makes a pulsar pulse: if the two axes were aligned, the beams would point along the spin axis and look the same at every moment (try the “Aligned axes” view). Nothing forces the two to line up. Earth's magnetic axis is about 10° from its spin axis, and Uranus's about 60°.
Seeing past the edge
Gravity bends light leaving the surface, so rays from points slightly behind the edge still reach you. For a typical neutron star you see about three quarters of the surface at once, which is why a hot spot stays in view for most of a turn. Switch on the grid to see how far round the far side the view reaches.
Closed field lines
Close to the star the field is that of an ordinary bar magnet, a dipole. Lines that leave one pole and return to the other within the light cylinder turn with the star like spokes on a wheel.
Open field lines
Lines from close to the poles reach the light cylinder before they can turn back. Beyond it they cannot turn with the star, so they trail behind it in spirals. The footprints of these open lines on the surface are the polar caps, which is why the hot spots sit there.
The light cylinder
Anything that turned with the star at distance r from its spin axis would move at speed Ωr. At r = c/Ω that speed is the speed of light. This radius sets the size of the whole magnetosphere: 67 km for the fastest pulsar, 1,590 km for the Crab.
The radio beam
Radio waves come from charges streaming out along the open field lines, tens to hundreds of km above the surface, beamed along their direction of motion. The beam is a cone around the magnetic axis. Each time it sweeps past you, you see a pulse.
The Pulses
What an Observer Records
Telescopes cannot resolve a neutron star; it is far too small and far away. What they record is brightness over time, folded on the spin period into a pulse profile. The “What an observer records” panel in the lower-left corner of the simulator (click its title to open it) shows three such profiles for whatever you have set up, and the vertical line marks the present moment.
X-ray brightness comes from the hot spots. Light bending keeps the spots visible for longer, which smooths the pulse: in the first experiment below, the brightness swings 38% either side of its mean with bending and 78% without. The amount of bending depends on the star's compactness, the ratio of its mass to its radius. Rotation adds Doppler shifts that make the pulse lopsided, and the size of that effect depends on the radius. Together, the two let a profile be fit for mass and radius separately. That is how NASA's NICER X-ray telescope measures neutron star radii:
| Pulsar | Mass (suns) | Radius | Analysis |
|---|---|---|---|
| PSR J0030+0451 | 1.34 (+0.15 / −0.16) | 12.71 (+1.14 / −1.19) km | Riley et al. 2019 |
| PSR J0437−4715 | 1.418 ± 0.037 | 11.36 (+0.95 / −0.63) km | Choudhury et al. 2024 |
| PSR J0740+6620 | 2.072 (+0.067 / −0.066) | 12.39 (+1.30 / −0.98) km | Riley et al. 2021 |
Ranges are 68% credible intervals. An independent team analyzing the same data found similar values (for J0030, 1.44 suns and 13.02 km; Miller et al. 2019). For J0030 the best fits put both hot regions in the same hemisphere, one of them a narrow arc, which a simple centered dipole cannot produce. The simulator's “same hemisphere” option shows a configuration of that kind but is not a fit to the data.
Radio brightness and polarization come from the beam. The radio waves are linearly polarized along the local magnetic field direction. As the beam sweeps past, the projected field direction rotates, and the polarization angle traces an S-shaped curve. Its steepest point marks the moment the magnetic pole passes closest to your line of sight. The shape of the curve gives the magnetic tilt and viewing angle. This is the rotating vector model of Radhakrishnan and Cooke (1969), shown dashed in the third plot.
The computed points differ from the dashed curve because the charges move with the star as well as along the field. Their beam is tilted forward by aberration, and light from the far side of the beam arrives later. Blaskiewicz, Cordes and Wasserman showed in 1991 that the polarization swing then lags the center of the pulse by 4h/RLC radians, where h is the emission height. This is one of the few ways to measure where in the magnetosphere radio waves are made. The simulator's numerical beam reproduces that shift to within 13% (details).
Controlled Experiments
Switch Off the Physics, One Effect at a Time
Each comparison below changes a single setting. All of them are available in the simulator above, and every image and curve on this page was produced by it.
Light bending
With straight-line light you see exactly half of the star. With general relativity you see 76% of it: the grid lines bunch up near the edge as the far side comes into view. The same effect flattens the X-ray pulse, because neither hot spot ever fully disappears.
Straight-line light: half the surface
General relativity: 76% of the surfaceThree magnetic fields
A static dipole turning rigidly is the textbook picture, and it is correct only far inside the light cylinder. The rotating dipole in vacuum is the exact solution of Maxwell's equations for a spinning point magnet in empty space. Deutsch (1955) solved the same problem for a magnetized sphere; outside the star the two differ by under 4%. Near the light cylinder its lines stop following the star and sweep back into spirals that carry energy away as electromagnetic waves. Real pulsars are not surrounded by vacuum but by plasma, which carries currents and reshapes the field. The plasma-filled sketch joins a dipole inside the light cylinder to Bogovalov's (1999) exact “split monopole” wind outside it. In that wind, field lines from the two magnetic hemispheres are separated by a rippled current sheet, purple here. The join between the two regions is our interpolation, not a calculation.
Static dipole
Rotating dipole, vacuum (exact)
Plasma-filled (sketch)Where you stand
The same star produces very different pulses depending on the magnetic tilt χ and your viewing angle ζ. When both are near 90° you see both hot spots in turn: a main pulse and an interpulse half a turn later. When both are small, one spot stays in view all the time and the pulse is shallow. Some pulsars are invisible in radio from Earth because their beam never sweeps across us.
Doppler shifts
At 716 turns per second the surface moves at up to 0.22 c. A hot spot moving toward you is beamed brighter, and one moving away is dimmer. Light-travel time adds to the effect: while a spot approaches, points near the edge show it as it was slightly earlier, so its image is stretched and covers more of the surface. In the example below, Doppler shifts raise the pulse fraction from 24% to 55%, move the peak about a tenth of a turn earlier than the moment the spot faces you, and make the pulse rise faster than it falls. NICER uses this asymmetry, along with the light bending, to separate a star's mass from its radius.
Radio emission height
Raising the emission height widens the beam, because the open field lines spread out with distance. It also moves the polarization swing later relative to the pulse.
The Lighthouse
What a Pulsar Beam Really Looks Like
Most people picture a pulsar the way textbooks and science videos draw it: a star with two narrow, straight searchlight beams sweeping around, like the left image below. Part of that is right. A radio pulsar does emit a beam around each magnetic pole, and the pulses come from that beam sweeping past Earth. Three things about the picture are not right.
The beam can't be a rigid searchlight. The textbook beams on the left reach 8 light-cylinder radii from the star. Turning with it, their far ends would move at up to 8 times the speed of light. Nothing actually moves that way. Each photon flies straight outward from where it was emitted while the beam's direction keeps sweeping on. At any instant the light already emitted lies along a spiral, as on the right, like water from a spinning lawn sprinkler. The spiral has one turn per c × P, about 420 km for the 716 Hz pulsar. The “light in flight” view plays at 0.25 turns per second by default, six times faster than the others, so you can watch the spiral being laid down.
A beam can't be seen from the side. Searchlight beams on Earth are visible because fog and dust scatter some of their light toward you. A pulsar has almost nothing around it to do that, so you only detect the beam when it points at you. The middle image draws the beam as a cone anyway, as a visual aid, in the same way as the field lines, and stops it at the light cylinder.
Real beams are wider and less tidy. The computed beam (middle and right) is the brightest ring of emission from the open field lines. It is tens of degrees wide for this pulsar, comes from just above the surface, and is tilted forward by the star's rotation. Real beams vary from a few degrees to a few tens of degrees across and are often hollow. A pulsar's gamma rays come from somewhere else again, farther out near the light cylinder.
Is this to scale? Yes. The star (12 km), the light cylinder (67 km) and the spiral (420 km per turn) are drawn at their true sizes. The spacing of the spiral is always 2π light-cylinder radii, because both are set by the spin period. The fastest pulsars are the only ones where the star is visible at that scale. For the Crab pulsar one turn of the spiral is 10,000 km, about 830 times the star's radius. For a pulsar that spins once a second it is 300,000 km, and the star would be far too small to see.
Textbook beams (not physical)
Cone to the light cylinder (visual aid)
Light in flight, at one instantTimescales
How Fast, and How Much Slowed Down
Periods are from the ATNF Pulsar Catalogue. The last column is how much the default playback speed (0.04 turns per second, one turn every 25 seconds) slows each one. The “light in flight” beam view plays at 0.25 turns per second, so it is slowed about 6 times less.
| Object | Spin period | Light cylinder | Equator speed | Slowed by |
|---|---|---|---|---|
| PSR J1748−2446ad | 1.40 ms | 67 km | 0.22 c | 18,000× |
| PSR J0740+6620 | 2.89 ms | 138 km | 0.13 c | 8,700× |
| PSR J0030+0451 | 4.87 ms | 232 km | 0.066 c | 5,100× |
| PSR J0437−4715 | 5.76 ms | 275 km | 0.052 c | 4,300× |
| Crab pulsar | 33.4 ms | 1,590 km | 0.0093 c | 750× |
| Vela pulsar | 89.3 ms | 4,260 km | 0.0035 c | 280× |
| Magnetar SGR 1806−20 | 7.56 s | 360,000 km | 0.00004 c | 3.3× |
Equator speeds use each star's NICER radius where one exists, otherwise 12 km, and are measured by an observer at rest on the surface. Magnetars spin slowly, and their light cylinders lie hundreds of thousands of kilometres out. At the scale of the star their field is a plain dipole, so the simulator shows a magnetar's field out to 60 stellar radii only. What makes magnetars distinctive, a field of about 1015 gauss that decays and powers bursts, is outside this model.
Limits
What This Model Leaves Out
- The real magnetosphere. The vacuum field is exact for a spinning magnet in empty space, but pulsars are surrounded by plasma. Plasma-filled (“force-free”) solutions for a tilted pulsar exist only as large numerical simulations (Spitkovsky 2006). The plasma option here is a labeled sketch: a dipole inside the light cylinder, an exact split-monopole wind outside it, and an interpolation in between.
- Gravity's effect on the field. The field lines are computed in flat space. Near the surface of a 1.4-sun, 12 km star, curved spacetime makes a dipole field 1.4 to 1.5 times stronger than in flat space and slightly changes its shape (Wasserman & Shapiro 1983). Light bending and redshift are fully relativistic.
- The star's own rotation in spacetime. Spacetime outside the star is treated as Schwarzschild (non-rotating). This leaves out frame dragging and the flattening of fast spinners: a 716 Hz star of 1.4 suns and 12 km is about 11% wider at the equator than at the poles (AlGendy & Morsink 2014), and a 205 Hz star like J0030 about 1%. NICER's analyses include the flattening; ours does not.
- Hot spot physics. Spots are uniform-temperature circles with sharp edges, emitting either as a blackbody or with a simple scattering-atmosphere limb darkening. Real analyses use detailed hydrogen-atmosphere models and fit the spot shapes.
- Temperature and color. Hot spots near a million kelvin glow in X-rays. The surface colors are a visible-light stand-in (8,000 K for the spots); brightness ratios and the shifts between them are computed exactly.
- Radio emission. How pulsars actually make radio waves is still not understood. The beam here is emission from a single height, beamed along the motion of charges on the open field lines, with a chosen brightness pattern across the cap. Each direction is spread over 1° to keep the computed curves smooth. The beam's light is treated as travelling in straight lines unless light bending is on.
- Field lines and beam cones are not light. Magnetic field lines do not glow, and a beam cannot be seen from the side. Both are drawn as if they glowed, so that bending and light-travel time apply to them as they would to anything luminous. Lines are drawn at constant width, so faint secondary images look thicker than real ones would. The “light in flight” spiral is the beam's actual light, traced in straight lines from the emission height; near the star its paths would bend slightly.
- Glow. The optional glow imitates scattering in a camera or eye and is the only cosmetic step.
Go Deeper
Method, Verification and Sources
The method page gives the equations behind each part: the light-bending table, the vacuum field, the pulse integration and the radio beam. It also lists the tests the code passes against independent calculations, and 27 references.
Built with Claude Opus 5.5: the physics design, the GPU code, the test programs and the write-up.
Use These Images
Free to Use, With Credit
Images and recordings you make with this simulator, and the figures on this page, may be used for any purpose, including teaching, publication and commercial work, without asking permission. We only ask that you credit Frosty Wren Studios, for example:
Pulsar simulation: Frosty Wren Studios, frostywren.com/pulsar“Save image” adds this line to the corner of each picture by default; you can switch it off under Display and output. The simulator’s source code is not covered by this permission.