Which of the two lightcurves shown above contains a planetary transit? If you’ve been following my blog then you’ll know the answer is the one on the right, since that’s the Kepler-5b graph from my last post about planetary transits! The graph on the left shows a detached eclipsing binary, which is consists of two stars orbiting eachother at a great enough distance that they are two distinct objects – as they orbit their centre of mass, the stars pass in front of eachother and reduce the total light coming from the system, which manifests as the dips in the lightcurve. So in that case, the transit is caused by a star and not a planet.
Detached eclipsing binaries are usually pretty straightforward to identify – if the period is short enough to fit in the 35-day window here then you should see repeating pairs of of transit dips, more often than not with different depths (as shown the graph on the left). You can take a look at my earlier “Making sense of the lightcurves” post here or at this PH blog post for more details about that.
However, as I discovered myself on a few occasions while looking at these lightcurves, occasionally the binary may be so widely separated that the transit from only one of the stars is visible in the 35-day dataset – so how can you tell if it’s a star or a planet then? Unfortunately, it’s not possible to to unambiguously identify whether an object is a star or a planet in this scenario – but it is still possible to glean some information from the single transit dip.
It turns out that (given a few assumptions) the depth of the dip in the lightcurve caused by the transit can be used to figure out the radius of the transiting object. We need to know the radius of the star as well, but that’s already provided in the supplemental data on the source pages for the stars (and in the public Kepler dataset). The planets that have been discovered so far don’t seem to get much bigger than about 1.8 Jupiter radii (1 Jupiter radius = 70,000 km), so if we calculate that the object is bigger than that then it’s probably a star instead – and if it’s a lot bigger, then it’s definitely a star). You can see the planet radii (and other data) for the known exoplanets at the interactive catalogue at the Extrasolar Planets Encyclopaedia (click twice on “Radius” there to list them in order of descending planet radius).
The equation to calculate the radius of the transiting object is Equation (1) from Seager and Mallen-Ornelas (2003) (click the PDF link in the top right to download a copy of the paper), which I’ve reproduced below:
where ΔF is the difference in flux between the top and the bottom of the dip (i.e. the depth of the dip on the graph), Rp is the radius of the planet, and Rs is the radius of the star. Since we know the transit depth, and we know the radius of the star (from the supplemental Kepler data), we can use this to estimate the radius of the planet!
As usual, there are a few things to keep in mind when using this equation:
- First, the equation is only valid if the transiting object is a planet, or is at least “dark” compared to the primary star. Since planets don’t emit light of their own, a planet passing in front of a star will block out a certain percentage of the star’s light, which is determined by the ratio of the planet’s radius to the star’s radius. If the transiting object is a star, then it’s emitting light of its own and the depth of the transit in the lightcurve will be reduced and give a misleading result. However, if the calculated radius for an unknown transiting object is still greater than 2 Jupiter radii, we can still be pretty sure that we’re looking at something that’s too big to be a planet and is actually a star, even though we can’t pin down its exact radius.
- Second, it’s not 100% accurate because it doesn’t take all the factors that can affect lightcurves into account. One thing is misses is limb darkening, which is the fact that a star is brightest in the middle of the disk and darker at the edges (the “limbs” of the star) because the viewer is looking through more stellar material towards the edges (see the Limb Darkening wikipedia page or this Oklo blog post for more info). However, it’s good enough to tell broadly whether you’re looking at a Jupiter-sized planet, or a Neptune-sized planet, or an Earth-sized one.
- Third, (and specifically for people using the Planethunters website) it should be noted that most of the lightcurves in the data on the PH website seem to be normalised to 1.008 instead of 1.000 (the datapoints for “Quiet” stars usually jitter around 1.008). This means that Fno transit in the radius equation is usually going to be 1.008.
Another interesting thing to note is that the distance between the planet and its star doesn’t affect the depth of the dip (ΔF) at all – from our distant perspective, a planet orbiting the star at 0.01 AU would cause the same depth of dip as one orbiting at 10 AU. The transit duration would be very different though, because one planet is much further out than the other – the width of the dip caused by the 10 AU planet would be much wider than the one caused by the 0.01 AU planet, and the inner planet would cross the star much more often – but the depth would be the same in both cases.
This equation can be used to create a table that shows the transit depth (ΔF) that you’d expect to see for various sizes of planet around different stars. I’m going to be a bit generous and say that the largest planet we can see will have a radius of 2 Jupiter radii (140,000km) and I’m going to call that “Twopiter” here. If the transit depth is greater than the ΔF value shown for Twopiter for that star, then the transiting object is too big to be a planet and must be a star instead. This allows us to quickly classify the system as either an eclipsing binary or one that may contain a transiting planet!
|140000 km||70000 km||58000 km||25000 km||6400 km|
The first column is the Star radius, in Solar radii.
The second column shows (roughly) what the spectral type of a Main sequence star with that radius should be.
The km values in the second row are the radii of the planets in that column in km.
The last five columns of the table show the transit depth (the difference in y-axis values from the top to bottom of the transit dip) for a planet of that size. Since the y-axis is normalised flux (to 1.008 anyway), this roughly corresponds to the fraction of the star’s light blocked by the planet during the transit.
One thing to notice from this table is that Earth-sized planets are going to be extremely difficult to see in the lightcurve data – most quiet stars seem to naturally vary in flux (y-axis value) by at least 0.001, and an earth-sized planet passing in front of 0.4 solar radius star (the smallest one listed here, which would be where such a transit would be most noticeable) would only produce a dip of 0.0005!
From the equation, we know that the fraction of light blocked by the transiting planet is essentially determined by the ratio of planet size to star size, so a large planet passing in front of a small star is going to produce a deeper transit (i.e. block out more light) than a small planet would. The difference is quite dramatic as illustrated in the images below, which show a Neptune (left), Jupiter (middle) and Twopiter (right) transiting in front of a 0.4 solar radius M0 V star, a 1.0 solar radius G2 V star, and a 3.0 solar radius B3 V star shown to the same scale. Looking at that, it shouldn’t surprise you that the Twopiter will block out a quarter of the M0 V’s light!
So, if you were looking at type G star with a radius of 0.9 sols, and you saw a transit with a depth of 0.010, then you’d look at the “0.9” row of the table and see that 0.010 is about halfway between the values for a Jupiter- and Saturn-sized planet – this means that you’re probably looking at a planet with a radius of around 64000 km (give or take 10% because of stellar limb darkening).
If on the other hand you see a transit with a depth of 0.5 around the same star, then as you can see this is much greater than the ΔF for Twopiter around a 0.9 solar radius star (which would be a transit depth of about 0.05). In fact, the transiting object would need to have a radius of about 450,000 km to produce a dip that big – far too big to be a planet – so in that case you would probably be looking at a transiting star that’s part of an eclipsing binary system (probably a K V companion star).
EDIT (30/12/10): I’ve discovered since I originally posted this article that it’s actually not quite that simple! For one thing, the radius provided in the Kepler data is likely to be accurate if you’re looking at a solo star with a planet, but if the star is a binary then that confuses matters since the derived mass and radius assume that it’s a single star! Further observations would be required to separate the two, once it’s obvious from the lightcurve that it’s a binary system.
So, keep in mind that while this equation and table are still handy if used in the right circumstances, you do need to be careful about when and where you use them!