Analytical Proof of the NS-Capture Theory



	The approach will be to have a brief description of the proof,
	 followed by a more detailed description.

	i.e. a high level description followed by a low level description.
	 (this page is a draft that starts w the low level description)

	The observations include directly seeing a supergiant(SG) experiencing
	 a supernova event (ex. SN1987A), plus we have observations of a post-
	 supernova environment (ex. the Crab Nebula and Pulsar) that shows
	 the supernova remnants (SNR) spread over several light years, with
	 a fast pulsar (FP) near the center of the SNR that is slowing down (FP(-)).

	Since the FP(-) is losing energy as it radiates pulses it will continue
	 to slowdown, becoming a slow pulsar (SP(-)) and eventually a solitary
	 neutron star that is not pulsing at all (NS(0)).

|              |  SG | => | SNE | => |FP(-)| => |SP(-)| => |NS(0)|   (1)
|                                    |+SNR |                         
|              |<-- SN 1987A -->|                                
|                         |<----- Crab: 1054 AD -> present ----->|

	Therefore we can represent the above process as: the SG "implies"
	 the SNE, meaning that a solitary SG has been observed to transform
	 via a supernova explosion (SNE) to become a collection of
	 supernova remnants.

	According to NS-Creation theory, a supergiant can spontaneously
	 explode in a supernova explosion producing a cloud of supernova
	 remnants plus a rapidly rotating Neutron Star Pulsar that is
	 slowing down in rotation (FP(-)). i.e. the Pulsar is created
	 as part of the SNE and is initially spinning fast, but slowing

	In the NS-Creation theory, the fast pulsar is created as a result
	 of the supernova explosion, and did not exist prior to that

	By contrast the NS-Capture theory claims that it was the fast pulsar
	 that caused the explosion, by disrupting the supergiant's atmosphere,
	 as it spun up from a non-rotating NS into a fast pulsar (FP),
	 by using the SG's material, itself, to fuel the whole system
	 to high energy, exceeding 100,000 times the luminosity of the Sun.
	The NS-Capture theory states that it was a passing neutron star, previously
	 unassociated w the SG, that collided w the SG and became bound to the SG
	 in a shrinking orbit that eventually destroyed the SG, while, itself,
	 as part of the process of destruction, spun up from a NS(0) to an FP(+).
	And once the SNE occurred and there was no longer a SG companion to
	 fuel the FP(+), that the FP(+) turned into an FP(-) and began to slow down
	 among the SG remnants (SGR).

	Basically, we have 2 claims:

	  1. NS-Creation:    SNE =>  FP
	  2. NS-Capture:      FP => SNE

	Both of these cannot be true, but one of them must be true.

	To be more explicit:

	  1. NS-Creation:     SG =>    SNE => FP+SNR == FP+SGR
	  2. NS-Capture:   FP+SG => FP+SNE => FP+SNR == FP+SGR

	i.e. 1. claims the SNE created the FP, along with the SNR, whereas
             2. claims the FP created the SNE.

	i.e. 1. also claims that the SG, by itself, caused the SNE,
	          which, in turn, created the FP, whereas
             2. also claims that it was the FP interacting w the SG that caused the SNE.

	The observations of the conditions under which a slow pulsar can spin up
	 to become a fast pulsar by being a part of a binary system with its companion
	 being a supergiant is observed in X-rays:

|              |NS(0)| -> |SP(+)| -> |FP(+)| -> | ??? | -> |FP(-)|   (2)
|              | +SG |    | +SG |    | +SG |    | ??? |           
|              |<- BeXrB's, Cen X-3, ... ->|  

	In the above diagram, the first three states are observed in X-rays,
	 while the 4th state is shown as ? marks to indicate that it is not
	 currently "known" what will become of this system.
	However, whatever becomes of the system will of necessity end up with
	 the SG being destroyed and a fast pulsar remaining intact, but slowing
	We can also fill in the ? marks with an obvious intermediate state
	 consisting of a fast pulsar that is no longer spinning up (FP(=))
	 and starting to spin down (i.e. equilibrium between the 2 states),
	 plus the remnants to the supergiant (SGR) that has been destroyed.

|              |NS(0)| -> |SP(+)| -> |FP(+)| -> |FP(=)| -> |FP(-)|   (3)
|              | +SG |    | +SG |    | +SG |    | +SGR|           

	If we now try to apply NS-Creation theory to explain the NS(0) in
	 the first state of (2) or (3), we need to start with a binary containing
	 2 supergiants, of which one explodes in an SNE leaving a fast pulsar
	 plus the other supergiant, whereby the fast pulsar subsequently
	 slows down, leaving us in state 1 of (2), which will then proceed
	 as the rest of process (2):

|              |  SG | -> | SNE | -> |FP(-)| -> |SP(-)| -> |NS(0)|   (4)
|              | +SG |    | +SG |    | +SG |    | +SG |    | +SG |      
|                      |<-------- not observed --------->|

	The NS-Creation theory predicts that the above sequence, (4), must exist,
	 but fails, because there are no observations of systems
	 in 2nd, 3rd, or 4th states of sequence (4).
	On the other hand, NS-Capture theory claims the 3rd and 4th states should not exist, because
	 an SP or FP cannot slow down in the presence of a supergiant (SG), which is
	 consistent w the fact that no observations have been found of these states.
	Given the number (~2500) of pulsars already observed, and the fact
	 that none of them fit the model of (4), leads us to conclude that
	 NS-Creation CANNOT explain the NS is state 1 of (3).

	Therefore based on the fact that either NS-Creation or NS-Capture
	 MUST be true, and the observed fact that NS-Creation is false,
	 we MUST conclude that NS-Capture is true.

	Once we have reached the conclusion that NS-Capture MUST be true,
	 then we can ask, where did the NS come from that resulted in the
	 initial capture in process (3).

	The answer is found by calculating how many NS's must be floating
	 around in order to produce a capture rate that matches the
	 observations. As we will see, that number is very large, to the
	 point that NS's must significantly outnumber the regular stars,
	 RS's, in the Milky Way Galaxy (MWG).


    1/19/20:  update to analytical proof

	The proof hinges on one key concept:

	 When NS-Creation theory creates a neutron star it creates it in the form
	  of a fast spinning pulsar.

	    This is evident in the Crab, plus all the supernova remnant pulsars
	      on the P/P-dot diagram: all these pulsars have spin periods much
	      less than 1 second, and they are all spinning down.

	    Therefore we have chosen the term FP(-) to describe the pulsar/neutron star
	     that is created by the NS-Creation theory.

	 We are also assuming that there is only one way to create a neutron star, i.e.
	  every neutron star we encounter must have been created in the same manner,
	  namely a giant star experiencing a supernova explosion, lighting up the
	  galaxy and crushing the core of the giant into a neutron star pulsar, FP(-).

	Therefore, the neutron stars found in the x-ray binary pulsar systems must also
	 have each been born as an FP(-), and somehow slowed down to have their current
	 spin periods of 1's to 100's of seconds, orders of magnitudes slower than any
	 of the FP(-)'s found in SNR's. We have therefore chosen the term "SP" for these
	 slow pulsars.

	However, unlike the isolated pulsars, that are all spinning down, the x-ray pulsars
	 are all spinning up. Therefore we designate them as SP(+).

	i.e. in a binary system where accretion is taking place, a neutron star cannont spin down,
	 it can only spin up.

	Therefore, if an FP(-) is created in a binary system with a giant companion, if accretion
	 is taking place, then it can only spin up further, i.e. as an FP(+).

	Therefore, we must conclude that in order for an FP(-) to spin down in a binary system,
	 then accretion cannot take place..

	Therefore, we should find several FP(-) or SP(-) pulsars in binary systems in order
	 to set the stage for the later spin up with accretion observed in the x-ray pulsars.

	No such systems are known to exist. i.e. there are no observed giant stars that have
	 a neutron star pulsar companion that is spinning down.

	In addition, it seems unlikely that if an FP(-) was created in the vicinity of a
	 giant star that the giant star would not be disrupted by its presence in which case
	 bursts of material from the giant would regularly be exploded from the giant in
	 some manner, which would, in turn, provide fuel for the FP(-) to accrete therefore
	 preventing it from ever slowing down.

	Therefore, we must conclude that the SP(+)'s found in the x-ray binaries must
	 have been introduced to those systems as basically non-spinning neutron stars
	 that started to spin up once they began having repeated encounters with their
	 giant star companion.

	Another aspect worthy of note, is that the x-ray pulsars not only are inconsistent
	 with the isolated FP(-) pulsars in terms of having been created by an SNE, they
	 also provide an observed mechanism for creating an FP(-), i.e. spinning up
	 from an SP(+) to an FP(+), then causing the giant companion to explode in
	 an SNE, and then continue as an FP(-) amongst the SNR.

	Therefore NS-Capture provides the only mechanism to explain both the binary
	 x-ray pulsars and the isolated radio pulsars.