# Pulsars and neutron stars/Testing theories of gravity

## Introduction

Stairs (2003) provided an introduction to testing theories of gravity using pulsar observations. The search for gravitational wave (a prediction of general relativity and other theories of gravity) is discussed elsewhere. Here, we consider verifying the predictions of various theories of gravity and the possible violation of equivalence principles using binary pulsars. The majority of binary pulsar orbits are adequately described by the five Keplerian parameters:

• the orbital period, ${\displaystyle P_{b}}$
• the projected semi-major axis, ${\displaystyle x}$
• the eccentricity, ${\displaystyle e}$
• the longitude of periastron, ${\displaystyle \omega }$
• the epoch of periastron, ${\displaystyle T_{0}}$

However, in some cases significant residuals are still present are fitting for the Keplerian parameters. In such cases one or more post-Keplerian (PK) parameters are required in the timing model.

## Tests with double neutron-star binary systems

In general relativity the equations the post-Keplerian parameters can be written as:

${\displaystyle {\dot {\omega }}=3\left({\frac {P_{b}}{2\pi }}\right)^{-5/3}(T_{\odot }M)^{2/3}(1-e^{2})^{-1}}$

${\displaystyle \gamma =e\left({\frac {P_{b}}{2\pi }}\right)^{1/3}T_{\odot }^{2/3}M^{-4/3}m_{2}(m_{1}+2m_{2})}$

${\displaystyle {\dot {P_{b}}}=-{\frac {192\pi }{5}}\left({\frac {P_{b}}{2\pi }}\right)^{-5/3}\left(1+{\frac {73}{24}}e^{2}+{\frac {37}{96}}e^{4}\right)(1-e^{2})^{-7/2}T_{\odot }^{5/3}m_{1}m_{2}M^{-1/3}}$

${\displaystyle r=T_{\odot }m_{2}}$

${\displaystyle s=x\left({\frac {P_{b}}{2\pi }}\right)^{-2/3}T_{\odot }^{-1/3}M^{2/3}m_{2}^{-1}}$

where ${\displaystyle T_{\odot }=GM_{\odot }/c^{3}=4.925490947\mu s}$, ${\displaystyle s=\sin i}$, ${\displaystyle M=m_{1}+m_{2}}$ where ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ are the masses of the pulsar and its companion. ${\displaystyle i}$ is the inclination angle of the orbit.