edoarad said:
Homework Statement
i need to find the mass of a new found planet at a given radius from the sun (R) and with a given cycle length (T)Homework Equations
Kepler and NewtonThe Attempt at a Solution
of course if R and T would fit with keplers third law then there would be no way to find the mass but because it doesn't fit (T is shorter than it should have been) i thought that mabye M is big enough so that it could enfluance the sun. i tried finding the center of mass but that didnt lead to anywhere..
hope someone can help me :)
You're on the right track. The Sun and any planet orbit their barycenter, or center of mass of the system. For instance, if they were of equal mass, they orbit a point halfway between them. For planets very small compared to the Sun, the barycenter is very nearly at the center of the Sun. In these cases you can safely ignore the mass of the planet, as it will make no noticeable difference.
When as the mass of the planet becomes greater, you start to see a difference.
In these cases, you have to take the position of the barycenter into account. A more massive planet will result in a shorter period.
Here's how you can do that:
Take the distance between Sun and planet and their respective masses to find the distance from planet to barycenter (Rb)
Determine the speed the planet would have to move in a circle with a radius of Rb so that the gravitational force between the Sun and planet is just enough to hold it in that circular path.
Determine how long it takes for the planet to travel a circle with a radius of Rb at that speed.
Since in your case you don't know the mass of the planet, you'll have to just use a variable for it as you go along. At the end you should get an equation that gives you the period of the orbit while taking the mass of the planet into account. Once you have this, you can plug in all the knowns and solve for the mass of the planet.
