1 Astrophysical Constraints on Dense Matter in Neutron Stars
11
combined to form the “mass function”, which is
f 1 (M 1 , M 2 ) =
K 3
1 P orb
2πG
(1 − e
2 )
3/2
=
M 3
2 sin 3 i
(M 1 + M 2 ) 2 .
(1.5)
Here M 1 is the mass of the star being observed, M 2 is the mass of the other star, and
i is the inclination of the binary orbit to our line of sight (i = 0 means a face-on
orbit, i = π/2 means an edge-on orbit). From this expression, f 1 is the minimum
possible mass for M 2 ; if M 1 > 0 or i < π/2 then M 2 > f 1 . If periodically shifting
spectral lines are also observed from the second star (and thus the binary is a socalled double-line spectroscopic binary), then the mass ratio is known and only i is
uncertain.
The inclination can be constrained for eclipsing systems. Particular precision is
possible in some extrasolar planet observations because of the Rossiter-McLaughlin
effect (in which the apparent color of the star varies in a way dependent on
inclination as a planet transits across its disk; see [147, 194]). This effect also
produces a velocity offset, which might have been seen in 2S 0921–63 [110]. The
inclination can also be constrained in systems for which the companion to a compact
object just fills its Roche lobe. This is because as the companion orbits it presents
different aspects to us, and the amplitude of variation depends on the inclination;
for example, a star in a face-on orbit looks the same to us at all phases, whereas star
in an edge-on orbit varies maximally in its aspect [13, 145]. In practice this analysis
is limited to systems that have low-mass companions (because Roche lobe overflow
from a high-mass companion to a lower-mass compact object is usually unstable;
see [81]) and that have transient accretion phases and hence have long intervals in
which there is effectively no accretion disk (because an active accretion disk easily
outshines a low-mass star and thus the binary periodicity is very difficult to observe).
Neutron star X-ray binaries might be less likely to be transient than black hole X-ray
binaries, although the data are ambiguous on this point, and their companions tend
to be much less massive and hence dimmer than the companions to black holes [81].
Thus despite the great success of this method for black hole binaries it has found
limited application for neutron star binaries.
1.3.2 Post-Keplerian Measurements of Pulsar Binaries
The most precise measurements of the masses of neutron stars in binaries are
made for systems in which additional parameters can be measured. The extreme
timing precision of pulsars makes pulsar binaries especially good candidates for
such measurements. The new effects that can be measured are:
• Precession of the pericenter of the system, ˙
ω.
• Einstein delay γ . At pericenter, the gravitational redshift from the system is
maximized.
11
combined to form the “mass function”, which is
f 1 (M 1 , M 2 ) =
K 3
1 P orb
2πG
(1 − e
2 )
3/2
=
M 3
2 sin 3 i
(M 1 + M 2 ) 2 .
(1.5)
Here M 1 is the mass of the star being observed, M 2 is the mass of the other star, and
i is the inclination of the binary orbit to our line of sight (i = 0 means a face-on
orbit, i = π/2 means an edge-on orbit). From this expression, f 1 is the minimum
possible mass for M 2 ; if M 1 > 0 or i < π/2 then M 2 > f 1 . If periodically shifting
spectral lines are also observed from the second star (and thus the binary is a socalled double-line spectroscopic binary), then the mass ratio is known and only i is
uncertain.
The inclination can be constrained for eclipsing systems. Particular precision is
possible in some extrasolar planet observations because of the Rossiter-McLaughlin
effect (in which the apparent color of the star varies in a way dependent on
inclination as a planet transits across its disk; see [147, 194]). This effect also
produces a velocity offset, which might have been seen in 2S 0921–63 [110]. The
inclination can also be constrained in systems for which the companion to a compact
object just fills its Roche lobe. This is because as the companion orbits it presents
different aspects to us, and the amplitude of variation depends on the inclination;
for example, a star in a face-on orbit looks the same to us at all phases, whereas star
in an edge-on orbit varies maximally in its aspect [13, 145]. In practice this analysis
is limited to systems that have low-mass companions (because Roche lobe overflow
from a high-mass companion to a lower-mass compact object is usually unstable;
see [81]) and that have transient accretion phases and hence have long intervals in
which there is effectively no accretion disk (because an active accretion disk easily
outshines a low-mass star and thus the binary periodicity is very difficult to observe).
Neutron star X-ray binaries might be less likely to be transient than black hole X-ray
binaries, although the data are ambiguous on this point, and their companions tend
to be much less massive and hence dimmer than the companions to black holes [81].
Thus despite the great success of this method for black hole binaries it has found
limited application for neutron star binaries.
1.3.2 Post-Keplerian Measurements of Pulsar Binaries
The most precise measurements of the masses of neutron stars in binaries are
made for systems in which additional parameters can be measured. The extreme
timing precision of pulsars makes pulsar binaries especially good candidates for
such measurements. The new effects that can be measured are:
• Precession of the pericenter of the system, ˙
ω.
• Einstein delay γ . At pericenter, the gravitational redshift from the system is
maximized.