Question

Furthermore, if we assume that the luminosity of a planet comes entirely from reflected starlight, then we can solve for the planet's luminosity as II. Direct Imaging Direct imaging is a technique for detecting extrasolar planets by observing them directly with telescopes. This technique is difficult to employ in practice because planets emit very little light compared to the very bright stars they orbit. In other words, there is a stark contrast ratio between stars and planets. The contrast ratio C is defined as the ratio of the planet's luminosity L, to the star's luminosity Ls, or Lp = Ls ( 34) = Ls (ako), where A, is the planet's projected area, Sorbit is the surface area of the sphere with a radius equal to the size of the planet's orbit, R, is the radius of the planet, and a is the semimajor axis of the planet. C = 7 The higher the contrast ratio of a planet-star pair is, the easier it will be to detect since that means the planet is brighter compared to the star. 4) (a) Calculate the luminosity of sunlight that the Earth reflects. Assume for simplicity that Earth reflects all of the sunlight it receives. (b) Calculate the contrast ratio between the Sun and the Earth.

Answer 1

a) for this problem,we have to calculate the luminosity of sunlight that the earth reflects and we assume that Earth reflects all luminosity it recieves. So once we calculate how much of the luminosity earth recieves,we will have our answer.

The luminosity recieved(L_{​​​​​​p} ) by the projected area A_{​​​​​​p} of the earth at an orbital distance (a)=1.5×10¹¹ m can be calculated by the given formula

De = 67

Where R_{​​​​​​p} is the radius of earth=6.4×10⁶ m and L_{​​​​​​s} is the luminosity of sun=3.828×10²⁶ W.

So let us plug in the values.

................(π cancels out)

L = 3.828 x 1026 ( (6.4 x 106) * 4x (1.5 x 101192)

L = 1.74 x 1017

So the luminosity reflected byEarth =1.74×10¹⁷ W.

b) the contrast ratio is just the ratio of luminosity of the planet with the luminosity of the star. In this problem,we have been instructed to assume that the luminosity of the planet is entirely due to the reflected starlight. So we apply the same assumption for earth. We have already calculated the luminosity of earth due to reflected sunlight to be 1.74×10¹⁷ W and we know that the luminosity of sun 3.828×10²⁶ W. So let's just take the ratio.

C = 42 1.74 101 3.828 x 1026 = 0.45 x 10-9

So the contrast ratio between earth and sun is 0.45×10^-9