The two-body problem - planets orbiting each other

I am taking a course in classical mechanics these days, which evolves the two-body problem. I need some clarity about the following questions, if possible.

Consider two stars orbiting around each other, one with mass $$M_1$$, the other with mass $$M_2$$.

Is this the case depicted here, where "2 bodies with a difference in mass orbiting around a common barycenter "?

Where is the C.M of the system?

Could this problem be treated so that one body is standing still and the other is orbiting it in an elliptical orbit?

If so, where is the focus of this ellipse?

Yes, the pictures you link are exactly this type of system. I note that the article gives the formula for determining the centre of mass in the section immediately preceeding the one you linked, so that answers your second question.

If you choose a reference frame centred on one of the masses then the frame is no longer inertial and you will have to take account of fictitious forces. In principle you could analyse the motion in this frame, but you would be turning a managable problem into an exceedingly hard one. It's hard to think of any reason why a sane physicist would do this.

In the centre of mass frame one of the foci of each ellipse is at the centre of mass. If you insist on using a non-inertial frame I have no idea whether the orbit is an ellipse and if so where its foci would be.

The motion is also discussed in the Wikipedia article on the gravitational two body problem. Other useful articles are the elliptical orbit and orbit equation.

• Re It's hard to think of any reason why a sane physicist would do this. This is exactly what sane physicists do. The reduced mass reduces the two body problem to a one body problem, in which one of the bodies is frozen at the origin and the other (the reduced mass) is orbiting that frozen body. Now you only have one equation of motion to analyze. This makes the problem much easier to solve. Jan 5 '15 at 16:43
• @DavidHammen: for two objects of similar masses? Jan 5 '15 at 16:58
• Absolutely. That the trajectory of one object with respect to another is a conic section is standard stuff in any classical mechanics text, e.g., Marion, Taylor, Goldstein, you name it. Showing that the trajectories of each of the objects about the barycenter are also conic sections follows. Jan 5 '15 at 17:28
• It's ambiguous. As stated, the question sounds like we are choosing a non-inertial frame that follows on of the physical bodies. John Rennie is correct, that's a nutty way to attack this problem. I suspect the author really meant something like the reduced mass approach David Hammen describes, where we choose the CM (not an actual body) reference frame. Jan 5 '15 at 22:57

Could this problem be treated so that one body is standing still and the other is orbiting it in an elliptical orbit?

Yes. That's the point of using reduced mass.

If so, where is the focus of this ellipse?

It's an ellipse with one of the foci at the body that is standing still.

There are a number of pathways to get to the same result. An easy one is via Newton's laws of motion. Without loss of generality, I'll work in the center of mass frame of the two objects (which presumably is an inertial frame). Denoting the particles as particle 1 and particle 2, the magnitude of the force that each exerts on the other is $$F=\frac{Gm_1m_2}{r^2}$$ where $r$ is the magnitude of the distance between the two particles. Denoting $\hat r$ as the unit vector in the direction of particle 2 from the center of mass, the accelerations of the two particles are \begin{aligned} \frac{d^2}{dt^2} \vec r_2 &= -\frac{Gm_1}{r^2} \hat r \\ \frac{d^2}{dt^2} \vec r_1 &= \phantom{-}\frac{Gm_2}{r^2} \hat r \\ \end{aligned} This is a bit pesky. The right hand sides are functions of $r$ rather than $\vec r_2$ and $\vec r_1$. Taking the difference between the yields $$\frac{d^2}{dt^2} (\vec r_2 - \vec r_1) = -\frac {G(m_1+m_2)}{r^2} \hat r$$ Denoting $\vec r = \vec r_2 - \vec r_1$, this yields $$\frac{d^2}{dt^2} \vec r = -\frac {G(m_1+m_2)}{r^2} \hat r = -\frac {G(m_1+m_2)}{||\vec r||^3} \vec r$$ Now we have a nice simple system. The $r$ and $\hat r$ on the right hand side are functions of the $\vec r$ on the left, specifically, $r = ||\vec r||$ and $\hat r = \vec r/r$. This vector $\vec r$ is the vector pointing from particle 1 to particle 2. In other words, the above describes the motion of particle 2 with respect to particle 1.

Note that this is a second order differential equation of the form $$\frac{d^2 \vec r(t)}{dt^2} = -\frac k {r^2} \hat r = -\frac k {||\vec r||^3} \vec r$$ The only solutions to such ODEs are conic sections: a circle, an ellipse, a parabola, or a hyperbola. Thus the trajectory of one body with respect to another is a conic section.

How about the trajectory of a single body with respect to the center of mass? Using $\vec r = \left(1+\frac {m_2}{m_1}\right)\vec r_2$, the differential equation becomes $$\frac{d^2}{dt^2} \vec r_2 = -\frac {G(m_1+m_2)}{\left(1+\frac {m_2}{m_1}\right)^3 ||\vec r_2||^3} \vec r_2$$ Note that this is still of the form $\ddot {\vec x} = -k\vec x/||\vec x||^3$. The motion of an individual body with respect to the center of mass is also a conic section.

Lagrangian formulation

The concept of reduced mass is motivated by looking at the kinetic energy of the system. Once again working in the center of mass frame, the kinetic energy of the system is $T = \frac12 m_1 \dot {\vec {r_1}}^2 + \frac12 m_2 \dot {\vec {r_2}}^2$. Expressing $\dot {\vec {r_1}}$ and $\dot {\vec {r_2}}$ in terms of $\dot{\vec r}$ yields \begin{aligned} \dot {\vec {r_2}} &= \phantom{-}\frac{m_1}{m_1+m_2} \dot{\vec r} \\ \dot {\vec {r_1}} &= -\frac{m_2}{m_1+m_2} \dot{\vec r} \\ T &= \frac12 \frac {m_1m_2}{m_1+m_2} \dot {\vec r}^2 \end{aligned} Defining the reduced mass $m_r = \frac {m_1m_2}{m_1+m_2}$ yields $T = \frac 12 m_r \dot {\vec r}^2$. This is the kinetic energy of a single particle of mass $m_r$ moving at velocity $\dot {\vec r}$. The Lagrangian for a pair of interacting particles in a central force field is thus $L = \frac 1 2 m_r v^2 - U(r)$ This once again reduces the two body problem to that of a single body. It is in studies of the above with $U(r) = -k/r$ (inverse square law) that the conic section nature of the solutions are uncovered. That takes multiple pages; I am not reproducing it here. You can find these derivations in practically any classical mechanics physics text. The books by Marion and by Taylor are reasonably good undergraduate texts. Goldstein is an introductory graduate level text.