Mass of an exoplanet

[alext87]

We will make some simplifying assumptions for this new planetary system:

 

 

the orbit of the planet is circular (e = 0)

 

the mass of the star is 1 solar mass

 

the mass of the planet is much, much less that of the star

 

we are viewing the system nearly edge on

 

we express everything in terms of the mass and period of Jupiter

 

 

 

We make these assumptions to simplify the equations we have to use for determining the mass of the planet. The equation we must use is:

 

Mass of planet = (P/12)^(1/3) x (K/13) x Mass of jupiter

 

P is orbiting time, K is the amplitude of radial velocity change.

 

P should be expressed in years (or fraction of a year), and K in m/s. Twelve years is the approximate orbital period for Jupiter and 13 m/s is the magnitude of the "wobble" of the Sun due to Jupiter's gravitational pull. Not all calculators will take the cube root of a number. Get help if yours does not. Put in your values for P and K and calculate the mass of this new planet in terms of the mass of Jupiter. That is, your calculations will give the mass of the planet as some factor times the mass of Jupiter (for example: Mplanet = 4 MJupiter).

 

Can anybody show me how to get to this equation?

[Martin]

this is a great question, alex!

 

it is a KEPLER THIRD LAW question

 

 

discussing this will be instructive and fun

there is a lot of meat in it, including wrestling with different systems of units if we want to get into that

 

I hope everybody should try to help clarify this (from swantsont and severian and atheist on down) so you get different viewpoints and attitudes.

 

I happen to be around so I will start

 

 

DO YOU KNOW THAT if the star mass is same as the sun's then THE PERIOD DETERMINES THE RADIUS OF THE ORBIT?

 

that is Kepler Third. the radius in AU equals the 2/3 power of the period expressed in years

 

Kepler saw this on the 8th of March 1618

Every 8th of March I take off my hat to that man and that moment in history and to the fact that one of us could actually SEE that pattern. The two thirds power of a number is very subtle. It is not like merely the square or cube or square root which are comparatively unsubtle.

 

What is the 2/3 power of 8?

 

I mean it, I want to make sure you understand. What is the 2/3 power of 27?

 

Another thing, suppose we adjust for the star average redshift so that ON AVERAGE it is staying the same distance from us. and we separate out the WOBBLE and just look at that.

 

DO YOU KNOW WHAT MOMENTUM IS? speed times mass

 

DO YOU KNOW THAT when the star is wobbling towards us the planet is swinging away from is WITH THE SAME MOMENTUM? so that the star's speed towards, times its mass, is the same size (but opposite sign) as the planet's speed away, times ITS mass.

 

this is the basic handle we have on the mass of the exoplanet. its period and (Kepler third) distance tell us its SPEED

 

but we can also measure the speed of the star as it comes towards us, and we know the mass of the star (from its luminosity and spectrum) so we know the MOMENTUM of the star as it comes towards us

 

and that is the same as the momentum of the planet going away from us because they have to balance (no net force on system, Newton)

 

so we know the speed of the planet and we know its momentum----simple algebra tells us the mass

 

your version of the equation has been ruthlessly simplified with approximate numbers like 12 and 13 and simplified units, to make the equation very easy to use-----what I have been talking about is the physical relations hiding inside the simplified equation.

 

before we try to derive the simplified equation, I suggest you reply to the questions i asked.

 

Do you understand the application of Kepler third? do you know the idea of momentum and why the momentum of the star has to balance that of the planet, in opposite directions.

 

the fat man does the polka with the skinny lady-----she must swing wider and circle around faster

[Martin]

alex did not come back, so maybe he did not like the style or manner of what I said. would anybody else like to try to answer his question and explain to him where that equation comes from?