## Search for Planets Around Detached Eclipsing Stars

SPADES is a long-term, wide-ranging project for finding exoplanets in a relatively new way that needs to involve amateurs as well as professionals. In addition to finding planets (or even where it doesn’t) it will yield valuable research data on EA type eclipsing binaries (hereafter EBs). The project was proposed by Dr Simon O’Toole of the AAO, and is now run as part of the Southern Eclipsing Binaries Programme (leader [cblink=”tom_richards”]dr Tom Richards[/cblink]).

#### Finding Exoplanets

Several methods are now in common use to find exoplanets. The most widely used ones are Doppler wobble, transit photometry, and microlensing. A relatively new technique, quite within the reach of amateurs as well as professionals, is to measure changes in eclipse times of EB stars. How does it work?

#### The “O-C Modulation” or “Eclipse Timing” Method

Consider a pair of EB stars rotating around their common centre of gravity (barycentre). Since we see eclipses, we lie close to their orbital plane, as illustrated in this simplified diagram.

*Figure 1. A simplified diagram of a partially eclipsing binary pair and their eclipse photometry. The yellow star is smaller, hotter and brighter, so its eclipse is deeper.*

The time of the primary (deeper) minimum can be measured very precisely, typically to ~10^{‑4} day (~9 seconds), even with amateur equipment. These times are always expressed as a Heliocentric Julian Date (Richards, “CCD Column” * *{jd_file file==5}). By measuring several such minima, an even more precise orbital period can be derived simply by linear regression on times of minima. The *Light Elements* (period P, and one such time of primary minimum, called the *Primary Epoch *or* E _{0}*) of the system can then be stated, commonly in the form that calculates a new time of minimum:

HJD_{min} = E_{0} + P × E

where E is an integral number of orbital cycles (aka epochs) elapsed since E_{0}.

Such a formula allows accurate predictions of times of minima well into the future. When a future minimum is observed, its observed time (labelled O) can be compared to the calculated time C by taking the difference O-C. A plot of (O-C) against cycle numbers (the Es in the equation) or years will show if the period is changing. If we look at a pair of *detached* stars, not touching and not transferring mass from one to the other or losing it from the system, P can be expected to stay constant over the years. The (O-C) diagram in Figure 2 shows, however, a system whose period is increasing linearly with time – observed minima are increasingly later than calculated minima, which explains the upward-opening parabola fitted to the data. (From Degirmenci *et al***1999A&AS..134..327D**.)

*Figure 2. An (O-C) diagram of an eclipsing system whose period is increasing linearly.*

Now consider the effect on eclipse times of a planet orbiting two eclipsing stars. We can model this as two bodies, namely the pair of stars revolving around each other, and the planet, both revolving around their common barycentre in coplanar circular orbits. See Figure 3.

*Figure 3. Schematic diagram of an eclipsing binary with a planet outside the binary orbit. Planet and binary pair revolve around the whole system’s barycentre (while the two stars are also revolving around each other). An observer in the plane of the orbits, lying in the direction shown, will observe a sinusoidal modulation of the times of the binary’s eclipses, with period equal to the orbital period of the planet.*

As the line joining the binary and the planet moves successively from AA’ to DD’ the binary moves away from the observer then towards the observer. When the binary is at B an eclipse signal arrives late by the time taken for light to travel from B to the system’s barycentre, and correspondingly early when at D. This generates a sinusoidal modulation to the (O-C) diagram of amplitude A_{b}/c, where A_{b} is the radius of the binary’s orbit around the system’s barycentre and c is the speed of light. If the total mass M_{b} of the binary equalled that of the Sun, and the planet’s mass M_{p} and orbital radius A_{p} were the same as Jupiter’s, the light time delay would be 2.6 s – the amplitude of the sine curve, and near the limit of detection. The formula for light time delay t is:

t = A_{p}M_{p}/cM_{b}.

Consequently the ideal search goal is for a massive planet far out from a not very massive binary pair. That will of course have a longer orbital period P given by Newton’s modification of Kepler’s Law:

P^{2} = 4p^{2}(A_{b}+A_{p})^{3}/G(M_{b}+M_{p}),

so (O-C) data on target systems needs to be gathered for many years!

But we don’t have to start that process from scratch. The literature on EBs contains a lot of (O-C) data on eclipsers gathered over many decades which we can use. Click here for a list of catalogues of light elements and physical data.

For a discussion of the O-C modulation method and how it can yield exoplanet discoveries, see the SPADES Science Case.