which pulsar has the longest spin period so far? Vela X-1 has a 283s spin period, while this one may have a 2.7hr spin period.
However neither of them is included in ATNF catalog.
They are not included in $p-\dot{p}$ diagram either. The longest period is just about 10s in this diagram.

My question is since Vela X-1 is definitely a pulsar, why is it not accepted? 
Which one has the longest spin period so far?



When this diagram is mentioned in textbooks, we do not say which kind of pulsars are included at all. Why?

$p-\dot{p}$ diagram only shows radio pulsars?  no!
you can see magnetars,radio quiet pulsars and pulsars which are only active in $\gamma-ray$

Except magnetars, there are no single pulsars only pulse in X-ray band?

If a pulsar pulses in X-ray band, that does not necessarily mean it is an X-ray pulsar?
Only pulsars in binaries and accretion-powered pulsars could be called X-ray pulsars? They should be radio-quiet?

In binaries, there are no pulsars which pulse in radio and x-ray band simultaneously?
Vela X-1 is an example of an accretion-powered pulsar. These emit pulsed emission because they are accreting material onto their magnetic poles. If the magnetic poles and rotation axis are misaligned this results in pulsed X-ray emission. The power for the pulsar comes from the infalling material.

The classic P-Pdot diagram you show is for radio pulsars, or rather rotationally-powered pulsars. These are objects that are pulsators by virtue of accelerating charged particles from their magnetic poles out along the field lines. These accelerated relativistic particles emit synchrotron and curvature radiation that is beamed and intensified. The energy ultimately comes from the rotational kinetic energy of the pulsar.

In other words, other than the fact that they both involve neutron stars, these are completely different phenomena.

In terms of radio pulsars, your diagram looks reasonably up to date. I think the longest period object on your diagram is PSR J2144-3933, which has a period of 8.51 seconds (Young, Manchester &amp; Johnston 1999).

Your diagram has a line marked as "graveyard". I believe this is a locus defined by Bhattacharya et al. (1992) and has the form
$$P = 2.42\times 10^{-6} B^{1/2}\ s,$$
where the pulsar magnetic field $B$ is in units of Gauss (typical pulsar values would be $B=10^{10}-10^{13}$ Gauss (as also marked on your diagram). The theory behind this "death line" is discussed by Ruderman &amp; Sutherland (1975). Briefly it arises from the requirement of a minimum potential difference to be generated such that accelerated particles produce energetic enough radiation to stimulate the production of further electron/positron pairs. If the magnetic field strength falls or the rotation period gets too long, then this mechanism fails and the pulsar is quenched.

PSR J2144-3933 appears to lie beyond this death line, but there are other ideas and models of how this death line may arise (of which I am not very familiar, but see for example Zhang et al. 2000).

Note that millisecond pulsars are thought to recycled accretion-powered pulsars. That is that they gain sufficient angular momentum from an accretion process that they spin up to become radio pulsars again.

Radio-quiet pulsars are thought to be rotationally powered but where the radio beam is narrower than say the beaming of gamma rays in which they are detected.

Magnetars are something different again. Their power comes from the decay of extraordinarily strong magnetic fields.
The soft gamma ray repeaters and anomalous X-ray pulsars fall in this category.

So you can see that the word "pulsar" might include all sorts of different types of objects and physics.