Extrasolar Planets: Physics and Detection Techniques Problem Set 1 | Exercises Astrophysics

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Course 12.425. Problem Set 1. Due 20 Sept. 2007

1. How long would it take for a spacecraft to get from Earth to the nearest star with an exoplanet?

List any assumptions you made and show your calculation.

2. Properties of known exoplanets. From the Extrasolar Planets Encylopaedia at (http://exoplanet.eu/), ﬁnd

the interactive catalog for planet candidates detected by radial velocity (http://exoplanet.eu/catalog-RV.php).

Construct three diﬀerent diagrams (either histograms or correlation diagrams). You may use any available

parameter for the x and y axes (e.g., planet mass, radius, semi-major axis, year of discovery, etc.) Identify any

trends that are or are not present. The intention of this exercise is both to update the planet property diagrams

presented in Lecture 1 and to explore exoplanet properties not discussed in class.

3. Exoplanet transit light curves for non-central transits. See Figures 1 and 2.

a. Show that the transit duration for a non-central transit is

� �2 � �2

tT = P R∗ 1 + Rp a cos i. (1)

πa R

− R

∗ ∗

Here tT is the total transit duration, Ris the stellar radius, Rp is the planet radius, a is the orbital semi-major

∗

axis, and i is the orbital inclination. Here i = 90◦ is ”edge on.”

b. Show that the duration of the ”ﬂat part” of the transit light curve, i.e. the time when the planet is fully

superimposed on the stellar disk, is

� �2 � �2

P R Rp a cos i

tF = πa

∗ 1 − R

− R. (2)

∗ ∗

c. Describe why the impact parameter b is

b = cos i. (3)

R∗

The impact parameter b is deﬁned as the projected distance between the planet and star centers during midtransit

in units of R∗.

4. Planet radius, star radius and mass, semi-major axis, and orbital inclination for HD 209458. For

this problem you will estimate Rp, R∗, a, i, and Mfrom a real exoplanet transit data set. You will need the

∗

equations shown in problem 3, as well as the following: Kepler’s Law

4π2a2

P 2 = ; (4)

GM∗

the stellar mass-radius relationship

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M R

∗ = ∗ , (5)

M R

where x will be deﬁned below; and the planet-star ﬂux ratio

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ΔF = Rp . (6)

R∗

a. Download the data for the star-planet system HD 209458 from the MIT Server. Plot the transit

light curve: relative intensity vs. phase.

b. Average the data into larger time bins. This will make the transit light curve more clear. For example, you

could average twenty adjacent points. Describe your method for averaging and approximately what duration

your time bins are.

c. Estimate ΔF , tT , tF . Use P = 3.52474859 days .

d. Using the equations in problem 3 and 4, part 4c, and and x = 2.61, derive Rp, R∗, a, i, and M∗.

e. Estimate or compute the scatter in the data. In other words, by how much do the points deviate from an

average value? Describe the method you used.

5. Planet radius, star radius and mass, semi-major axis, and orbital inclination for GJ 436.

a. through d. as in Problem 4. Use P = 2.64385 days and x = 1.

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Extrasolar Planets: Physics and Detection Techniques Problem Set 1, Exercises of Astrophysics

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Download Extrasolar Planets: Physics and Detection Techniques Problem Set 1 and more Exercises Astrophysics in PDF only on Docsity!

1. How long would Courseit take for^ 12.425. a spacecraft^ Problem to get^ Set from^ 1. Earth^ Due to^20 the^ Sept. nearest^2007 star with an exoplanet?^1