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## Homework Statement

If NASA wants to put a satellite in a circular orbit around the sun so it will make 2.0 orbits per year, at what distance (in astronomical units, AU) from the sun should that satellite orbit? The earth's orbit is 1.0 AU from the sun.

[itex]T_E = \frac {1.0orbit}{yr}[/itex]

[itex]T_s = \frac {2.0orbit}{yr}[/itex]

[itex]r_E = 1.0AU[/itex]

## Homework Equations

[itex]\frac {T_E^2} {r_E^3} = \frac {T_s^2} {r_s^3} => r_s = \sqrt[3]{\frac {T_s^2 r_E^3}{T_E^2}}[/itex]

## The Attempt at a Solution

Using Kepler’s 3rd law of planetary motion, we know that the orbital period squared is proportional to the radius of the orbit cubed. Since the proportion is approximately the same for all objects orbiting relative to the Sun, then: [itex]\frac {T_E^2} {r_E^3} = \frac {T_s^2} {r_s^3}[/itex]

Solving for the radius of the satellite and plugging in the values, we get:

[itex]r_s = \sqrt[3]{\frac {T_s^2 r_E^3}{T_E^2}} = \sqrt[3]{\frac {(\frac {2.0orbit}{yr})^2 (1.0AU)^3}{(\frac {1.0orbit}{yr})^2}} = \sqrt[3]{4.0AU^3} = 1.5874AU \sim 1.6AU[/itex]

Thank-you

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