The following multipart problem asks you to derive a number ofcharacteristics of an extrasolar planetary system. Assume that theplanet has been detected by Kepler with the transit method, andthat the transits are periodic (as shown below, a dip in thelightcurve indicates that a planet has moved in front of thestar).
(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, theorbital period of the planet around its star is twice the orbitalperiod of the Earth around the sun Tp = 2.0T⊕). To make thingsinteresting, let’s imagine that the planet is in an ellipticalorbit with eccentricity e = 0.3. What is the perihelion of theextrasolar planet to it’s star in a.u.? (Remember that the radiusof the Earth’s orbit around the sun is a⊕ = 1 a.u., it might helpto eliminate some constants).
(b) If the star has its maximum emission (observed flux per unitwavelength interval) at a wavelength of λp,∗ = 250nm what is thetemperature of the star T∗? (Hint: note that the sun has it’s peakemission λp,⊙ = 500nm and has a temperature of T⊙ = 5, 800K, useratios!)
(c) If the star has twice the radius of the sun R∗ = 3.0R⊙, whatis the luminosity of the star relative to that of the sunL∗/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 thesun, d⊕, and the average distance of the planet to its star, dpcalculate the no-greenhouse temperature of the planet as follows:First, assume that all of the properties of the atmosphere and theplanet’s surface (i.e., the emissivity, absorptivity, pollution,etc.) are the same as those of the Earth. Also assume that theradius 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 theEarth (Tp = T⊙). Then, use the average temperature of the Earth T⊕= 256K to find Tp. Do you want to live on this planet? (Note: theno- greenhouse temperature is that temperature for which the totalpower absorbed by the planet, equals the total power re-radiatedinto 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 isnecessary to have a diagram to solve the problem.
