5. Triton is the largest satellite of Neptune. It is very unusual in that Triton is in a retrograde orbit around Neptune (i.e., it orbits in a direction opposite to the planet's rotation). Tidal braking on Triton by Neptune is causing its orbit to decay, and in about 3.6 billion years from now, Triton will cross Neptune's Roche limit and will be destroyed. Densities of Triton and Neptune are 2060 kg/m³ and 1640 kg/m³, respectively. Calculate the Roche limit of Neptune. Express your answer in meters. Neptune's radius is 2.48 × 104 km. As a satellite gets closer to a planet, eventually tidal forces stretching the satellite in the radial direction becomes greater than its own gravity holding itself together. Thus, tidal effects impose a minimum permissible orbit size on the satellite, known as the Roche limit. Consider two identical, small spherical masses m and a planet with mass M. The masses are touch- ing each other, and their centers are on a radial line from the planet. The distance between the centers is Ar, and the distance from the center of the planet to the midpoint between the masses is r as shown below. R M Tidal force on the two masses by the planet is equal to 2GMm AF = Ar, p3 while the self gravity holding the two masses together is equal to F = - Gm² Ar2. 4 ||| m m As r gets smaller, there is a critical distance at which the magnitude AF which separates the two masses becomes greater than the magnitude F which holds the two masses together. This distance is called the Roche limit, rR. 2GMm Gm² Ar = Ar2' TR= 2M 1/3 Ar. m It is more common to express the Roche limit in terms of the mass density of the objects involved: 4π M = 3 R³ PM, 3 4π Ar m = Pm, 3 2 TR = 160м Pm 1/3 1/3 PM R≈ 2.52 R. Pm A more exact calculation assumes the satellite as a deformable fluid: :44 (PM) 1/3 TR≈ 2.44 Pm R.

5. Triton is the largest satellite of Neptune. It is very unusual in that Triton is in a retrograde orbit around Neptune (i.e., it orbits in a direction opposite to the planet's rotation). Tidal braking on Triton by Neptune is causing its orbit to decay, and in about 3.6 billion years from now, Triton will cross Neptune's Roche limit and will be destroyed. Densities of Triton and Neptune are 2060 kg/m³ and 1640 kg/m³, respectively. Calculate the Roche limit of Neptune. Express your answer in meters. Neptune's radius is 2.48 × 104 km. As a satellite gets closer to a planet, eventually tidal forces stretching the satellite in the radial direction becomes greater than its own gravity holding itself together. Thus, tidal effects impose a minimum permissible orbit size on the satellite, known as the Roche limit. Consider two identical, small spherical masses m and a planet with mass M. The masses are touch- ing each other, and their centers are on a radial line from the planet. The distance between the centers is Ar, and the distance from the center of the planet to the midpoint between the masses is r as shown below. R M Tidal force on the two masses by the planet is equal to 2GMm AF = Ar, p3 while the self gravity holding the two masses together is equal to F = - Gm² Ar2. 4 ||| m m As r gets smaller, there is a critical distance at which the magnitude AF which separates the two masses becomes greater than the magnitude F which holds the two masses together. This distance is called the Roche limit, rR. 2GMm Gm² Ar = Ar2' TR= 2M 1/3 Ar. m It is more common to express the Roche limit in terms of the mass density of the objects involved: 4π M = 3 R³ PM, 3 4π Ar m = Pm, 3 2 TR = 160м Pm 1/3 1/3 PM R≈ 2.52 R. Pm A more exact calculation assumes the satellite as a deformable fluid: :44 (PM) 1/3 TR≈ 2.44 Pm R.