Orbital Velocity

An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has a mass of ##M_1## = 1.90 x 10##^{27}## kg and radius ##R_1## = 7.14 x 10##^7## m.
A. If the radius of the planned orbit is ##R## , use Newton’s laws to show each of the following.

• The orbital speed of the planned satellite is give by ##v## = ##\sqrt{\frac{GM_1}{R}}## .
• The period of the orbit is given by ##T## = ##\sqrt{\frac{4\pi^2R^3}{GM_1}}##

B. The explorer wants the satellite’s orbit to be synchronized with Jupiter’s rotation. This requires an equatorial orbit whose period equals Jupiter’s rotation period of 9 hr 51 min = 3.55 x 10##^5## s. Determine the required orbital radius in meters.
C. Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs form the desired value in each of the following ways, sketch the resulting orbit on the figure. ( J is the center of Jupiter, the dashed circle is the desired orbit, and ##P## is the injection point.) Also, describe the resulting orbit qualitatively but specifically

• When the satellite is at the desired altitude over the equator, its velocity vector has the correct direction, but the speed is slightly faster than the correct speed for a circular orbit of that radius.
• When the satellite is at the desired altitude over the equation, its velocity vector has the correct direction but the speed is slightly slower than the correct speed for a circular orbit of that radius.