3. Pulse Arrival Time Analysis

The regular pulsations of X-ray binary pulsars provide us with accurate clocks that can be used to probe the dynamics of the system. The time at which the n^th pulse is emitted is

$$\prime$$ (38)

where

$${1\over 2} \dot{P}$$ (39)

takes into account variations in the intrinsic pulse period.

As the pulsar moves in its orbit pulse arrival times will be delayed or hastened by up to the light travel time across the projected semi-major axis of the orbit a_xsin i. This is $\sim$ 40 s for Cen X-3. The arrival time t_n of the n^th pulse is

$${a_{\rm x}\sin i\over c} \omega \tau \nu$$ (40)

where the function F(e,$\omega$,$\tau$,$\nu$) describes the orbit of the neutron star about the center of mass. The orbit is elliptical with eccentricity e. The longitude of periastron $\omega$ is the angle between the line of sight and the major axis. The time of periastron passage is $\tau$. The position of the neutron star in its orbit is described by the mean anomaly $\theta$ = 2$\pi$(t - $\tau$)/P_orb. The function F has the form

$$\omega \tau \nu {\sin(\nu+\omega)\over1+e\cos\nu}$$ (41)

where the true anomaly $\nu$ may be found from the relations

$${\nu\over2} \sqrt{1+e\over1-e} {E\over2}$$ = (42)

$$\theta$$ E - esin E = (43)

For a circular orbit the arrival time t_n of the n^th pulse is

$${a_{\rm x}\sin i\over c} \left(\vphantom{{2\pi(t_n^\prime - T_{\rm ecl})\over P_{\rm orb}}}\right. {2\pi(t_n^\prime - T_{\rm ecl})\over P_{\rm orb}} \left.\vphantom{{2\pi(t_n^\prime - T_{\rm ecl})\over P_{\rm orb}}}\right)$$ (44)

where T_ecl is the time of mid-eclipse, and P_orb is the orbital period. If the eccentricity of the orbit is small this circular ephemeris may be used to obtain an initial estimate of the orbital parameters. Then if terms of order e² and higher are ignored the residuals from the fit to the arrival times versus pulse number are ()

$$\delta \delta \delta {1\over 2} \dot{P}_{0}^{}$$ = (45)

$$\delta \left(\vphantom{{a_{\rm x}\sin i\over c}}\right. {a_{\rm x}\sin i\over c} \left.\vphantom{{a_{\rm x}\sin i\over c}}\right) {2\pi\over P_{\rm orb}} {a_{\rm x}\sin i\over c}$$ (46)

$${2\pi\over P_{\rm orb}^2} {a_{\rm x}\sin i\over c} \delta \prime$$ (47)

$${3\over2} {a_{\rm x}\sin i\over c} \omega {1\over 2} \omega$$ (48)

$${1\over 2} {a_{\rm x}\sin i\over c} \omega$$ (49)

Here l_n = 2$\pi$(t_n^$\prime$ - E₀)/P_orb + ${\pi\over2}$ is the mean orbital longitude at time t_n^$\prime$ and E₀ $\approx$ T_ecl is a time when the mean orbital longitude is 90^o. With the substitutions g = esin$\omega$ and h = ecos$\omega$ the expression becomes linear in the parameters omitted from the circular ephemeris ($\dot{P}_{0}^{}$, g, and h) and in the differential corrections to the orbital parameters ( $\delta$t₀, $\delta$P₀, $\delta$$\left(\vphantom{{a_{\rm x}\sin i\over c}}\right.$${a_{\rm x}\sin i\over c}$ $\left.\vphantom{{a_{\rm x}\sin i\over c}}\right)$, $\delta$$\tau$, and $\delta$P_orb). The arrival time residuals may be fitted to Equation 1.54 to obtain a better estimate of the orbital parameters. This may be compared with the arrival times to obtain a new set of residuals. The process can be iterated until convergence is obtained.

The arrival times provide information about the orbit and changes in the orbital period. Intrinsic variations in the spin period trace the angular momentum accretion rate $\dot{J}$.

By fitting the arrival times, the mass function of the primary

$${4\pi^2(a_{\rm x}\sin i)^3\over G P_{orb}^2}$$ (50)

can be found. If there are optical measurements of Doppler shifts in lines from the companion the companion's mass function can be found. Then the mass ratio is

$$\equiv {M_{\rm x}\over M_{\rm c}} {K_{\rm c}P_{\rm orb}\sqrt{1-e^2}\over 2\pi a_{\rm x}\sin i}$$ (51)

where K_c is the amplitude of the Doppler velocity curve measured from optical lines. The main uncertainties are usually in K_c and $\theta_{e}^{}$. $\theta_{e}^{}$ can be difficult to measure if the stellar wind has a significant optical depth. If $\theta_{e}^{}$ is overestimated it will result in a value for M_x that is too low. K_c can be difficult to measure because of the extended nature of the companion and its wind.