A classification of compact stars, depending on the electron distribution in velocity space and the density profiles characterizing their magnetospheric plasma, is proposed. Fast pulsars, such as NP 0532, X-ray sources such as Sco-X1, and slow pulsars are suggested as possible evolutionary stages of similar objects. The heating mechanism of Sco-X1 is discussed in some detail

Coppi, B.

Treves, A.

English

NASA Technical Reports Server

CLASSIFICATION OF RADIATING COMPACT STARS
by
B. Coppi and A. Treves*
Massachusetts Institute of Technology
ABSTRACT
A classification of compact stars depending on the electron
distribution in velocity space and the density profiles character-
izing their magnetospheric plasma is proposed. Fast pulsars, such
as NP 0532, X-ray sources such as Sco-Xl and slow pulsars are
suggested as possible evolutionary stages of similar objects.
The heating mechanism of Sco-Xl is discussed in some detail.
I. Introduction
The radiation emitted by the magnetosphere of a collapsed star depends
primarily on the profile of the distribution function f (vl ) of the plasma
electrons, where vil is the particle velocity along the magnetic field. We
suggest that three main types of distribution function fe account for
different objects which can be associated with compact stars.
One type corresponds to small electric fieldsEll , along the magnetic
field, so that the distribution fe is a low temperature Maxwellian with a
small drift velocity ueli (Fig. la). This regime is associated with slow
pulsars.
Another type corresponds to larger electric fields Egl so that the
distribution fe is a broad high temperature Maxwellian (Fig. lb). This can
be considered as a regime of turbulent heating and is associated with sources
such as Sco X-l.
* Permanent address: Istituto di Fisica dell'Universita, Milano, Italy.
N72-13844 (NASA-CR- 1248 3 13 ) O -P
ACLASSIFICATION O- -- nRADIATING COmPACT STARS B. Coppi, et al
0n0cl0as (sassachusetts Inst. of TechOP 1971 16 pUnclas CSCL 03A ' 90010
(NASA CR OR TMX OR AU NumIcej
https://ntrs.nasa.gov/search.jsp?R=19720006195 2020-03-11T21:47:26+00:00Z
-2-
A third type corresponds to values of the particle density, temperature
and El; field for which the distribution function acquires a tail of fast
superthermal particles (Fig. lc). We can call this a runaway regime and
consider it typical of stars such as NP 0532.
We refer to compact objects, typically neutron stars, whose mass M=M
33 dAmA 6
2 x 10 gr.,A radius a 10 cm. The magnetic field is assumed dipolar on
12
the surface, with an intensity B = 10 gauss. We assume, for simplicity,
that the dipole axis coincides with the axis of rotation and we consider a
region defined by closed magnetic field lines which are contained within the
speed of light cylinder (of radius RSL = c/o, where wo is the star angular
velocity). In this region, the magnetic field is dipolar, there are no
electric fields and no currents along the magnetic lines, the plasma
C-rntt'--> , .J1.with the star. .nd no emiscion occurc. In a second
region the lines of force are open, an electric field E can be maintained
along them and emission processes can be sustained. This region is within
1/2
the latitude 6c = (aw o/c) 2, and we can argue that the width of the
C
active region is of order ° 0 a = (a 3 c/
o c o
The potential difference, seen in an inertial frame between a pole and
a point at about the latitude 0 is (Deutsch 1955; Goldreich and Julian
1969).
I 4 a 2 B (cos 20 -1)
For the parameters considered above
~ O 1.7 x 1011 2volt
0
-3-
We assume that the magnetic lines of force in the active regions are connected
with the external field of the nebulawhkch may surround the considered star,
or of the interstellar plasma. Then it is reasonable to expect that the
electric field ElN along the active magnetic field lines will result from
a potential drop of the order of AD.
An important limit for the electric field Ellis the classical runaway
field ER (Gurevich, 1962) defined as ER = V.m v /e where V eis the
electron-ion collision frequency and vth the electron thermal velocity.
This gives the threshold above which the plasma, under the electric field
influence and in the absence of collective effects,would lose its cohesiveness
and the electrons (so called runaway electrons) would tend to undergo almost
free acceleration. In reality when E > ER, plasma collective modes are
excited and their effects are equivalent to that of a stronglv enhanced
collision frequency. Thus, the electron distribution function is kept from
running away and a strong anomalous resistivity sets in. This involves
a considerable ohmic heating of the plasma, which is commonly called turbulent.
In laboratory experiments, a tail of runaway electrons has been observed
to appear when the electric field is slightly below the field E
R
so that the
effects of a strong turbulence which tends to destroy this tail, are not
T
important. It has been conjectured that a turbulent runaway field ER
should exist with ER >> ER, so that for E > ER particles are not affected
by two body collisions nor by interactions with plasma collective effects
and undergo free acceleration. However, there has not been, as yet, a
clear experimental observation of such a threshold.
-4-
II. Three Types of Collapsed Stars
Slow Pulsars. Slow pulsars (all pulsars but NP 0532) are
observed only at radio-wavelengths. The shape of the pulse is
not stable and sometimes varies greatly from pulse to pulse.
We assume that, for this class of objects, the electron distri-
bution function in velocity space is a narrow Maxwellian slightly
displaced by the electric field El . In particular, there are no
fast electrons which could emit in the higher frequency bands.
The density profile of the magnetosphere corresponds to a shell
of plasma surrounding the star with such a volume that the
overall thermal optical emission is not appreciable. Assuming
for example that the emission spectrum is a black body at
6temperature T X 10 K, we find that an object at a distance of
100 pc, with a magnetosphere extending for 10 stellar radii,
7 m
R '% 10a '" 10 cm would have a visual magnitude m X 27 and thus
could not be seen.
Radio emission can be produced by the coupling of self-
excited, low frequency plasma modes (micro-instabilities of the
inner magnetosphere) with electromagnetic modes. These plasma
modes are likely to be of electrostatic type and to propagate,
along the open lines of force on which they are excited, from
the shell of dense plasma close to the star up to a region
where the plasma density is so low that electromagnetic modes
are not strongly damped. It is then reasonable to expect that
the pulse irregularities observed in the radio band depend on
the characteristics of the regions in which propagation and
-4a-
coupling of plasma modes occurs.
The pulsed emission can be associated with a lack of
symmetry of the star magnetic configuration about the axis of
rotation.
Thermal X-Ray Sources. We consider Sco-X1 as the best known of
the collapsed stars which belong to the second class of objects
as indicated earlier. In these stars, the main emission is
assumed to result from strong turbulent heating of the magneto-
spheric plasma, which is supposed to occupy a much larger volume
than the one of the collapsed star, that is R >> a. In the case
of Sco-X1, the spectrum between 1 and 40 KeV corresponds to a
bremsstrahlung emission of a plasma with temperature T = a1 x 5 x 107 °K
(Gorenstein et al. 1968, Meekins et al. 1969) where al accounts
for an experimental uncertainty. The received intensity is
- 5 -
-7 2
I = 2 x 3 x 10 erg/cm sec
20
The star distance d is still uncertain and we shall consider d = a3 x 6 x 10 20cm,
a value deduced by proper motion observations (Sofia et al. 1969).
14
The flux Fv measured at infrared frequencies (v % 10 Hz) was
interpreted by Neugebauer et al. (1969) as being due to black body emission
of the plasma at the same temperature as deduced from the X-ray spectrum.
In particular according to Neugebauer et al. (1969)
F c4 1x 10 V erg/Hzcm sec
By equating this flux to the one emitted by a spherical black body with
radius R and at a distance d we obtain
(F M/lj2 1/2 
R = cd = 2.7 x 108L 4 23 cm (1)
The ULt=iissLL'ah1Ulng IIIissiviLy of a plasma is given by (Ginzburg 1i67):
E =£ 1.6 x 10 27n2T1 / 2 erg/cm3sec
where n is the electron density. The observed value of the emissivity can
be taken as:
3Id2
R
which corresponds to a total power
36 2
obs = 1.3 x 10 2 ergsec
Then we have,
1 2 c11 X 1/2 
ne= 3.8 x 10
1
6 2 cm (2)
4 3
-6-
The thermal power associated with the current flowing along the star
magnetic field is
Ptot voumeEOljl1dV (3)
We assume that the toroidal field BT produced by the current is smaller
than the star's dipolar magnetic field B . We also notice that, given the
high values of the magnetic field in the neighborhood of the star and for
reasonable assumptions on the values of ne and T, the transverse component
of the current density J1 has to be much smaller than J11. This in order
not to generate forces J x B that would not be compensated by gravity,
pressure gradient and centrifugal effects. Indicating by Jp and JT the
poloidal and toroidal components of the current, the previous requirement
is expressed by the relation
p T T p
The charge conservation equation V · J = 0, then implies J = aB
with B * Vct = O0. Then if we assume that BT < B in the vicinity of the
star, we have Jll c -3 since in a dipolar magnetic field,the cross
r
section of a tube of force increases near the poles as r . Therefore,
neglecting the spatial variation of the resistivity so that Eil ~ JEI,
we obtain
·f· r1-6 r.o2r3 2
P 2E J ai ro dr = ra 2E J (4)tot o a Ol o 11 °
where J = J l(r=a), 2r r dr is the volume element calculated supposing
that J follows the magnetic lines of force and represents the linear
dimension of the active region of the magnetosphere which we assume of the
order of 10 cm. The toroidal component of the magnetic field at the poles
can be obtained from the Ampere theorem
-7-
BoT X 2_ Jj (5)
so that (4) can be written as
Ptot aoBo TE (in Gaussian units) (6)tot 2 0 T
Assuming BoT ~ 0.3B = 3 x 10 gauss from (5), we have
Jl = 4.8 x 106 ampere/cm2
16 -3
and taking the average density at the poles n 4 x 10 cm , we deduce
a mean velocity of the electrons in the direction of the magnetic field
ue = 7.5 x 108 cm/sec. which is a fraction of the electron thermal velocity
V = ( Te)/2= 4.4 x 109 cm/sec
In these conditions and for values of the plasma frequency w less
pe
than the electron cyclotron frequency e' the resistivity has been shown to
increase proportionally to the electric field and the average electron
drift velocity to saturate at a fraction of the electron thermal velocity
(Coppi and Mazzucato 1971; Coppi et al. 1970). In addition, electric field
ratios Eo/ER well in excess of 103 have been applied on laboratory plasmas
with w > Qe without destroying their cohesiveness that is, create ape e
global runaway process (Hamberger and Friedman 1968, Hamberger and Jancaric
1970).
On this basis, we assume that large amplitude electrostatic oscillations
are induced by the applied electric field so that electrons are trapped in
the resulting electrostatic wells. If 0 is the oscillating electrostatic
potential, this can be a8 large as
kTe
~th e ~ 4.3 KV (7)th e
We recall that the Debye length for the conditions mentioned above is about
-8-
AD = 4.2 x 10 4cm. So the uppermost limit for the oscillating electric fields
1 7
seen by the particles is of the order th 107 volt/cm. On the otherAD th
hand, the electric field which is necessary to emit the radiated power
can be obtained from (6), for the assumed plasma and magnetic field
parameters and equating Ptot to the observed value P 1.3 x 10 -36erg/sec so
E 2 bs = 10 volt/cm (8)
oil aocBoT
The value of E therefore is well below th/AD and we can argue that the
direct electric field is not sufficient to overcome the particle trapping
and to generate a runaway process. We also notice that the corresponding
collisional runaway field is ER = 40V/cm and that in our case E >> ER.
The non thermal tail observed for energies above 40 KeV (Agrawal et al.
1971) can be associated with a very small tail of highly energetic electrons
("Super-runaway") riding over the strong electric field fluctuation.
The radiation emitting volume, which was assumed to have a radius R,
corresponds to that part of the magnetosphere which is heated by the high
thermal conductivity along the magnetic field lines, the energy source
being localized in the active regions near the poles. Thus even if the
magnetic axis is not aligned with the rotation axis and the emitting regions
extend to large latitudes, no beaming of radiation will be present and no
pulsation is to be expected.
For the sake of finding an order of magnitude of the angular velocity
of the star, we can assume that the emitting regions have a radius of the
-9-
same order of the speed of light radius PR RSL. We may justify this assumption
by arguing that the radial thermal conduction will tend to drop as, on
approaching the light speed cylinder, the magnetic field lines acquire an
increasingly larger toroidal component. We have then
X 'X 100 sec
-1
and the rotational energy of the star is
3 1.2 49
E = ·w M ' 10 erg
rot 5 o
In the absence of any indication of the variation of with time,
o
a rough estimate of the life-time of the source is given by:
tot 4T = 'I 3 x 10 years,
Pobs
a time long enough to justify the absence of a shell around this source.
The energy output corresponds to a loss of angular momentum i
Q §t Prot u got 33
The angular momentum can leave the star in the form of relativistic particles
and photons and be communicated to the plasma beyond the light cylinder.
Sco-Xl is also known to exhibit bursts of radio, optical and X-ray emission
(flares) (see for example Evans et al. 1970). For this we notice that the
plasma in the regions close to the light speed cylinder can be subject to
instabilities, for instance of the type transforming magnetic energy into
kinetic energy. When the associated perturbation extend to modify the plasma
density up to the polar regions, the total overall X-ray emission can also
mainly optical
be affected. Otherwise the instabilities will/induce a change of/emission
as in the case of solar flares. (Coppi and Friedland 1970).
- 10 -
Fast Pulsars We consider NP 0532 (in the Crab Nebula) as the typical
representative of this class. Unlike all other known pulsars, this object
exhibits emission in the optical, X and y-ray domains.
So it is proposed that the high frequency part of the spectrum, from
the optical up, is associated with the tail of fast electrons characterizing
T
the distribution function when E > E R. We notice that most of the
measured radiation energy is emitted in this part of the spectrum which,
in addition, can be described by a power law. Assuming that a synchrotron
or inverse Compton scattering radiation process is involved, the spectrum
profile reflects a power law of the electron distribution tail. It is
possible to argue that the equilibrium distribution results from a balance
between the loss of electron energy perpendicular to the magnetic field and
gain due to a scattering process of parallel energy into perpendicular
energy (Coppi and Ferrari 1970). Iferring to the comparison made between
the electric field E and the fluctuating field kt1P considered for the case
oil
of Sco X-1 (here k], denotes the typical wave number of electrostatic
fluctuations along the magnetic field), we notice that in the present case Eo
is likely to be larger, while th/AD ought to be smaller because of the
lower temperature of the Maxwellian body of the distribution. Therefore a
runaway process is likely to set in, and electron distribution function is
likely to acquire the profile sketched in Fig. l(c). In fact, we may take
T as
the so-called turbulent runaway field ER /of the order of to th/D
III. Discussion
We have identified three classes of radiating collapsed stars which are
distinguished by the regimes characterizing their magnetospheric plasma.
-11-
These regimes depend primarily on the magnetic field, on the
electric field parallel to it, and on the electron density
distribution. Since these three quantities can change during
the evolution of the star, it is possible that the same star will
evolve through the three mentioned classes.
At the beginning of the evolution, when the angular velocity
is highest, the electric field can be large enough for free
acceleration of particles to take place. As was mentioned
earlier, this can occur when the electric field is larger than
E T
R'
T
We can think of the lowering of the electric field below E R
as being associated with the star slowing down and progressive
heating of the Maxwellian body of the distribution. The object
in this stage will be similar to Sco X-1 if the radius of the
magnetosphere is many times greater than the radius of the star,
R > 102a.
Finally, when the electric field is no longer large enough
to produce sufficiently large ohmic heating, and the plasma
density distribution around the star is reduced to a relatively
cold shell the radius of which is not much larger than the star's
radius, R X a, the star can emit only at low frequencies via
plasma collective effects and is observed as a slow pulsar.
We wish to point out that an evolution which takes into
account only the frequency of rotation may suggest that PSR 0833
(Vela pulsar), which has a period three times larger than
NP 0532, is an object similar to Sco X-1. On the other hand, no
-lla-
optical counterpart of this radio pulsar has been detected.
Thus consideration of the rotation frequency alone appears to
be insufficient for a classification of the emission character-
istics of a given star. As indicated previously, the possibility
of different plasma density profiles around the star and of
different electron distribution functions, in addition to a
variety of magnetic configurations, should be taken into account
in assessing its evolutionary stage.
I
References
Agrawal, P.C., Biswas, S., Gokhale, G.S., Iyengar, V.G., Kunte, P.K.,
Manchanda, P.K., Greekantan, B.V. 1970, to be published in Astroph.
Space Science.
Coppi, B., Dobrowolny, M. and Santini, G. 1970, International Center for
Theoretical Physics, Report IC/70/26. (I.A.E.A., Trieste).
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Coppi, B. and Friedland, A. 1970, submitted to Ap. J.
Coppi B. and Mazzucato, E. 1971, Phys. Fluids, 64, 134.
Deutsch, A.J. 1955, Ann. Astrophys., 18, 1.
Evans, W. D., Belian, R.D., Conner, J.P., Strong, I.B. Hiltner, W.A.,
Kunkel, W.E. 1970, Ap. J. (Letters), 162, L115.
Ginzburg, V.L. 1967, in High Energy Astrophysics, ed. C. DeWitt, E.
Schatzman, and P. Veron, (Gordon and Breach).
Goldreich, P. and Julian, W.H. 1969, Ap.J., 157, 869.
Gorenstein, P., Gursky, H. and Garmire, G. 1968, Ap. J., 153, 885.
Gurevich, A.V. 1961, Soviet Phys. - J.E.T.P., 12, 904.
Hamberger, S.M. and Friedman, M. 1968, Phys. Rev. Letters, 21, 674.
Hamberger, S.N. and Juncarik, J. 1970, Phys. Rev. Letters, 25, 999.
Meekins, J.G., Henry, R.C., Fritz, G., Friedman, H., and Byram, E.T.
1969, Ap. J., 157, 197.
Neugebauer, G., Oke, J.B. Becklin, E., and Garmire, G. 1969, Ap. J., 155, 1.
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Acknowledgements
One of us (A.T.) has been supported by a joint ESRO-NASA fellowship,
and this work was sponsored in part by the U.S. Atomic Energy Commission
(Contract AT-30-1 39 80) and by American Science and Engineering (NASA
Contract NAS5-9041)'.
Caption to Figure 1
Typical distribution functions characterizing three types of collapsed
stars.
fe (vi )
e I
VII
(b)
fe (vl ) fe (v I )
VI I
(a)
VI I
(C)
.,


Classification of radiating compact stars

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