Solar System Center of Mass

Right now scientists are searching for new planets. They have telescopes pointed at distant stars carefully measuring how fast they move and looking for tiny deviations—or wobble—in their speed. This wobble tells us that something must be orbiting the star. When two objects are in orbit, they both orbit their respective center or mass, called the barycenter.

In the case of solar systems the sun is so much more massive that the rest of the planets that the barycenter is very nearly at the center of the sun. This is also helped by the fact that on any given day the planets are likely scattered randomly about and to some degree cancel each other out. Think of an evenly matched tug-of-war. But what if all the planets were on one side of the sun? Then how far off center would the barycenter be? I guessed it would be still very near the middle of the sun, but it turns out it's about 800,000 kilometers outside the sun!

Don't believe me? I didn't believe it when I first read it so I had to calculate it myself. How does one go about calculating the center of mass of the solar system?

First lets set up the situation. Like all great physics problems, this is a vast simplification. We can safely make some favorable assumptions. Put the sun at the center of a coordinate system, and everything is on the x-axis, and only count the 8 most massive objects other than the sun.

Center of mass diagram

To find the center of mass \(\boldsymbol{R}\) we just add up the masses \(m_i\) times the distances \(\boldsymbol{r_i}\) for everything and divide by the total mass.

\[\boldsymbol{R} = \frac{\displaystyle\sum\nolimits_i m_i \boldsymbol{r_i}}{\displaystyle\sum\nolimits_i m_i}\]

Lets actually look at some numbers:

Body Distance from Sun[1] [cm] Mass[2] [g] % Mass of the Solar System
Sun 0 1.99×1033 99.866
Mercury 5.78×1012 3.3×1026 0.000017
Venus 1.08×1013 4.87×1027 0.00024
Earth 1.50×1013 5.98×1027 0.00030
Mars 2.28×1013 6.5×1026 0.000033
Jupiter 7.79×1013 1.90×1030 0.095
Saturn 1.43×1014 5.70×1029 0.029
Uranus 2.88×1014 8.7×1028 0.0043
Neptune 4.50×1014 1.00×1029 0.0050

Again you can really see just how massive the sun is. It accounts for about 99.866% of the mass in the solar system! If we use the equation above and plug in all the distances and masses and add it up we get:

\[\boldsymbol{R} = \displaystyle\frac{2.9969\times 10^{44} \, [\mathrm{cm} \cdot \mathrm{g}]}{1.9917\times 10^{33} \, [\mathrm{g}]}\]

\[\boldsymbol{R} = 1.51\times 10^{11} \, [\mathrm{cm}]\]

Or in other words, the center of mass of the solar system, with all the planets to one side, is about 1.5 million kilometers from the center of the sun. The sun itself is about 0.7 million kilometers across, so that puts the center of mass about 800,000 kilometers off the surface of the sun.

Just something to think about. But don't forget, given the scale of the solar system, 800,000 kilometers isn't very far. In fact it's about 70 times closer to the sun than Mercury, the closest planet.

References

  1. Universe today — How Far are the Planets from the Sun? http://www.univer...
  2. Ask a Scientist — Planets and Their Mass http://www.newton.dep.anl.gov/...
This entry was posted in Science and tagged , . Bookmark the permalink. Post a comment or leave a trackback: Trackback URL.

8 Comments

  1. mother
    Posted August 25, 2009 at 4:49 pm | Permalink

    fix paragraph 4 "safely make"

  2. mother
    Posted August 25, 2009 at 4:49 pm | Permalink

    why, again, is it important to know the center of mass?

  3. Posted August 25, 2009 at 4:55 pm | Permalink

    It's not important, it's interesting. I just thought it would be a lot closer to the sun than it ended up. There is sometimes a difference between what we feel intuitively to be true, and what actually is.

  4. Adam
    Posted August 31, 2009 at 12:31 pm | Permalink

    Isn't the CM also supposed to be ABOVE the plane of axis?

  5. Posted August 31, 2009 at 12:38 pm | Permalink

    Probably. I was being super simplistic.

  6. mervin
    Posted July 21, 2010 at 1:56 pm | Permalink

    it may be the cm is supposed to be above the axis because plantes are moving and when they are moving thier weigth also change

  7. Manny Cruz
    Posted January 24, 2011 at 10:15 pm | Permalink

    In reality the center of mass of the Solar System is very close to center of the Sun because planets spend most of their time on opposite sides during their orbits, canceling each other out.

  8. harsha vardhan
    Posted May 30, 2012 at 7:04 pm | Permalink

    one doubt from years that the center of mass of our solar system is changing every day even every sec right but if we consider our solar system as one system all the forces come under internal forces, but the center of mass changes due to external forces not due to internal forces , so is there any external force??? If not the our solar system as a whole must move to nullify the change in center of mass???

Post a Comment

Your email is never published nor shared. Required fields are marked *

*
*

You may use these HTML tags and attributes <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <strike> <strong>