Kepler's third law

Question

Kepler's third law, T = √(Kr³), can be used to calculate the time it takes for a planet to complete one orbit of a star. The average radius of the planet's orbit in km is r, and K is a constant that changes depending on the mass of the star.

a) Calculate the value of K if it takes Earth on year to go around the Sun, and Earth is, on average, 1.49 x 10⁸ km from the sun.

b) Use your answer from 2.a to calculate how many Earth years (rounded to the nearest tenth) it takes for Neptune to complete an orbit if Neptune's average orbital radius is 4.46 x 10⁹ km

Thank you

Answer 1

a) T=1 year r=1.49 x 10⁸

 

T = √(Kr3)

 

1=√(K(1.49×10⁸)³)

 

K=3.023×10^-25

 

b) T = √(Kr3)

 

T=√(3.023×10-25)(4.46 x 10⁹)³)

 

T=163.8 years

 

Answer 2

 T = √( Kr^3) =

 

T^2 = K r^3   / Square both sides.

 

K = T^2/ r ^3

 

  =  1/ (1.49 * 108) ^3

 

   =  2.4 X 10^ - 7

b)

     T = √( 2.4 * 10 ^ -7) ( 4.46 X 109) ^ 3=

        = 

Answer 3

First we want to solve for K. Since earth conveniently has a T = 1 orbit, this simplifies our equation to:

1 = sqrt( K * e^3 )

but

1 = sqrt( e^3 / e^3) = sqrt( 1 ),

So K = 1/e^3.

Plug this in for Neptune we have

T = sqrt( n^3 / e^3 ) = sqrt( (n/e)^3 )
= sqrt( (4.46e9 / 1.49e8)^3 )
= sqrt( (44.6/1.49)^3 ) = sqrt ( 29.93289^3 )
=163.7657. (sig digits 164 years).

Compare this to Wikipedia which says Neptune’s orbit is 164.79 y. Pretty close! We only had 3 significant digits to start the problemand we are almost accurate to 3, this is about what you should expect.

Answer 4

T = sqrt(K*r^3)

Square both sides:
T^2 = K * r^3

Divide by r^3:
K = T^2 / r^3
= (1 E-yr)^2 / ((1.49 km)^3*(10^8)^3)
= (10^(-24) / 1.49^3) E-yr^2/km^3
= 0.302302121344676 * 10^(-24) E-yr^2/km^3
= 3.02302121344676 * 10^(-25) E-yr^2/km^3

T_N = sqrt(3.02302121344676 * 10^(-25) * 4.46^3 * 10^9^3)
= sqrt(268.19197031151317 * 10^(2))
= 16.37656772072565 * 10
= 163.8 E-yr

Answer 5

My answer is 163.74 years, but my calculations didn't show up yet. The constant K = 2. 995x 10 -19 s²/m³.

If T is measured in Earth's years and r - in astronomical units, then K = 1 for all planet in the solar system.

Answer 6

a) Solve for K,

K = T^2/r^3 = 1^2/(1.49*10^8)^3 = 3.02*10^-25 Earth years^2/Km^3

 

b) T = √(Kr^3) = √(3.02*10^-25*4.46^3 * 10^27) = 164. Earth years