Question

The following multipart problem asks you to derive a number of characteristics of an extrasolar planetary system. Assume that the planet has been detected by Kepler with the transit method, and that the transits are periodic (as shown below, a dip in the lightcurve indicates that a planet has moved in front of the star).

(a) The star has 3 times the mass of the sun (i.e., M∗ = 3.0M⊙) and the period of the transits are 2.0 Earth years (i.e, the orbital period of the planet around its star is twice the orbital period of the Earth around the sun Tp = 2.0T⊕). To make things interesting, let’s imagine that the planet is in an elliptical orbit with eccentricity e = 0.3. What is the perihelion of the extrasolar planet to it’s star in a.u.? (Remember that the radius of the Earth’s orbit around the sun is a⊕ = 1 a.u., it might help to eliminate some constants).

(b) If the star has its maximum emission (observed flux per unit wavelength interval) at a wavelength of λp,∗ = 250nm what is the temperature of the star T∗? (Hint: note that the sun has it’s peak emission λp,⊙ = 500nm and has a temperature of T⊙ = 5, 800K, use ratios!)

(c) If the star has twice the radius of the sun R∗ = 3.0R⊙, what is the luminosity of the star relative to that of the sun L∗/L⊙?

(d) Now, using the relative luminosity of the star to the sun, L∗/L⊙, from part (c), the relative distances of the Earth to the sun, d⊕, and the average distance of the planet to its star, dp calculate the no-greenhouse temperature of the planet as follows: First, assume that all of the properties of the atmosphere and the planet’s surface (i.e., the emissivity, absorptivity, pollution, etc.) are the same as those of the Earth. Also assume that the radius of the planet is equal to twice that of the Earth Rp = 2R⊙. Solve for the ratio of the temperature of the planet to that of the Earth (Tp = T⊙). Then, use the average temperature of the Earth T⊕ = 256K to find Tp. Do you want to live on this planet? (Note: the no- greenhouse temperature is that temperature for which the total power absorbed by the planet, equals the total power re-radiated into space assuming the planet is a perfect black-body)

note: there was no diagram provided to me after the line thst says "as shown below" in this question but I do not think it is necessary to have a diagram to solve the problem.

Answer 1

Solution:

M*= mass of star; Ms=mass of sun; e=eccentricity; T*= orbital period of star; Te= orbital period of earth

a) given: M* = 3Ms; T* = 2Te; e= 0.3

we know, III kepler's law,

substituting the given values,

we get, semi major axis, a = 2.727e6 a.u

perihelion P = a(1-e) = 2.727e6 * (1-0.3) = 1.91e6 a.u

b) given,

using Wein's displacement law at maximum intensity,

500 * 5800 = 250 * T*

T* = 11600 k

c) given, R*=2*Rs

using the luminosity relation,

d) given: radius of planet,Rp = 2Re; Re= radius of earth

using luminosity equation,

Now consider the no greenhouse temperature,

taking Tp and Ts and dividing them,

Tp = 256.000 k