How long is it until Phobos hits the surface of Mars

[leonne]

Homework Statement

The semimajor axis is decreasing at a rate of 20 cm yr−1. At that rate, how long is it until Phobos hits the surface of Mars
the semimajor axis is 9380 km

Homework Equations

The Attempt at a Solution

Well first i found the distance from Mars to phobos which was like 5900km and i got 29 million years but says on wiki like around 11 million years. so i am guessing that its not at a constant rate . Was thinking of using tidal acceleration and something with the Roche limit. formula for tidal accel is is a=(2GMR)/r^3 and the roche limit is at 5540km.
Any hints will be helpful thanks

 

[diazona]

Are you sure the answer you found online is the answer you're supposed to obtain for the problem?

 

[leonne]

i was thinking that, but they had the same info like every year it has a decreasing rate of 20 cm from wiki "Because Phobos's orbital period is shorter than a Martian day, tidal deceleration is decreasing its orbital radius at the rate of about 20 metres (66 ft) per century. In an estimated 11 million years it will either impact the surface of Mars" and they have the semi major axis 9,377.2 km which is pretty close to what i have so my answer should not be that much off from what they have.

 

[diazona]

But are you sure that the calculation they did to obtain the answer in the article is the same as the calculation you're supposed to be doing? I ask because I wonder if it's possible that you're being asked to solve a simplified version of the problem.

How about this: what class is this problem for, and what sort of material have you most recently been learning in it?

 

[leonne]

Hey its a junior level physics course , astrophysics, so not a intro course. like this week we covered Lagrange points; asteroids; close binaries. This problems is a 4 part problem first was to find density of Mars and the moon, than find the Roche limit and find the orbital period of the moon, now how long till the moon crashes into mars.

 

[diazona]

OK, sorry about that. The wording of the problem seemed like the kind of thing that might be asked in an intro-level physics class - I thought that perhaps "at that rate" meant you were supposed to assume the given rate of change in orbital radius was a constant.

Anyway, the first thing I'd try would be to account for the variation in gravitational force as Phobos gets closer to Mars. You could probably write a differential equation for r(t) and integrate it to find the time it would take for the moon to reach the planet's radius.

 

[leonne]

lol its ok... yeah was just thinking of that thxs ill try it out and see what i get

 

[leonne]

would i use the tidal acceleration formula? and make it r^3=(2GMR/a) then integrate that to find t? well going to try it out later.