Pulsars and neutron stars/Using pulsar timing to study (and navigate) the solar system

Introduction
The pulsar timing method relies on knowledge of the telescope position on the Earth (or in space) and the position of the Earth with respect to the Solar System Barycentre (SSB). If these terms are not known precisely then the error in these parameters will lead to induced timing residuals. It is therefore possible to fit for these positions as part of the standard timing process. This provides the opportunity to localise a telescope anywhere in space (i.e., for spacecraft navigation) and also for identifying errors in the assumed position of the SSB.

The solar system ephemerides
The pulsar timing software packages require knowledge of the position of the Earth, Sun and planets with respect to the SSB. These ephemerides are constructed by numerical integration of the equations of motion to fit data from various measurements of solar system objects - such as optical astrometry, observations of transits and radar ranging of the planets.

The JPL ephemerides
The following JPL ephemerides are in common use in pulsar data analysis:

Two special ephemerides were made for the Champion et al. (2010) paper. These ephemerides DE990 and DE991 are used to study the effect of a small error in the mass of Jupiter.

INPOP
Fienga et al. (2011) described the INPOP10a planetary ephemeris and the corresponding TT-TDB time transfer.

Comparison between the ephemerides
Hilton & Hohenkerk (2011) provided a comparison between the EPM2008, DE421 and INPOP08 ephemerides. They highlighted the largest differences as:


 * EPM2008 includes pluo + 20 trans-Neptunian objects, whereas the DE421 and INPOP08 ephemerides only include pluto
 * The DE421 ephemeris include 343 main belt objects whereas both EPM and INPOP include around 300 objects, but also a ring of masses.
 * The EPM2008 ephemeris has a significant offset in the barycentric position because of the inclusion of other trans-Neptunian objects than pluto. Over small time-scales this represents a fixed offset from the centre of mass of the Sun and other solar system bodies.

Searching for unknown objects in the solar system
The effect of an error in the planetary ephemeris on the timing residuals for the i'th pulsar is:

$$r_i(t) = \frac{1}{c}(\vec{e}(t) \cdot \hat{k_i}) $$

where $$c$$ is the vacuum speed of light, $$\vec{e}$$ is the time-dependent error in the position of the SSB with respect to the observatory and $$\hat{k}$$ is a unit vector pointing towards pulsar i.

Tiburzi et al. (2015) provided a method for searching for errors in the solar system ephemeris. They suggested the following procedure:


 * 1) Fit for the three components of the error vector in the solar system barycentre position as a global fit to all available pulsars. These vector components can be rotated into the ecliptic plane. If an error is detectable and is in the ecliptic plane then a new fit can be carried out purely for the two vector components in the ecliptic plane.
 * 2) It is then useful to take a power spectrum of each vector component and search for excess power at the period of a known planet.
 * 3) If there is excess power at the period of a known planet then the Champion et al. method (see below) can be used to make an optimal measurement of the planetary mass error.
 * 4) It may also be useful to search for errors that correspond to particular orbits (such as prograde or retrograde orbits). If true, then a fit to such errors can be carried out.

Measuring planetary system masses
Champion et al. (2010) presented the first determination of planetary system masses using high-precision millisecond pulsar timing observations. The basic concept is straightforward. A small error in the mass of a planet will mean that there will be an "error vector" between the actual position of the solar system barycentre and its assumed position given the solar system ephemeris. That error vector will point towards (or away from; depending upon whether the assumed mass is too high or too low) the planet. Of course, the planet is moving and so that vector will trace the orbit of the planet. The induced timing residuals will therefore be a sinusoidal signal with amplitude related to the error (and also depends upon the pulsar's position) in the assumed mass and the wavelength being the orbit period of the planet. Champion et al. (2010) updated the tempo2 software package to enable such sinusoids to be searched for as part of a normal tempo2 fit.

Note that his method assumes that the mass error is small. It also assumes that there's no significant error in the position of the planet in its orbit.

Pulsar-based spacecraft navigation
It is possible to assume that the position of the SSB is known sufficiently well and fit for the position of the telescope (with respect to the SSB). The telescope need not be on Earth and so this provides a method to localise a space-craft travelling through the solar system. The pulsars can be thought of as providing a global GPS system. Of course, it is unlikely that a massive radio telescope (such as the Parkes telescope) will be launched into space. Most research into this topic has therefore considered observations of pulsars using X-ray telescopes.

History
The first attempt to use pulsars as navigational devices were the Pioneer plaques (placed on board the 1972 and 1973 Pioneer 10 and 11 spacecraft. In the plaques the relative position of the Sun to the centre of the Galaxy and 14 pulsars are displayed. The pulsar lines contain binary numbers which stand for the pulse period of the pulsars using the hydrogen spin-flip transition frequency as the unit. In theory, this, not only, gives the position of the solar system, but also the epoch at which the spacecraft was launched (as the pulsars are slowing down). The lengths of the lines show the relative distances of the pulsars to the Sun. A tick mark at the end of each line gives the Z coordinate perpendicular to the galactic plane. More details can be found on Wikipedia.



Method
The basic concept for pulsar-based navigation is straight-forward. Pulsars emit a stable pulse train allowing the pulse arrival times at a given telescope to be predicted. Consider a pulsar with a period of exactly P seconds. We can predict that a pulse will arrive at time t. Note that other pulses will be arriving at t+P/c, t-P/c, t+2P/c, etc. in the direction of the pulsar and, of course, the pulses are not numbered and so we cannot determine our position exactly with just a single pulsar. However, if we also observe two more pulsars that are in different directions then we can use a triangulation procedure to determine the telescope position. Of course, the telescope is never stationary. It is either on the moving Earth, or onboard a moving spacecraft. During the observation, or between observations, the telescope will have moved. This must be accounted for in the navigation methodology. Also, pulsars are not perfectly stable rotators. They glitch, have timing noise and the pulses are affected by the interstellar medium. Such affects must be accounted for in the navigation procedures.

There are five main types of navigation:


 * 1) Determining the position of a telescope on the surface of the Earth
 * 2) Determining the position of a spacecraft on an approximately known trajectory in the solar system
 * 3) Determining the position of a satellite in orbit around the Earth or other object
 * 4) Determining the position of e.g., a rover on a distant planet
 * 5) Determining the position of a spacecraft on an unknown orbit in distant space