# Model used for exoplanet data

1. Feb 24, 2005

### Dracovich

I've been googling like crazy the past few hours trying to find something i can use :) I find a lot of general explenations, but i'm having a hard time finding a concrete model to what is used to get the final values that seem to be extracted from the data gained from doppler/radial velocity measurements.

So you get a grab with radial velocity and time, but what do you do with this data to deduce the size of a planet, it's distance from the star etc. As far as i can see, you get the total period of the planets orbit, and the radial velocity of the star at some given time in the orbit. But i'm unsure of how this translates into the values of mass and radius of orbit.

If you could explain, or point me in the right direction :) it'd be greatly appreciated.

2. Feb 24, 2005

### marcus

I read some of the first papers on this (by Geoff Marcy and Paul Butler)
http://astron.berkeley.edu/~gmarcy/
in the 1990s and I vaguely recall the proceedure. Maybe someone here will have a source online, but I don't. I can tell you what I remember, more or less.
Here's some general info
http://exoplanets.org/

I assume you know what CURVE-FITTING is all about.

if there is only one planet and the orbit is essentially circular then you start by fitting SINEcurve to the doppler data taken over sufficiently long time

if the data over several years looks like a superposition of several sinewaves then maybe there are several planets. maybe the wobble is a sum of wobbles caused by several planets so you have to subtract out and fit curves to separate components

sinecurve fitting only works for circular orbits. (eccentricity zero)
Suppose you watch the star over a long time, like several years, and plot the radial velocity. Suppose the plot looks like a distorted sinewave with the peaks skinny and the valleys broad-----or the hills broad flat-topped and the valleys more narrow like canyons. Then you are looking at wobble caused by a planet with ECCENTRIC orbit

Look at number 68 on this list
http://exoplanets.org/almanacframe.html
It is HD28185
the period is 383 days and the eccentricity is 0.07

That would be an easy one to fit curve to. the eccentricity is small, only 0.07. The plot of radial velocity over several years (to be sure) would
be almost a sinewave.

But look at some of the other planets on the same list. Eccentricity 0.2 and 0.3 and 0.4. Terrible.

So the first thing is to take data over several years (or at least orbit periods) and to pick period and eccentricity numbers so that the curve fits!

The next thing for you to understand is how you would estimate the planet MASS and SEMIAXIS from that curve, but for starters assume that it is zero eccentricity nice sine curve. Then the arithmetic is very easy to understand.
I just went thru this with someone else in another thread. Maybe I will just give a brief sketch of the idea and then sit this out and let someone else talk.

Last edited: Feb 24, 2005
3. Feb 24, 2005

### marcus

you know the mass M of the star by its color (more massive stars are hotter)

the mass M and the period P tells you the semiaxis A
by Kepler's Third Law.

now you just need to calculate the planet mass m.

this is found with the assumption that line of sight is in the orbit plane so that the maximum radial velocity of the star is the true maximum and not some fraction of it (as in case where orbit plane is tilted)

the planet mass comes from this proportion

planet mass/star mass = star speed/planet speed

BECAUSE YOU KNOW THE SEMIAXIS AND THE PERIOD you know the planet speed
BECAUSE YOU HAVE THE DOPPLER DATA you know the star wobble speed
BECAUSE YOU KNOW THE COLOR you know the star mass

so by this proportion you learn the planet mass.

if the planet speed turns out to be 1000 times the star speed, then the planet mass must be 1/1000 of the star mass

but this is only a lowerbound on the planet mass. if the orbit plane is actually tilted then the star's true wobble speed is greater and that means the planet is more massive

4. Feb 24, 2005

### Chronos

Well done marcus. You sit back often, it does not go unnoticed.

5. Feb 24, 2005

### marcus

Thanks, Chronos!

what the questioner may really want is to have a formula written down.
this would essentially be to write down Kepler's Third.
maybe someone will do this and say how, knowing mass of star and period, one can deduce semiaxis

6. Feb 24, 2005

### Dracovich

ok sounds good :) Thanks for a good explenation.

Is there any way to find out what the tilt of the orbit is? Or is that just something to be left to speculation?

7. Feb 24, 2005

### tony873004

I don't think so, unless the planet transits the face of the star, in which case it is 90 degrees.

On the following web site:
http://exoplanets.org/almanacframe.html

the table that gives planetary data has a column labeld M sin i M(jupiter)

I'm not quite sure how to use the data in this column, but the number in that column represents the minimum mass the planet can be. But like Marcus said, any change in inclination will result in a more massive planet.

8. Feb 25, 2005

### Chronos

It's a 3D geometry problem - tilt of the rotational axis. You assume it is perfectly aligned with our line of sight and derive a maximum mass of the body in question.

9. Feb 28, 2005

### Dracovich

Ok, some slight followup :)

You say i'll find the semiaxis (or in the case of a circular orbit, the radius) from keplers third law, which i kinda expected. But what's bothering me is that it's $$\frac{T^2}{r^3}=Constant$$ But how do i know what that constant is? After some googling i read that a special case of Keplers law was that $$T^2=R^3$$ which would of course be easy enough to use :) But i saw no further explenation of it (and can´t seem to find it in my classical mechanics book.

So assuming i've found r, then i got $$v=sqrt{\frac{GM}{r}}$$ for the planets velocity, and as you said the the extremums of the radial velocity would be the stars velocity. Then you said that $$\frac{M_p}{M_s}=\frac{v_p}{v_s}$$, which sounds good :) But this also seems to be missing from any of my testbooks. And this is something i'd like to use for a project so i'd need to either use references or derive the formula, and i can't seem to find anything resembling this in my textbooks (granted it's just one chapter on planetary motion etc). A reference to some other book would be great as well :) Think i can find most books at the library.

10. Feb 28, 2005

### marcus

$$\frac{r^3}{T^2}=Constant$$

Hi Draco, I turned your fraction upside down.
the constant is related to the mass of the star

It is essentially (up to some order-one factor) just GM.

if the T in your fraction is equal to Period/2pi
then it would be GM itself

that is why one must look at the color of the star, to tell its mass

if the T in your fraction is orbit Period

then there is an extra factor of (2pi)^2 to take care of and one
must say

$$\frac{r^3}{T^2}=\frac {GM}{4\pi^2}$$

BTW kepler law is not perfect, it is best when the masses of the planets are small compared to the star, like in solar system

Last edited: Feb 28, 2005
11. Feb 28, 2005

Staff Emeritus
Or not too close to its primary, causing GR factors to come into play.

12. Mar 2, 2005

### Dracovich

Didn't just want this to drop without recognition. Just wanted to throw a quick thank you in here :) I´ve now got the theory part for my project and seen how you can derive those formulas. So, again, thanks.

13. Mar 6, 2005

### Dracovich

Sorry to bother you even further :) but how would one go about switching this to an elliptical system? Obviously this is possible judging from the data in planet catalogs. I can see how the minor semiaxis can be found from the equation for the period of an elliptical orbit, but i'm having a hard time seeing how other values are found. I'm guessing that the eccentricity is found by the plot, looking at the plots of very eccentric orbits i can't quite see what function they use to fit to their data, but i'm assuming that you get eccentricity from there (since the plot gets quite a unique shape the more eccentric it is). But i'm having a hard time how you would determine anything else.

I'm looking at some equations for elliptical orbits, and they all seem to rely heavily on $$\theta$$ (it's position in the elliptial orbit), but we don't know how the ellipse looks like exactly, or in which direction it is pointing (where we are looking at it). So all the data (velocity etc) must depend on where in it's elliptical orbit the star/planet is, but i can´t see how we could possibly know anything about this.

14. Mar 6, 2005

### Chronos

Use spherical coordinates.