How to much mass does it take to change Neptune's Orbit?

[masterpiece5]

Every time Neptune scatters bodies to Jupiter, Neptune gains energy and its
orbit becomes larger. How much mass would Neptune have to scatter to Jupiter
for Neptune’s orbit to have changed from a circular orbit at 22AU to a circular
orbit at 30AU? Give the answer in terms of Neptune’s mass and assume that all
scattered objects approached Neptune on circular orbits but left on orbits with
aphelia of 26AU (average between 22 and 30) and perihelia of 5 AU.

 

[JANm]

Every time Neptune scatters bodies...

...aphelia of 26AU (average between 22 and 30) and perihelia of 5 AU.

Hello masterpiece5

The energy of bodies is given by the semimajor axis. Which is here 15,5. I suppose you mean that Jupiter captures them when they are near to the perihelium of 5 AE?

The lost masses go from somewhere in between 22..30 AU to 15,5 AU that is the energy Neptune loses when the bodies are captured by Jupiter, variing form 2/3 to 1/2 of the original energy. Mean of 2/3 and 1/2 is 7/12.

On the other hand gain of the remaining Neptune has to be: 30/22=15/11

Mass of Neptune = Mass_remaining + Mass_captured

Total energy = 15*Mass_remaining/11 +7*Mass_captured/12

Two linear equations with two unknowns must be solvable.

By the way greetings to this site

Janm