2.2 Pulsar Timing
2.2.3 Timing Analysis
2.2.3.5 Mass Function and post-Keplerian parameters
The Keplerian parameters can be used to estimate the mass of the companion using the mass function. The mass function (e.g. Lorimer & Kramer,2004) is:
f(mp, mc) = (Mcsini)3 (Mp+Mc)2 = 4πx 3 TPb2 , (2.19)
wheremp and mcare the masses of the pulsar and its companion in units of solar mass (M),
and T = (GM/c3) = 4.925490947µsis the solar mass (M) in time units (c is the speed of
light and G is Newton's gravitational constant). If no other parameters are available, we can 7http://www.jb.man.ac.uk/pulsar/Resources/tools.html
50 Chapter 2. Pulsar Searching and Timing Methods 0 10 20 30 40 50 60 70 Time of Observation(MJD) +5.633×104 45.765 45.770 45.775 45.780 45.785 45.790 45.795 O bs er ve d P er io d( m s ) PSR J0453+16
Figure 2.9. Observed Doppler-shifted spin period of newly discovered pulsar, now known as PSR J0453+1559, (black dots) are plotted against time with a tted red line that best describes the orbit of the system.
assume a mass of a pulsar (Mp = 1.35 M) and a maximum and median orbital inclinations
(i= 90◦,60◦). An estimation of the minimum and median companion masses can be obtained
with the mass function. But we are still left with two unknowns in Equation2.19, which can be determined by measuring two or more post-Keplerian (PK) parameters.
Unfortunately, precise mass measurements in general are hard to achieve because suitable systems are rare. Up to now, only 41 pulsars have precise estimates of their neutron star mass8.
One way to directly measure the masses of a pulsar in a binary is by measuring two or more PK parameters, which describe relativistic corrections. Each of the PK parameters produce their own characteristic timing residual signatures (Damour & Deruelle,1986), for that reason it can be taken into account by the process of pulsar timing. In GR and other gravity theories (e.g. Blandford & Teukolsky, 1976; Damour & Deruelle, 1985, 1986), the ve important PK parameters are as described by Lorimer & Kramer(2004):
• The rate of advance of periastron, ω˙, is the rate which the orientation of the orbit
rotates through space, in GR, this is given by
˙ ω = 3T2/3Pb 2π −5/3 1 1−e2(Mp+Mc) 2/3. (2.20) 8https://www3.mpifr-bonn.mpg.de/sta/pfreire/NS_masses.html
2.2. Pulsar Timing 51
• The time dilation and gravitational redshift parameter, γ, referred as the Einstein delay, which describes the relativistic dilation experienced by the pulsar signal as it traverses the gravitational potential of the binary system:
γ =T2/3 Pb 2π 1/3 eMc(Mp+ 2Mc) (Mp+Mc)4/3 . (2.21)
• The orbital period decay,P˙b, is the rate at which the orbital period decreases due to
the emission of GWs, given in GR by:
˙ Pb=− 192π 5 T 2/3 h1 + (73/24)e2+ (37/96e4) (1−e2)7/2 i MpMc (Mp+Mc)1/3 . (2.22)
• The Shapiro delay has two parameters, the range, r, and shape, s, which describe the extra time required by the pulsar signal to travel through the curved space-time containing the companion, given in GR by:
r=TMc, s= sin(i) =T−1/3 Pb 2π xp (Mp+Mc)2/3 Mc . (2.23)
The relativistic eects described by these PK parameters depend on the Keplerian parameters and the masses of the pulsar and companion. Measuring two of these eects provides two more equations which together with mass function allows a solution of the 3 unknowns (i, Mp, Mc).
This process can be visualized by plotting them in a mass-mass diagram, where two or more of these PK parameters intersect, which result in measuring of individual masses for the pulsar and the companion. The degree of how precisely these PK parameters are measurable varies depending upon the binary system itself. For example, a pulsar in a circular orbit will make the measurement ofω˙ more dicult due to the low precision available on bothω and T0. Also,
the measurement of the Shapiro delay parametersr and sdepends upon the binary being close to edge on in its inclination for the binary companion to pass close to the line of sight during superior conjunction. In the following Chapters 3, 4, 5, there will be mass-mass diagrams of pulsar in binary systems that have mass measurements.
The pulsar timing software analysis TEMPO9 has many timing models used to describe the
binary motion and relativistic corrections for any type of binary system. The models used throughout this thesis are the DD model, DDGR model, DDH model and ELL1 model. The DD model described by (Damour & Deruelle,1985,1986), incorporates the required relativistic corrections for all the PK parameters presented in this thesis in a theory-independent fashion, which allows for tests of multiple gravitational theories to be conducted. The DDGR model described by (Taylor,1987; Taylor & Weisberg,1989), assumes the correctness of GR and en- sures consistency between the PK parameters according to their GR denitions, only tting for individual masses of the pulsar and the companion. The DDH model described by Freire & Wex (2010), was designed to allow for easier separation and constraint of the Shapiro delay by re-parameterizing it as the orthometric amplitudeh3 and orthometric ratioς. These orthometric
parameters relate to the standard Shapiro parametersr and sby: ς =
r
1−¯c
1 + ¯c (2.24) 9http://tempo.sourceforge.net
52 Chapter 2. Pulsar Searching and Timing Methods where ¯ c=p1−s2 = cos(i), (2.25) and h3 =rς3. (2.26)
Finally, the ELL1 model described by (Lange et al., 2001), parameterizes the pulsar's binary motion in a way that is particularly well suited to describe orbits with small eccentricities; it uses the Laplace parameters described in Section2.2.3.3. A summary of these and other binary models can be found in the literature (e.g. Damour & Taylor, 1992; Edwards et al., 2006, and references therein).