Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  On gravitational wave radiation changing orbital parameters

+ 2 like - 0 dislike
586 views

Two celestial objects of mass $m_{1}$ and $m_{2}$ are orbiting around each other on a very elongated ellipse($\epsilon\approx1$). The system is isolated, there is no other proximate celestial body. How would the parameters of the elliptical orbit change as a consequence of radiation of gravitational waves. How can the time of the process be estimated when the orbit changes from an ellipse to a circle?

Here is my work:
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 :
$$\tau=\frac{5}{256}\frac{c^5}{G^3}\frac{a_{0}^4}{\mu M^4}$$

I am having problems in converting this concept to that mentioned in the question, I.e., for a system of binaries of highly elliptical orbit(in my calculations, I have assumed a circle) and how would the elliptical parameters change with time. I also require help in calculating the time of the process when the orbit changes from an ellipse to a circle.

I have obtained the following integral for finding the decay time of the orbits( the orbits would eventually decay into a circular one):
$$T(a_{0},e_{0})=\frac{12(c_{0}^4)}{19\gamma}\int_{0}^{e_0}{\frac{e^{29/19}[1+(121/304)e^2]^{1181/2299}}{(1-e^2)^{3/2}}}de\tag1$$
Where $$\gamma=\frac{64G^3}{5c^5}m_{1}m_{2}(m_{1}+m_{2})$$
For $e_{0}$ close to $1$ the equation becomes:
$$T(a_{0},e_{0})\approx\frac{768}{425}T_{f}a_{0}(1-e_{0}^2)^{7/2}\tag2$$
Where $$T_{f}=\frac{a_{0}^4}{4\gamma}$$
Here $T$ depends on both $a_{0}$ an de $e_{0}$ as the former determines the latter.
And $\gamma$ has the same value as define do above.
I am stuck with $(1)$ and $(2)$; any help would be appreciated.
**My approach:**
I have tried to use Appell hypergeometric integration and obtained the following result:(let $x=e$)
>$$\boxed{\int\frac{x^(\frac{3}{2}+y)[1+Ax^2]^{\frac{1}{2}+z}}{(1-x^{2})^\frac{3}{2}}dx=2x^{y+\frac{5}{2}}\frac{F_{1}(\frac{y}{2}+\frac{5}{4};\frac{3}{2},-z-\frac{y}{2}+\frac{9}{4};x^{2},-Ax^2)}{2y+5}}$$
Where $y=\frac{55}{38},z=\frac{63}{4598},A=\frac{121}{304}$
Here $F_{1}(a,\beta,\beta^{'},c;x,y)$ is appells function and I read that it can be expressed as the following integral:
>$$\boxed{F_{1}(a,\beta,\beta^{'},c;x,y)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}u^{a-1}(1-u)^{b-a-1}(1-ux)^{-\beta}(1-uy)^{-\beta^{'}}du}$$
Is this right? How am I to solve this integral? And also how am I to solve for $e_{0}$ in equation (2)?

asked Oct 29, 2016 in Astronomy by Naveen (85 points) [ revision history ]
edited Oct 31, 2016 by Naveen

I have to say it is not clear to me at all how does equation (1) come about. For instance, do you take relativistic precession into account, or do you assume Newtonian trajectories at every point of the decay? (I.e., what is your approximation, Newtonian+linearized gravity waves, post-Newtonian...) The loss of eccentricity can be essentially seen from the slightly different behavior of energy and angular momentum - how would that look in your case?

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysicsOverflo$\varnothing$
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...