Orbits of planets and stars decay due to gravitational wave radiation. An elliptical orbit would become more circular with time. This is especially observed well in binary systems.

Taking the example of binaries, we can compute the rate of change of eccentricity of the orbit and the rate of change of the semi-major axis as a function of time. If we consider a highly elliptic orbit of a binary, with eccentricity, $\epsilon\rightarrow{1}$, we can prove that with time the orbit would eventually become a circular one, as follows:

The power emitted by gravitational waves is given by: $$P_{GW}= \frac{c^5}{G}\left(\frac{GM}{c^5l}\right)^5$$ Very compact binaries will lose energy rapidly by GW radiation. If we assume that the two bodies making up the binary lie in the $x-y$ plane and their orbits are circular ($\epsilon=0$), then only non-vanishing components of quadrupole tensors are: $$\boxed{Q_{xx}=-Q_{yy}=\frac{1}{2}(\mu)a^{2}\cos2\Omega t}$$ and $$\boxed{Q_{xy}=Q_{ya}=\frac{1}{2}(\mu)a^{2}\sin2\Omega t}$$ Where $\Omega$ is the orbital velocity, $\mu=\dfrac{m_{1}m_{2}}{m}$ is the reduced mass and where $m=m_{1}+m_{2}$ The luminosity of the system can be deduced as: $$L^{GM}=\frac{32}{5}\frac{G}{c^5}\mu^2 a^4\Omega^6=\frac{32}{5}\frac{G^4}{c^5}\frac{M^3 \mu^2}{a^5}$$ The latter part is obtained from Kepler third law: $\Omega^2=\frac{GM}{a^3}$ As the gravitating system loses energy by emitting radiation, the distance between the 2 bodies shrinks at a rate: $$\boxed{\frac{da}{dt}=\frac{64}{5}\frac{G^{3}M\mu}{c^5 a^3}}$$ The binary would hence colase at a time : $$\tau=\frac{5}{256}\frac{c^5}{G^3}\frac{a_{0}^4}{\mu M^4}$$

By applying a similar treatment to elliptical orbits , the following can be computed: $$\left<\frac{da}{dt}\right>=-\frac{64}{5}\frac{G^{3}m_{1}m_{2}(m_{1}+m_{2})}{c^{5}a^{4}(1-e^{2})^{\frac{7}{2}}}\left(1+\frac{73e^2}{24}+\frac{37e^4}{96}\right)$$


Solving this system of ODE's ultimately results in the equation:

$$T(a_{0},e_{0})=\frac{12(c_{0}^4)}{19\gamma}\int_{0}^{e_0}{\frac{e^{29/19}[1+(121/304)e^2]^{1181/2299}}{(1-e^2)^{3/2}}}de\tag1$$ Where $$\gamma=\frac{64G^3}{5c^5}m_{1}m_{2}(m_{1}+m_{2})$$ Upon solving this one can find the time taken for the orbit to decay into a circle from an ellipse due to gravitational wave radiation.

A similar treatment can be done for the orbital decay of the planets of our solar system. Now my question is if we apply time reversal symmetry, the orbits become more and more elliptical and eventually tend to become a straight line(an elliptical orbit with $\epsilon\rightarrow{1}$) with a chunk of mass(moving in the straight line) from which all the planets and the Sun of our solar system arose. Now the Milky Way "orbits" the Local Group, which in turn "orbits" the Virgo Supercluster. Beyond that, the expansion of the universe starts to dominate over gravitation.

So reversing time to a space after the Big Bang, there must have been many galaxies and solar systems that were forming; so according to the above result the mass chunks from which galaxies, stars and planets arose must have had a highly elliptic orbits. Thus, the spread of matter in space must have been along approximate lines(which were orbits for the massive mass chunk a that formed them). This moves towards a linear arrangement of galaxies, contrary to what is observed. Can anyone please resolve my doubt. Any help is appreciated.

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    $\begingroup$ I'm having difficulty understanding this question. I don't follow the reasoning that leads to "orbits become more and more elliptical and eventually become a straight line". The release of gravitational radiation is not a significant factor in the formation and evolution of the solar system. And I don't understand how you conclude that "the spread of matter happened along a line". This seems to be a complete non-sequitor $\endgroup$
    – James K
    Nov 24, 2016 at 18:47
  • $\begingroup$ @JamesK I have added some results to explain what I meant by those statements, and also changed the question a bit as what I wanted to ask was not conveyed properly in the previous edit. $\endgroup$ Nov 24, 2016 at 19:31
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    $\begingroup$ It's not at all true that objects in the Solar System originally had highly elliptical orbits. That contradicts everything we know about planet formation, both observationally and theoretically. $\endgroup$
    – HDE 226868
    Dec 6, 2016 at 18:23
  • $\begingroup$ @HDE226868 But aren't they elliptical at some point of the orbital lifetime? $\endgroup$ Dec 6, 2016 at 18:32
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    $\begingroup$ @NaveenBalaji Well, yes, but nowhere near orbits where $e\approx1$. $\endgroup$
    – HDE 226868
    Dec 6, 2016 at 18:36

1 Answer 1


When you go back in time, take collisions into account. A very highly elliptical orbit primarily leads to chunks colliding at the very first periapsis.

Even if they are not chunks but very dense and small objects (point-like), if you extrapolate back in time so far as to the straight line "orbit", what does it mean? Extreme case of objects with zero relative initial velocity going against each other on a collision course - ups, this "orbit" does not have a chance to "decay due to gravitational wave radiation". Two smaller chunks collide and form a larger chunk, which has now a non-zero velocity (probably, if the chunks weren't exactly equal momentum). The chunk won't enter a straight line "orbit" around (towards) anything, but either an elliptical or hyperbolic.

Your extrapolation does suggest that initial "chunks" hadn't ideally equal momenta (masses*velocities); that's all.


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