How NS-Capture Theory explains the P/P-dot diagram

This page describes how NS-Capture Theory is used to explain the P/P-dot diagram.

The following is an example of the standard P/P-dot diagram taken from the book “Essential Radio Astronomy” by James J. Condon and Scott M. Ransom (2016), see book excerpt at: “Chapter 6: Pulsars“, which also includes in depth information about the diagram.

Figure 0. The standard P/P-dot diagram

The above diagram, figure 0, gives data points of spin period (P) and rate of change of spin period (P-dot) for most of the pulsars known in 2016.

This page describes at a high level of how the NS-Capture Theory explains the major features of the P/P-dot diagram. More information about the details of data on the P/P-dot diagram is described on the “Classification of the Pulsars” page.

This page also makes the assumption that the NS-Capture theory is correct and that there are probably more neutron stars than regular stars in the Milky Way Galaxy, and uses that as a basis for explaining the properties of the P/P-dot diagram.

Spinning-down isolated pulsars

We will start by describing at a high level the groups of pulsars on the standard P/P-dot diagram, Figure 0, above.

The following diagram is a summary sketch of the P/P-dot diagram which shows the major clusters of points and how to characterize the properties of each cluster.

Figure 1. Spinning-down pulsars

Note that all the pulsars on the standard P/P-dot diagram are “spinning down” pulsars. i.e. the length of time for one full rotation, the period, is getting longer as the pulsar slows down. That means that the pulsars are all moving from left to right on the diagram as time goes by. In addition, the pulsars appear to be moving vertically downward with time, which may be due to pulsar’s magnetic field weakening caused by residual accretion of material left over from the supernova explosion. The rate at which they are slowing down is on the vertical axis, with the pulsars that are slowing down the fastest are at the top of the diagram, and those that are hardly slowing down at all are at the bottom.

It is believed that the slow-down rate is a function of the strength of the magnetic field associated with the pulsar with the fastest slowing pulsar having the strongest magnetic field and the slowest slowing down pulsars having the weakest magnetic field. This is because a pulsar with a strong magnetic field will emit more energy with each rotation at a specific period, than a pulsar with a weak magnetic field rotating at the same period. i.e. a pulsar at a specific “P” will have a weaker magnetic field than a pulsar at the same “P”, but directly above it on the diagram.

Spinning-up pulsars: part of a binary star system

There is a second type of pulsar that is also found, which is quite different from the spinning down pulsars, but these pulsars do not appear on the standard P/P-dot diagram. These are the neutron star spinning-up pulsars, which are always found as part of a binary system, unlike the spinning down pulsars which are not in binaries, except for a specific class of millisecond pulsar that will be discussed later. These are spinning-up pulsars are high energy x-ray pulsars that emit on the order of 100,000 times the energy emitted by the Sun. The emission is not coming from the neutron star, itself, but from the material falling into its gravitational and magnetic fields, which is drawn in from the atmosphere of the companion star.

The process of pulling material from the companion to the neutron star is called “accretion”. During the accretion process, material is pulled by the neutron star’s gravitational field and directed along the magnetic field lines toward the pole axes (north and south), so that it flows in two funnels: one in toward the north pole, and the other toward the south pole. During the fall in the material, which is made of charged particles, is accelerated both in a spiral motion with a declining radius by the magnetic field and a vertical motion toward the NS surface by the gravitational field. Radiation is emitted out of both funnels in a directed beam of x-rays which can be observed at great distances, such as at Earth, when the beam passes through the Earth’s line of sight.

The combination of the material flowing in along the funnels, and the general orbital motion of the NS around the companion results in a torque pulling on the two funnels, which causes the NS to begin to rotate with a spin axis perpendicular to the magnetic axis which results in a lighthouse-like beam of radiation rotating in and out of an observer’s line of sight. This is how a pulsar is created from an otherwise non-rotating neutron star. (At least non-rotating with respect to its magnetic field axis.)

The following diagram, figure 2, while not as detailed as the standard p/p-dot spinning down diagram, is the “p/p-dot spinning-up diagram” which we will see below has an important correspondence to the spinning-down diagram in figure 1.

Figure 2. Spinning-up Pulsars

Under the assumption that the NS-Capture Theory is correct, we expect there to be occasional binding collisions between some random neutron star and a random regular star, which result in a bound binary star system consisting of the neutron star plus the regular star it collided with. During the collision, enough energy was lost from the neutron star’s orbit with respect to the target regular star, such that the neutron star (NS) did not have enough energy to achieve escape velocity and thus remained bound to the regular star (RS).

Upon becoming bound by this initial collision, the newly bound neutron star, begins its life as a pulsar at the lower right of the diagram in figure 2. Its initial spin period will effectively be infinity, but once the magnetic field begins interaction with the atmosphere of the regular star companion, the NS will begin to be twisted by the atmosphere of the RS interaction with magnetic field of the NS. Once the twisting begins, the NS will have a non-zero rotation rate with a component perpendicular to the magnetic field pole axis, which gives it a finite spin period that gets faster (shorter) as the magnetic field of the NS continues to interact with the atmosphere of the companion RS.

At this point the NS will keep colliding with the RS each time the orbit reaches the distance of closest approach, during which time the NS loses more energy to the RS through tidal interaction, and the elliptical orbit becomes smaller, but the NS will keep returning to the distance of closest approach and lose more orbital energy each time. The NS will also have an increasing spin rate perpendicular to the magnetic pole axis which is the initial stage of the NS becoming a measurable pulsar.

After many such orbits and collisions the orbit will become nearly circular with the radius between the NS and RS continuing to shrink as the NS continues to lose orbital energy while in constant collision in the atmosphere of the RS.

During this process the NS will emit x-rays when it is in contact with the RS through the process of accretion, which is when the particles of the RS atmosphere are pulled toward the NS by the NS’s gravitational field. In addition, the particles interact with the NS’s magnetic field and spiral down to the NS surface along the magnetic field lines which brings the particles to the north and south magnetic poles of the NS. The x-rays are typically emitted by a synchrotron or bremsstrahlung process where decelerated charged particles can emit x-rays.

The streams of infalling particles act as a torque on the NS causing it to begin rotating as a pulsar. As the process continues the pulsar spins faster and faster.

We can view the spin rate of the pulsar as a function of time and therefore we can have a p/p-dot diagram where because the spin is increasing, the change in period is decreasing, therefore p-dot, or dp/dt, is negative which places the p/p-dot points below the p-axis of the p/p-dot diagram.

Correlation between spinning-up and spinning-down pulsars

We now have 2 collections of spinning pulsars: the usual collection of pulsars on the positive p-dot axis, above the line between the 2 collections that are all spinning down, and a new collection below the line that are spinning up in binary systems.

Note that the spinning up pulsars first become visible at much longer rotation periods than the spinning down pulsars. This is because there is much more energy emitted when the pulsar hits its companion at the distance of closest approach and the x-rays are visible by satellite, while at other points of the orbit, the emission is primarily by radio waves and is not generally strong enough to be observed.

The following diagram, figure 3, shows both the spinning-up and spinning-down pulsars on a single graph with a shared spin period, P, axis. Note that both the top half and bottom half of the diagram are logarithmic P-dot values, and both the upper and lower P-dot axes approach zero as they hit the midline joining the 2 collections. In absolute terms the positive and negative parts of the P-dot axis will never meet, but asymptotically get smaller and smaller as they approach each other. We therefore simply truncate the axes at the spin-down and spin-up rates of 10^-24 sec/sec. There are no measurements currently at smaller P-dot values, so we do not lose any known data by truncating each half axis at this point. We simply assume that there is a non-zero gap between the 2 halves of the diagram, but that nothing is measurable nor physically significant within that gap. Another way to look at this is to imagine a spinning-up pulsar loses its source of accretion fuel. It will then have no force causing it to spin up any further and will begin to spin down.

Figure 3. Correspondence between spinning-up pulsars and spinning-down pulsars

We then have the phenomenon that when a pulsar stops spinning up, that it will instantaneously start spinning down, and thus cross the gap to the top half of the diagram joining the other spinning down pulsars. Therefore, on the far left of the diagram, we have binary millisecond pulsars, some of which may be slightly spinning up and some slightly spinning down. Some may intermittently go thru spin-up and spin-down phases due to the presence or absence of fuel from the mostly burned out companion. This would result in a “hovering” above and below the “P-dot = zero” horizontal axis. This axis is the asymptotic limit of both the logarithmic positive and negative axes along the y-direction. One could make an analogy with plotting the diminishing negative acceleration of a ball thrown in the air to the point where the acceleration becomes positive when the ball just passes its peak and starts to fall down.

Transitions between spinning-up and spinning-down pulsars

This section will explain the full picture of the both the upper (positive dP/dt) spinning-down isolated pulsars and the lower (negative dP/dt) spinning-up binary pulsars. As we will see, all pulsars begin their existence upon being captured by a regular star (RS) on the lower right portion of Figure 4, which contains the spinning-up binary pulsars.

The final diagram, Figure 4,  shows how a spinning up binary pulsar transitions to a spinning down pulsar. There are 2 distinct cases:

  1. A spinning up binary pulsar can cause its companion to blow up in a supernova explosion at which time it will become a spinning-down isolated pulsar amidst the supernova remnants.
  2. A spinning up binary pulsar can exhaust the companion’s atmosphere leaving only a dense core behind while remaining in close binary orbit with the core. Since there is no more fuel or very little fuel left, and the long period of accretion has caused the neutron star’s magnetic field to decay, it will very slowly slow down as a millisecond binary pulsar.

The main difference between these cases is that the pulsars on the far left of the diagram generally remain as binary pulsars with pulsar component spinning at millisecond rates and its companion being simply a burned-out RS of which nothing is left except its core.


How NS-Capture explains the Magnetic Fields of Pulsars

The general theory of pulsars makes a correlation between the magnitude of the spin-down rate (dP/dt) of pulsars and the intensity of the pulsar’s magnetic field. It says that the stronger the magnetic field corresponds to the fastest spin-down rates, which appear at the upper right of the standard P/P-dot diagram, Figure 0. The question is then asked why different pulsars have different magnetic fields. In particular, the binary millisecond pulsars (MSP) in the lower left of the spin-down portion of the diagram have the weakest fields.

The generally accepted reason for the weakened magnetic fields is that the accretion process that spins the MSP’s up to their extremely fast spin rate causes the magnetic field to weaken, so that by the time the pulsar has accreted all the material available to it from its companion, that the magnetic field has been so weakened that the spin-down rate is extremely slow. This then explains why the MSP’s are expected to last hundreds of millions, or billions of years as MSP’s retaining the high spin rate, because the magnetic field radiates so little energy with each rotation that the change in spin rate is tiny and therefore the MSP will remain with a high spin rate for an extremely long time.

The NS-Capture Theory is consistent with this explanation, and, in fact, takes the explanation further as will now be described.

NS-Capture assumes that all neutron stars, that have never been in a binary system, have extremely strong magnetic fields, on the order of 10^14 Gauss or more as shown in the upper right of the standard P/P-dot diagram, Figure 0.

Therefore, when an NS is first captured by an RS, the NS’s magnetic field is at its strongest. However, once captured and accretion begins, the magnetic field starts to weaken as a result of the accretion, as is commonly expected to be the case. Therefore, in the lower half of the dual P/P-dot diagram in Figure 4, the NS will enter at the lower right with a very slow initial spin rate and very long period (P).

(Note: spin rate is rotations per second, and spin period is seconds per rotation, as distinct from P-dot, which is rate of change of spin period, measured in seconds per second, i.e. the difference in seconds that the period has from one second to the next.)

As time goes by, the spin rate increases and the spin period decreases, which causes the spinning-up pulsar to move to the left of the lower diagram. In addition, as the process of accretion of material from the RS to the NS continues with each orbit of the NS around the RS, the magnetic field will begin to decay (become weaker). As the magnetic field becomes weaker, the rate of change of the spin rate (P-dot, or dP/dt) will also decrease, causing the spinning-up pulsar to move up on the lower half diagram.

We therefore end up with a general tendency on the lower half diagram for the pulsar spin properties (P, dP/dt) to move up and to the left consistent with a decaying magnetic field.

Now comes the interesting part:

The RS’s that become bound in a binary with an NS can be any kind of RS: normal stars like the Sun or giant or super-giant stars. The general rule is that the bigger and more massive that the RS is, the more likely it is that it will become bound to a passing NS, simply because its larger gravitational field will increase the cross-section that a passing NS needs to enter in order to become bound.  Basically, the bigger the target, the easier it is to hit.

Figure 4 shows 3 wide vertical arrows pointing from different positions on the lower half diagram to positions on the upper half diagram.

These arrows generally correspond to 3 major categories of RS’s that are captured to a binary system with an NS.

  1. The first category is the right hand vertical arrow, which corresponds to an RS that once bound with an NS, rapidly becomes unstable and explodes in an SNE leaving a pulsar with a strong magnetic field (has not done enough accretion to decay very much) and a long spin period (P). Such a pulsar will appear at the upper right of the upper half of Figure 4, where we find the magnetars with their strong magnetic fields and very slow spin periods.
  2. The second category is the middle vertical arrow, which corresponds to more stable RS’s, which include giants and normal stars of various sorts, that enter into long term close binary systems typical of the eccentric orbit BeXB’s and circular orbit, such as Cen X-3 and Her X-1, which represent a supergiant companion and an RS companion, respectively. These companions explode in major SNE’s leaving behind a well spun-up pulsar with a partially decayed magnetic field that places the now-isolated pulsar in the main center section of the standard P/P-dot diagram, figure 0, with the initial being in the upper left of the spin-down group with periods between 0.01 and 1 second. As the pulsar spins down it will continue to accrete from the supernova remnants (SNR’s) which will weaken the magnetic field and cause the pulsar to move down in the upper half of the diagram, and also the spin period will become longer, which will cause the pulsar to move to the right of the upper half of the diagram.
  3. The third category is the millisecond pulsars, which continue to accrete and spin up until the material for accretion has been used up at which point it will be at the upper left of the lower half diagram, and begin to spin down very slowly which will cause it the then be part of the upper half diagram as shown by the third vertical arrow on the left side of figure 4.

In summary, the above descriptions are how the NS-Capture theory explains both the standard P/P-dot diagram for spinning-down isolated pulsars, Figures 0 and 1, plus the spinning-up binary pulsars shown in figure 2.