Two suns at two foci of an ellipse?

[asjaradja]

I am doing my extended essay in the IB program in mathematics and my question is "How did Newton prove Kepler’s conjecture that planets move in elliptical orbits, can there exist a solar system with two or three suns?" If our sun is at one focus of an ellipse then can there be another sun at the other focus? Would they be attracted to each other? What if there was a third sun? Can anyone give me any input? It would be greatly appreciated!

 

[jgreimer]

I thought Kepler derived his laws from Tycho Brahe's data thus the data itself proved his laws.<br /><br />http://en.wikipedia.org/wiki/Kepler's_laws<br /><br />Generally n-body systems are unstable and tend to eject their planets but there are some special cases that are stable. One is the system you described with two stars each at one focus of an ellipse. This system would only be stable if the two stars are orbiting around their common center of gravity. The more stars in a system the less likely it is to be stable because there are more ways for the stars to interact.

 

[PistolPete]

<blockquote><font class="small">In reply to:</font><hr /><p>How did Newton prove Kepler’s conjecture that planets move in elliptical orbits<p><hr /></p></p></blockquote><br />As stated before, Kepler proved his laws empirically using Tycho Brahe's data. By the time of Newton, Kepler's laws were pretty well accepted, they just didn't know why planets orbited elliptically. What Newton did was to prove that gravity was the force that caused planets to revolve around the Sun by using his measurements of gravity to mathematically predict the orbits of the planets. His predictions mirrored Kepler's laws proving that gravity not only affected terrestrial objects, but celestial ones as well.<br /><blockquote><font class="small">In reply to:</font><hr /><p>can there exist a solar system with two or three suns?<p><hr /></p></p></blockquote><br />Of course. Look up into the night sky. Over half the stars you see are actually two or more stars orbiting one another. Statistically, one star solar systems are the minority.<br /><blockquote><font class="small">In reply to:</font><hr /><p>If our sun is at one focus of an ellipse then can there be another sun at the other focus?<p><hr /></p></p></blockquote><br />If the two stars were of equal mass, then I think what would happen is that each star would orbit in an ellipse around the other, creating two ellipses with one focus from each ellipse overlapping. It's kind of a confusing explanation so I will include an illustration: <i>a</i> is the semimajor axis, <i>b</i> is the semiminor axis, <i>c</i> is the distance between the center of the ellipse and either of the foci, and <i>F</i> stands for either foci (they are numbered). As you can see the foci overlap. How the stars would revolve around one another is as one star moves around its ellipse, the other star maintains an exact opposite mirror image. For example, star A is on the left most end of the left ellipse and star B is on the right most end of the right ellipse. As star A moves to the bottom of the le <div class="Discussion_UserSignature"> <p> </p><p><em>So, again we are defeated. This victory belongs to the farmers, not us.</em></p><p><strong>-Kambei Shimada from the movie Seven Samurai</strong></p> </div>

 

[yevaud]

Pretty much (if I catch your description correctly), that's one of the long-term, quasi-stable orbits for two bodies; they literally will move in a figure-8. <br /><br />Sorry, memory's faulty. That's a three-body quasi-stable orbit.<br /><br />In expiation, here is a really interesting site I found back when on some very unusual orbits. <div class="Discussion_UserSignature"> <p><em>Differential Diagnosis:  </em>"<strong><em>I am both amused and annoyed that you think I should be less stubborn than you are</em></strong>."<br /> </p> </div>

 

[PistolPete]

Wow, that's an interesting web site. I'm definitely bookmarking that one.<br /><br />Yes, it's a lot like a figure 8. I was also assuming that the two stars were of the same mass. If they are different, then it begins to get complicated. Also, most orbits are not as eccentric as shown in the illustration (0.5). If you have a system like Alpha Centauri where you have two stars of similar mass orbiting each other in a low eccentric orbit, then they orbit each other in almost a perfect circle, with the stars opposite of one another, their distance varying slightly. <div class="Discussion_UserSignature"> <p> </p><p><em>So, again we are defeated. This victory belongs to the farmers, not us.</em></p><p><strong>-Kambei Shimada from the movie Seven Samurai</strong></p> </div>

 

[vogon13]

The center of mass of the two stars will be at one focus of an elliptical orbit of a body travelling about both of them.<br /><br />The other focus will be empty.<br /><br /><br /> <div class="Discussion_UserSignature"> <p><font color="#ff0000"><strong>TPTB went to Dallas and all I got was Plucked !!</strong></font></p><p><font color="#339966"><strong>So many people, so few recipes !!</strong></font></p><p><font color="#0000ff"><strong>Let's clean up this stinkhole !!</strong></font> </p> </div>

 

[yevaud]

If you've seen the estimated tracks for the orbits of such stars (and their associated planetary bodies), they can be...odd, and certainly complicated. <div class="Discussion_UserSignature"> <p><em>Differential Diagnosis:  </em>"<strong><em>I am both amused and annoyed that you think I should be less stubborn than you are</em></strong>."<br /> </p> </div>

 

[PistolPete]

I want to revise what I said about the orbits of Alpha Centauri system. The orbit of the two stars are actually very eccentric, roughly 0.519, and there is about a 20% difference in the mass between Alpha Centauri A and B. At their farthest, A and B are about 36 AU and 11.4 AU at its closest. If the semimajor axis of one of the elipses in my origional illustration is about 12 AU, then the illustration is serendipitously close to the actual orbits of Alpha Centauri A and B. <div class="Discussion_UserSignature"> <p> </p><p><em>So, again we are defeated. This victory belongs to the farmers, not us.</em></p><p><strong>-Kambei Shimada from the movie Seven Samurai</strong></p> </div>

 

[brellis]

Welcome to SDC - mind-bending question for sure.<br /><br /><font color="yellow">can there exist a solar system with two or three suns?</font><br /><br />Sorry, the Planetary Society site is down right now, but Emily Lakdawalla wrote a beautiful piece called "Double Sunsets and Sunlit Nights". There are some charts of planetary and protoplanetary disks surrounding a binary system.<br /><br />I'll leave Newton's laws to the pro's <img src="/images/icons/cool.gif" /> but according to recent observations it is not only possible but more likely than not that planets can form in a binary system. <div class="Discussion_UserSignature"> <p><font size="2" color="#ff0000"><em><strong>I'm a recovering optimist - things could be better.</strong></em></font> </p> </div>