76
M. Burgay et al.
Finally, the last term is the Shapiro delay [70], which measures the extra time
required for an electromagnetic wave to travel through the curved gravitational field
of a celestial body. The amplitude of this effect is a function of the angle θ formed
by the radius from the pulsar to the telescope with the radius from the telescope to
the third body.
For the Sun, the formula is:
ΔS = −
2GM
c 3 ln (1 + cos θ ) ,
(2.11)
where θ is the angle at the telescope between the pulsar and the Sun.
If the pulsar is in a binary system, the pulsarcentric ToAs (i.e. the ToAs in
pulsar proper time) must also be corrected and transformed at the Binary System
Barycenter (BSB). Equation 2.7 hence has to include four further terms:
t
Bin
i,bary = t i,bary + ΔR Bin + ΔE Bin + ΔS Bin + ΔA Bin
(2.12)
In a Newtonian framework, fitting for the orbital modulations of the ToAs
induced, mainly, by the Roemer delay, allows us to derive five Keplerian parameters
of the orbit: the binary period P b , the projected semi-major axis x = a sin i, the
eccentricity e, the longitude of the periastron ω and the epoch of the passage at
periastron T 0 . From these parameters we can derive the mass function:
f (M) =
(M c sin i)
3
M p + M c
2 =
4π 2 (a sin i)
3
GP 2
b
(2.13)
where M p is the pulsar mass and M c is the companion mass. Assuming a standard
value for M p and for an edge-on orbit (i = 90 ◦ ), we can calculate a lower limit for
M c .
The other terms 4 in Eq. (2.12) describe the deviations from classical physics that
occur in a pulsar which experiences strong gravitational fields and/or high orbital
velocities in a relativistic binary system. A full description will follow in Sect. 2.5.
From the equations above, we can understand how the timing procedure allows
us to also derive, besides the rotational parameters of Eq. (2.5), the astrometric
parameters (position and proper motion, through the transformation to the SSB)
and orbital parameters, in the case of a binary system (through the transformation
to the BSB) of a pulsar.
As mentioned above, the timing procedure is made through a multi-parametric
fit of all the spin, positional and orbital parameters, with the aim of minimising
the χ 2 of the timing residuals. From a practical point of view, we first create a
4 ΔA Bin , a parameter describing the changing aberration along the orbit, is actually a classical
effect. The parameters describing it, however, are degenerate with some relativistic parameter,
hence it is usually described in the context of Post-Keplerian effects.
M. Burgay et al.
Finally, the last term is the Shapiro delay [70], which measures the extra time
required for an electromagnetic wave to travel through the curved gravitational field
of a celestial body. The amplitude of this effect is a function of the angle θ formed
by the radius from the pulsar to the telescope with the radius from the telescope to
the third body.
For the Sun, the formula is:
ΔS = −
2GM
c 3 ln (1 + cos θ ) ,
(2.11)
where θ is the angle at the telescope between the pulsar and the Sun.
If the pulsar is in a binary system, the pulsarcentric ToAs (i.e. the ToAs in
pulsar proper time) must also be corrected and transformed at the Binary System
Barycenter (BSB). Equation 2.7 hence has to include four further terms:
t
Bin
i,bary = t i,bary + ΔR Bin + ΔE Bin + ΔS Bin + ΔA Bin
(2.12)
In a Newtonian framework, fitting for the orbital modulations of the ToAs
induced, mainly, by the Roemer delay, allows us to derive five Keplerian parameters
of the orbit: the binary period P b , the projected semi-major axis x = a sin i, the
eccentricity e, the longitude of the periastron ω and the epoch of the passage at
periastron T 0 . From these parameters we can derive the mass function:
f (M) =
(M c sin i)
3
M p + M c
2 =
4π 2 (a sin i)
3
GP 2
b
(2.13)
where M p is the pulsar mass and M c is the companion mass. Assuming a standard
value for M p and for an edge-on orbit (i = 90 ◦ ), we can calculate a lower limit for
M c .
The other terms 4 in Eq. (2.12) describe the deviations from classical physics that
occur in a pulsar which experiences strong gravitational fields and/or high orbital
velocities in a relativistic binary system. A full description will follow in Sect. 2.5.
From the equations above, we can understand how the timing procedure allows
us to also derive, besides the rotational parameters of Eq. (2.5), the astrometric
parameters (position and proper motion, through the transformation to the SSB)
and orbital parameters, in the case of a binary system (through the transformation
to the BSB) of a pulsar.
As mentioned above, the timing procedure is made through a multi-parametric
fit of all the spin, positional and orbital parameters, with the aim of minimising
the χ 2 of the timing residuals. From a practical point of view, we first create a
4 ΔA Bin , a parameter describing the changing aberration along the orbit, is actually a classical
effect. The parameters describing it, however, are degenerate with some relativistic parameter,
hence it is usually described in the context of Post-Keplerian effects.