Calculating Orbital Period of Asteroid Between Mars & Jupiter

[Havenater23]

Homework Statement

An asteroid is located between Mars and Jupiter. It is thought that a planet once orbited here but was somehow destroyed and broken up into small chunks(perhaps by getting hit by a comet or asteroid). If an asteroid in this belt has an average distance from the sun of 500 * 10^6 km what would the orbital period be?

Homework Equations

The Attempt at a Solution

I know you use the keplers formula

t^2 / r^3

and make it proportional. I know the distance between Mars and Jupiter and their orbital periods , how can that help me?

 

[SammyS]

Homework Statement

An asteroid is located between Mars and Jupiter. It is thought that a planet once orbited here but was somehow destroyed and broken up into small chunks(perhaps by getting hit by a comet or asteroid). If an asteroid in this belt has an average distance from the sun of 500 * 10^6 km what would the orbital period be?

Homework Equations

The Attempt at a Solution

I know you use the Kepler's formula

t^2 / r^3

and make it proportional. I know the distance between Mars and Jupiter and their orbital periods , how can that help me?

Hi Havenater23.

Right: T²/R³ is the same for objects orbiting the sun.

What is Earth's orbital period and average distance from the sun?

Note: 1 Astronomical Unit ≈ 150 × 10⁶ kilometers.

 

[Havenater23]

Average distance : 149.6 * 10^6 km

Period : 1.0 Earth years

 

[SammyS]

average distance : 149.6 * 10^6 km

period : 1.0 Earth years

T²/R³ = ? for earth. It's the same for the asteroid.

 

[Havenater23]

How come ?

 

[SammyS]

How come ?

What is Kepler's 3^rd Law?

 

[Havenater23]

That the period of orbital is squared and that radius is squared, and this proportional between two planets , I believe ?

So it would actually work with distances and period orbitals of other planets correct ?

 

[SammyS]

That the period of orbital is squared and that radius is squared, and this proportional between two planets , I believe ?

So it would actually work with distances and period orbitals of other planets correct ?

So, it will work for all objects orbiting the sun.

 

[Havenater23]

I need someone to check my work can you please solve this and tell me what you get?

I have it set up like so

t^2 / (500*10^6)^3 = 3.35*10^24

 

[SammyS]

T² / (500*10^6)^3 = 1 / (3.35×10²⁴)

Now solve for T.

 

[Havenater23]

Okay I got the answer, but I don't think it's as long as I calculated. I got something like

2.05*10^25

I can see it being 2 years, but what am I doing wrong?

 

[SammyS]

T² / (500*10^6)^3 = 1 / (3.35×10²⁴)

Now solve for T.

This is:

$$\frac{T^2}{(500\times10^{\,6})^3}=\frac{1}{3.35\times10^{\,24}}$$