If its orbit is circular, what is its radius? ?

Question

Suppose a planet were found with an orbital period of 64 years. How might you estimate its distance from the Sun? If its orbit is circular, what is its radius?

Answer 1

Kepler's III law says the radius vector of the orbit sweeps equal areas in equal time. Taking a cue from this, the circular and the elliptical orbuts should have the same area for a period of 64 years.

If circular orbit has radius 'r' and elliptical orbit has semi major & semi minor axes have 'a' & 'b', their areas are

pi ab = pi r^2

r = sqrt(ab).

Hence, radius would be the geometric mean of semi major & semi minor axes.  

Answer 2

to find the radius you must use calculus. the circumference of an orbit is 64 years.  circumference = C) diameter =D)  radius = R) PI=TT)                     C/ TT) = D).  D/2 = R)

64 C) / 3.14159 TT)= 20.371 D..  20.371 D) /2 = 10.185 R)

radius = distance to the sun in years.  R= 10.185 years at the planets speed. but it might have a slow orbital period for a number of reasons. so it is not a vary acurit system of mesure.  

Answer 3

You would either need the mass of the object and it's sun, or the circumference of it's orbit to calculate the radius.  And this seems to be working backwards.  Typically, you would be able to find the radius of this "circle" (by measuring the distance of the planet from the sun) to figure out the circumference of the orbit (Circumference = 2 * Radius * Pi^2).  Once you had that number, you would be able to calcualte it's velocity around it's star, because the circumference would be distance, 64 years would be time, and speed = time / distance.

<edit>My above statement is incomplete.  After looking deeper, Kepler's third law states that it is possible to deterimine an orbit based on its orbital period.

Kepler's third law states that the square of the period, P, is proportional to the cube of the semi-major axis, a. In equation form Kepler expressed the third law as: P^2=ka^3. k is the proportionality constant. To Kepler it was just a number that he determined from the data. Kepler did not know why this law worked. He found it by playing with the numbers.

Answer 4

Kepler's third law says that the square of the periods of two planets are in the same ratio as the cubes of their orbit radii. So comparing the hypothetical planet to Earth, you have

64^2 = T^3 or T = 16.0 years.