Help with Modeling Solar Wobble for Pre-Calc Project

[robotoop]

I'm a junior in high school and was recently assigned an open ended project in my pre-calculus class to research how precalc (or the basic calculus we have started) can relate to the real world. I have a strong interest in astronomy and physics and therefore desire to look at modeling the wobble of stars due to the gravitational effects of an orbiting body. I was thinking this should be a sinusoidal function (showing the velocity of the star) and perhaps a rotated conic graph to show the path of the planet. Unfortunately, other than understanding the concepts (at least I think I do) I don't have much of a clue where to begin. I have been able to locate some graphs online which look useful but I was hoping for more help.

I still have a good amount of time left on this project so if I have taken on a challenge well beyond my capabilities it wouldn't be the end of the world. However, this is something which greatly interests me and hope I can get some help so I can learn more about it.

 

[harcel]

Hi robotoop,


What you said seems totally right to me (an astrophysics PhD). If there is an object around a star (or the Sun) and it is pulling it due to their mutual gravity and we re watching it from a distance (this is an important point, for the Sun it is more complicated because we are in the solar system as well! For exo-planets life is easier).

Let's assume that we are talking about a system far away, and that we are talking about a star+ planet, in which the planet is invisible to us and that its orbital plane is perpendicular to our line of sight and that it moves around in circle (so from here, the complete orbit looks like a line, with the star in the middle). That star now also moves in a circle, in fact they both move in circles around their mutual center of mass.

One step in the right direction would be to realize that we will always see only the component of the velocity (assuming that we cannot see its position changing, which is true for almost all stars with planets, or binary star companions) that is along the line of sight, not perpendicular to it. The star itselfs goes around in a little circle (assuming a circular orbit for the companion). Of the circular motion we can only see the component in our line of sight which sometimes is the whole velocity component (in the part of the orbit where it moves directly towards or away from us), and sometimes this is 0 (in the part of the orbit where the star moves 'along the plane of the sky').

It will greatly help you to make a drawing that looks at the system earth, exo-planet, star from above. Try it and let me know if you need help!

Cheers, Marcel

 

[robotoop]

Harcel,

Thanks for the clarification. As for my project, I think the optimal situation would involve me using a known period and radial velocity (<- correct term?) to make a graph to compare with one created from actual astronomical measurements. Is there anywhere you know of where I can get data for this? I'm open to other suggestions on how to work some math into demonstrating these "wobbles" if anyone has some.

 

[Drakkith]

Why not do a model of how the Earth affects the sun? Just ignore the moon and other planets and use the mass of the Earth and the sun along with the distance and speed of the Earth's orbit.

Or you could take an object with the mass of Jupiter and put it at the distance Earth is and calculate that. That would still be real world uses since its entirely possible we could find a Jupiter size planet around a star about the mass of the sun.

 

[tony873004]

There's a group that studies radial velocities to try to pull planets out of the data. You can read about them here. They have a downloadable program you can use on a Windows or Mac, where you can download some radial velocity data and try to find planets yourself.
http://oklo.org/

You can also use my program, Gravity Simulator (Windows only), to create your own solar systems, then output the radial velocity of the star as viewed from a great distance, then import this data into the above-mentioned program and see if you can find the planets you created.

Or you can simply use data created by Gravity Simulator to make your own graphs. Here's how you would do it:

Open anyone of the many existing simulations that contain real solar system data (or make your own solar system).

Create an additional object 10 light years away from the Sun on the x-axis (so the sun and the planets can't significantly affect your object)

File > Output data. Choose to make a text file of the radial velocity of the 10 ly object (choose Vx). This will show you the velocity of the sun moving towards and away from you.
Import this data into Excel, or use it any other way you wish, to make a graph of radial velocity.

They you can experiment with hypothetical systems: hot jupiters, elliptical orbits, multiple planets, ect.

This method uses numerical integration, which is a topic you might cover in calculus when you do differential equations.

Here's a page I wrote about the effects the planets have on the Sun's position, as it wobbles around the barycenter of the solar system.
http://www.orbitsimulator.com/gravity/articles/ssbarycenter.html

 

[robotoop]

Thank you everyone for answering, just one more question: how is planetary mass and orbital distance determined if the only knowns are orbital period and stellar wobble and mass?

 

[tony873004]

There are several methods for discovering exosolar planets. Since you're interested in a star's wobble, the following info is for the radial velocity method, where you measure a star's radial velocity based on its red/blue shifts.

The wobble will correspond to the the planet's period, P. Orbital distance (semi-major axis, a), can be computed with a=cuberoot(P^2*G*M/(4*pi^2))

For planetary mass, you can only determine a range of possible masses. So what they do is compute the planet's mass with the assumption that it orbits its star edge-on to Earth (similar to the way we view the moons of Jupiter orbiting Jupiter from Earth). Here's a page that gives you all the math: http://www.physics.fsu.edu/users/ProsperH/AST3033/ExtraSolar.htm

 

[robotoop]

Ok thanks, that's exactly what I was looking for. Are there required (or just standard) units for these equations to work?
Also, is that "M" representing planetary or stellar mass? If it is stellar than I should be good (correctly calculated the orbital distance of jupiter).

 

[tony873004]

In the formula P^2*G*M/(4*pi^2)), the units for M depend on the value you use for G. The most common way to express G is 6.673 * 10-11 m^2/(kg s^2). So if this is the value you are using for G, then you must express mass in kilograms, your distances in meters, and your units of time in seconds. It's common to express mass in terms of Jupiter masses. But G must cancel out of your equations before doing so.