Question

(i) Assume the Earth's orbit to be a perfect circle. Now consider an asteroid on a circular orbit of

radius 16 A.U. Use Kepler's Second Law to gure out the time period P of the asteroid's orbit.

(ii) If the same asteroid were in orbit around the star Antares, with a mass of about 16 solar masses,

and its orbit had the same radius, what would its period be?

(iii)Would it receive more or less energy from its star?

Answer 1

i) we are going to calculate the orbital period P of an asteroid having a circular orbit of radius (a) 16 AU.

We can use a simpler version of Kepler's third law for this calculation. It is expressed as follows

Where a= radius of orbit in AU=16 AU,

K is the Gaussian gravitational constant=0.017.

K^{​​​​​​2}​​​ Has the same dimensions as gravitational constant.

M= mass of the star in solar masses(Mo). For this problem,M=1Mo

P= period in days.

Now we have a formula with only one unknown. Let us plug in the values.

So the orbital period of the asteroid is about 64.77 years.

ii) now we replace the sun with Antares and repeat the calculation. I looked up the mass of Antares. It is in the range 15-18 Mo. I'm going to pick 15 Mo for this calculation.

Mass of Antares= 15 Mo

The orbital radius of the asteroid is the same i.e.a=16 AU

So the orbital period of the asteroid around Antares is about 16.72 years.

iii) This asteroid is at a distance of 16 AU from Antares which is 15 Times more massive than the sun. So the energy output of Antares is obviously more than that of the Sun. So it is clear that the asteroid would recieve more energy around Antares as opposed to it's orbit around the Sun.