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Astro 2 - Exoplanets transits

Dylan Price

13.1.15 to 20.1.15

This experiment show how we discover and get data on exoplanets just from their transits across their

parent star. By taking careful measurements of the luminosity dip when the planet transits across the star

and plotting a light curve we can find out its orbital time period, the radius of the orbit, the size of planet

and more. We take these measurement using photo-sensitive telescopes which can look out into deep spaceand give us accurate readings of light. We used data gathered from such telescopes published by NASA to

work these out. Using this data we got values quite close to the accepted values for the planets.

1 Introduction

When a planet transits across their parent star theyblock out some of the light emitted by the star. Us-ing very photosensitive measuring devices like a largetelescope we can take readings of this planet and workout its basic features. This does however require theplanet to have a favourable orbit (i.e. it is the onlybody transitting across when it does as to not give

false readings therefore not all exoplanets found willhave data as good as the ones used in this paper.Equations we have used frequently in this paper

will be these:

L= 42r2T4 (1)

Luminosity equation for a black body. Where L isluminosity, r is the radius of the black body, sigma isthe Stefan-Boltzmann constant and T is the effectivetemperature.

A= r2 (2)

Area of a circle

R3 =T2GM

42 , R3 T2 (3)

Keplars third law. Where R is radius of object, T isthe orbital period of the object, G is newtons grav-

itational constant, M is the mass of the body theobject is orbitting.

2 Method

Using the NASA exoplanet archive[1] we found thedata for the system of Kepler-12, and of planetKepler-12-b from here we opened the light curve forthe planet. This is the data for when the planet tran-sits across its parent star.

The screenshot on the next page shows thelightcurve, I have selected to only show 1 of the curvesusing the button that looks like stacked paper, select-ing only the top row. This gives the graph shown.When this was up and I was satisfied with the graphI normalised the y-axis bu using the 3rd tab shownin the picture. The normalised y-axis shows the lu-minosity dip to be from 1.000 to 0.985, showing thedip as a fraction and not values of flux, this makescalculation easier. The graph also shows us the timeperiod between each transit so we can get a value for

its orbital period.

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Screenshot of the light curve with settings.

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2.1 Radius of Kepler-12b

Using the dip in luminosity we can find out theplanets radius. For Kepler-12 it was found to befrom 1 to 0.983. This means the planet was blocking1.7% of the stars luminosity as it transits across. Weassume that we see a circlular disc from the star andthe planet transitting for this we can use equation 2to work out the area of the star and planet. We alsoassume that the planet radiates as a black body sothe Luminosity equation (equation 1) can be used forit. From the equations shown in the Introduction sec-tion we can see that L r2 and thatA r2 thereforeA L and so A L. This means that the ratiobetween the area of the planet and the star is same as

the dip in luminosity when the planet transits acrossthe star, in mathematical terms

Aplanet

Astar= 1.7% for

Kepler-12b. Putting this through gives the value ofthe radius of the planet as 1.34108m . The acceptedvalue is 1.3 108m, therefore we are very close withour values, only 3% out.

2.2 Orbital Radius of Kepler-12b

To work out the orbital radius of the planet you firstneed to find out the orbital period of the planet, for-tunatly we can find this out from the time periodbetween the two dips in the image on page 2 we also

need the mass of Kepler-12 which is 1.166x the massof the sun. The time period is worked out to be about4.4 0.025 days which is 3801609s and the mass ofthe star is 2.321030, using these values we can inputit into equation 3:

Rorbit = 3

38016092 6.67 1011 2.32 1030

42(4)

This all comes out to be 8.19 109m which is 0.050.006AU.The accepted value for the orbital radius iswithin our errors at 0.0556AU.

3 Radius and orbits of planets

Kepler-40b and KOI-217b

We apply the same method to these two other exo-planets and get these results:

3.1 Kepler-40b

Time period: 6.87 0.27 days; Luminosity dip per-centage: 0.4%

Radius of planet = 8.67 107m, actual is 8.17107m, 5.76% out but still very close.

Radius of orbit = 1.19 1010 4.76 108m =0.08 0.003AU. This is the same as the actualvalue.

3.2 KOI-217b

Time period: 3.91 0.1 days; Luminosity dip per-centage: 2.375% .

Radius of planet = 9.18 107m, actual is 7.76107m which is 15% out

Radius of orbit = 7.97 109 1.594 108m =0.050.001AU, the actual value was 0.06AU butit had no error I could find, making my value 16%out.

4 Analysis

My measurements are very close to the actual values

shown in the confirmed planets table[2]

. This showsthat the transits method works very well to get truevalues for the planets data. If I were to use more thanone light curve in my measurement I believe it wouldimprove the accuracy of my results as they would giveme an average and an easier way to spot mistake inother readings but it is rare to find a second lightcurve which gives readings which are useable for thesame planet using this method. A difficulty I facedwhen finding two other planets was finding a lightcurve with useable data, most of the time the planethad a light curve which showed an imperceptable dipand when checking these planets I noticed they had

larger AU than the ones I used in the end.

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5 Conclusion

The transit method works very well in working outvalues for all sorts of planetary data but I believethe method is biased towards planets with a smallAU as all planets we found with a good light curveall had 0.1AU this leads me to believe that planetswith larger AUs dont block enough light from theirparent stars for the transit method to work properly.The method also favours planets with more circularorbits.

References

[1] http://exoplanetarchive.ipac.caltech.edu/index.html

[2] http://exoplanetarchive.ipac.caltech.edu/cgi-bin/TblView/nph-tblView?app=

ExoTbls&config=planets

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