Two-body problem in orbiting frame

[Tetrapoil]

TL;DR Summary

What is the acceleration acting on object 2 in the orbiting frame centered on object 1?

Hi,

I am interested in a system where two point mass objects 1 & 2 are orbiting each other. I want to compute the trajectory of object 2 in the orbiting but non-rotating frame centered on object 1. For some reason this confuses me...

Let r₂ be the position of object 2 in such orbiting frame centered on object 1. The acceleration of object 2, in the orbiting frame, is

a₂ = - (GM₁/|r₂|³) r2 - (GM₂/|r₂|³ )r2 .

The first term being the acceleration of object 2 from object 1, and the second term being the acceleration of object 1 relative to the inertial
frame. Of course, something is wrong as in this case object 2 will necessarily fall on object 1. Am I missing something?
Thanks!

 

[PeroK]

Why do you think the objects must collide? The gravitational force is always in the direction between the two masses, so the acceleration of each object is always in this direction. e.g. the Earth is always accelerating towards the Sun.

 

[Tetrapoil]

Thanks for your reply. I agree, and this is what confuses me...
Let's say that I want to compute the trajectory of object 2 in such frame, I would have something roughly like that:

r2(t+1) = r2(t) + vr2*dt
where
vr2(t+1)= vr2(t) + ar2(t)*dt .

As vr(t=0) = 0 and ar2 < 0, r2 decreases, whatever the initial orbital velocity of object 2... Right?

 

[PeroK]

Thanks for your reply. I agree, and this is what confuses me...
Let's say that I want to compute the trajectory of object 2 in such frame, I would have something roughly like that:

r2(t+1) = r2(t) + vr2*dt
where
vr2(t+1)= vr2(t) + ar2(t)*dt .

As vr(t=0) = 0 and ar2 < 0, r2 decreases, whatever the initial orbital velocity of object 2... Right?

In principle what you have is not that different from uniform circular motion. It's clearly not necessary that the radius decreases.

You may be making a common mistake of taking $dt$ to be some finite interval over which the acceleration vector is constant. That does make a mess of circular motion!

Instead, circular motion requires calculus to be done properly, with $dt$ as a differential and not a finite time interval.

 

[vanhees71]

It's of course not a good idea to try to describe the motion of one object in the rest frame of the other, which is a non-inertial frame to begin with. The most simple frame is the center-of-mass frame. The equation of motion for the relative vector $\vec{r}=\vec{r}_1-\vec{r}_2$ is
$$\mu \ddot{\vec{r}}=-\frac{G m_1 m_2}{r^3} \vec{r}$$
and not the strange formula you wrote in#1, where $\mu=m_1 m_2/(m_1+m_2)$ denotes the "reduced mass".

The solution of this "Kepler problem" is that either (a) the trajectory of $r$ is a conic section with one of its foci in the origin (i.e., the center of mass of the two-body system) or (b) straight lines.

 

[Tetrapoil]

In principle what you have is not that different from uniform circular motion. It's clearly not necessary that the radius decreases.

You may be making a common mistake of taking $dt$ to be some finite interval over which the acceleration vector is constant. That does make a mess of circular motion!

Instead, circular motion requires calculus to be done properly, with $dt$ as a differential and not a finite time interval.

I see what you mean. I am in fact using a leapfrog integrator to numerically compute the position of object 2, with dt << 1, so in theory it should work (but it doesn't..).

 

[Tetrapoil]

It's of course not a good idea to try to describe the motion of one object in the rest frame of the other, which is a non-inertial frame to begin with. The most simple frame is the center-of-mass frame. The equation of motion for the relative vector $\vec{r}=\vec{r}_1-\vec{r}_2$ is
$$\mu \ddot{\vec{r}}=-\frac{G m_1 m_2}{r^3} \vec{r}$$
and not the strange formula you wrote in#1, where $\mu=m_1 m_2/(m_1+m_2)$ denotes the "reduced mass".

The solution of this "Kepler problem" is that either (a) the trajectory of $r$ is a conic section with one of its foci in the origin (i.e., the center of mass of the two-body system) or (b) straight lines.

Thanks. Unfortunately, I have to consider the problem in the non-inertial frame orbiting with M1, for technical numerical reasons. I believe that your equation is the same as the one I wrote in #1. My problem may thus be mostly related to my numerical resolution of the equation...

 

[PeroK]

The acceleration vector is continuously changing, so you would need to take its average over a small time interval $dt$, not simply its value at the start of the interval.

That is probably your error and what is causing the iteration to go wrong.

 

[Tetrapoil]

The acceleration vector is continuously changing, so you would need to take its average over a small time interval $dt$, not simply its value at the start of the interval.

That is probably your error and what is causing the interaction to go wrong.

That makes sense, indeed. Thanks a lot!

 

[vanhees71]

Again: I'd not choose the rest frame of one particle as the computational frame. It's much more simple to work in the center-of-mass frame, which is an inertial frame. That holds also true for numerical solutions of the equations of motion.

 

[Tetrapoil]

Again: I'd not choose the rest frame of one particle as the computational frame. It's much more simple to work in the center-of-mass frame, which is an inertial frame. That holds also true for numerical solutions of the equations of motion.

The thing is that I am actually interested in the gas dynamics, assuming that these two objects orbit in some stratified medium. So I also account for the effect of the mass distribution on the trajectories of both particles (i.e. of particle 2 and of the non-inertial orbiting frame) as well as the fluid's self-gravity.
For numerical reasons, the (spherical) computational domain must be bounded by fixed (in the considered frame) and non-zero minimum and maximum radius. If I were to work in the center of mass frame, the latter would be inside an empty sphere of radius rmin, on which I would have to impose arbitrary unrealistic boundary conditions, impacting the fluid flow. In the rest frame of a particle, on the other hand, one can imagine that such a sphere of radius rmin is the actual physical object (e.g. star or planet), and somewhat realistic boundary conditions can be imposed.

 

[Tetrapoil]

The thing is that I am actually interested in the gas dynamics, assuming that these two objects orbit in some stratified medium. The code I use requires a domain that is bounded by fixed minimum and maximum radius. If I were to work in the center of mass frame, the latter would be inside an empty sphere of radius rmin, on which I would have to impose arbitrary unrealistic boundary conditions. In the rest frame of a particle, on the other hand, one can imagine that such a sphere of radius rmin is the actual physical object (e.g. star or planet), and somewhat realistic boundary conditions can be imposed.

The thing is that I am actually interested in the gas dynamics, assuming that these two objects orbit in some stratified medium. So I also account for the effect of the mass distribution on the trajectories of both particles (i.e. of particle 2 and of the non-inertial orbiting frame) as well as the fluid's self-gravity.
For numerical reasons, the (spherical) computational domain must be bounded by fixed (in the considered frame) and non-zero minimum and maximum radius. If I were to work in the center of mass frame, the latter would be inside an empty sphere of radius rmin, on which I would have to impose arbitrary unrealistic boundary conditions, impacting the fluid flow. In the rest frame of a particle, on the other hand, one can imagine that such a sphere of radius rmin is the actual physical object (e.g. star or planet), and somewhat realistic boundary conditions can be imposed.

On a side note, I have rewritten my leapfrog "kick-drift-kick" integrator, and everything seem to be working as intended.