174
A. Patruno and A. L. Watts
Pulse phase can be measured as a function of time and each harmonic analyzed
separately. We drop the subscript k from the phase symbols, since the following
equations are valid for each harmonic. The pulse phase series can be decomposed
into several terms encoding different physical effects:
φ (t) = φ L + φ Q + φ orb + φ A + φ M + φ N
(4.16)
• φ L = φ 0 + νt is the linear term due to a constant spin of the NS (φ 0 is an initial
reference phase)
• φ Q =
1
2 ˙
νt 2 is the quadratic variation due to constant spin up or spin down
• φ orb contains the effect of the orbital motion of the NS around the companion
and, as a first order approximation, can be calculated by measuring the delays in
the time of arrival of photons:
t em = t arr −
A 1
c
sini sin
2π
P b
(t arr − T asc )
(4.17)
where A 1 is the semi-major axis of the NS orbit, c the speed of light, i the
inclination of the orbit with respect to the observer, T asc the time of passage
through the ascending node, P b the orbital period and t em and t arr the photon
emission and arrival time [238]. In this expression we have assumed that the
orbit is perfectly circular, a good first order approximation in all AMXPs. More
sophisticated models for nearly circular orbits can be found for example in [180].
• φ A gives phase variations related to uncertainty in astrometric source position,
which introduces a spurious frequency and frequency derivative offset. These
offsets can be expressed as [123]:
Δν = ν 0 (a ⊕ cosβ/c) (2π/P ⊕ ) cosτ
(4.18)
Δ˙ ν = −ν 0 (a ⊕ cosβ/c) (2π/P ⊕ )
2 sinτ
(4.19)
Here ν 0 is the true pulse frequency, the position error parallel to the plane of the
ecliptic, β the ecliptic latitude of the AMXP, a ⊕ and P ⊕ the Earth semi-major
axis and orbital period and τ = 2π t/P ⊕ the orbital phase of the Earth. Phase
zero is defined as the point where the Earth is closest to the AMXP and for order
of magnitude estimates can be assumed such that cosτ = sinτ = 1.
• φ M refers to unavoidable phase wandering due to measurement errors, and is
normally distributed with an amplitude predictable by propagating the Poisson
uncertainties due to counting statistics.
• φ N is a subtle term covering residual phase variations that do not fall into any
of the previous categories. These are usually called “X-ray timing noise”, by
analogy with the timing noise often observed in radio pulsars. Note that if the
spin-up (or spin-down) process is not constant in time we do expect terms higher
than the quadratic (φ Q ) and these are considered, in our definition, as part of φ N
A. Patruno and A. L. Watts
Pulse phase can be measured as a function of time and each harmonic analyzed
separately. We drop the subscript k from the phase symbols, since the following
equations are valid for each harmonic. The pulse phase series can be decomposed
into several terms encoding different physical effects:
φ (t) = φ L + φ Q + φ orb + φ A + φ M + φ N
(4.16)
• φ L = φ 0 + νt is the linear term due to a constant spin of the NS (φ 0 is an initial
reference phase)
• φ Q =
1
2 ˙
νt 2 is the quadratic variation due to constant spin up or spin down
• φ orb contains the effect of the orbital motion of the NS around the companion
and, as a first order approximation, can be calculated by measuring the delays in
the time of arrival of photons:
t em = t arr −
A 1
c
sini sin
2π
P b
(t arr − T asc )
(4.17)
where A 1 is the semi-major axis of the NS orbit, c the speed of light, i the
inclination of the orbit with respect to the observer, T asc the time of passage
through the ascending node, P b the orbital period and t em and t arr the photon
emission and arrival time [238]. In this expression we have assumed that the
orbit is perfectly circular, a good first order approximation in all AMXPs. More
sophisticated models for nearly circular orbits can be found for example in [180].
• φ A gives phase variations related to uncertainty in astrometric source position,
which introduces a spurious frequency and frequency derivative offset. These
offsets can be expressed as [123]:
Δν = ν 0 (a ⊕ cosβ/c) (2π/P ⊕ ) cosτ
(4.18)
Δ˙ ν = −ν 0 (a ⊕ cosβ/c) (2π/P ⊕ )
2 sinτ
(4.19)
Here ν 0 is the true pulse frequency, the position error parallel to the plane of the
ecliptic, β the ecliptic latitude of the AMXP, a ⊕ and P ⊕ the Earth semi-major
axis and orbital period and τ = 2π t/P ⊕ the orbital phase of the Earth. Phase
zero is defined as the point where the Earth is closest to the AMXP and for order
of magnitude estimates can be assumed such that cosτ = sinτ = 1.
• φ M refers to unavoidable phase wandering due to measurement errors, and is
normally distributed with an amplitude predictable by propagating the Poisson
uncertainties due to counting statistics.
• φ N is a subtle term covering residual phase variations that do not fall into any
of the previous categories. These are usually called “X-ray timing noise”, by
analogy with the timing noise often observed in radio pulsars. Note that if the
spin-up (or spin-down) process is not constant in time we do expect terms higher
than the quadratic (φ Q ) and these are considered, in our definition, as part of φ N