Power Radiated by orbiting bodies:

As two or more bodies orbit each other, some amount of energy is released from the system via gravitational waves. This in turn causes an inspiral or decrease in orbit. So after losing all energy the two objects fall in too each other! This same is happening with the earth and sun system at a rate of 200W (I didnt bother to calculate this as it was done on Wikipedia ;) ) but as per wiki it will take 10^13 times the current age of universe for the earth to fall in and before that ever happens the sun will eat us in red giant phase(or will it...).

So in the formula of radiated power via G-waves we take two masses orbiting around each other $m_1,m_2$. For simplicity we can take standard keplarian orbits in x-y plane or in 2-D plane.

This is our formula:

where $\frac{dE}{dt}$ is the derivative or rate of change of energy with respect to $t$.

$G$ is our gravitational constant and $G=6.673*10^{-11}m^3kg^{-1}s^{-2}$

$c$ is speed of light and $c=300000000m/s$

$r$ is the separation between the two orbiting bodies

(note: The answers will be minus because it is power lost!)

First of all I am going to make a 2-D graph in the x-y plane where y-axis will be $P$ and the x-axis will be $m_1$, we will take $r$ as to be constant, so we can take $r=1.5*10^11meters$. $m_2$ will also be constant and we can take that as earths mass so $m_2=6*10^{24}kg$

In this graph with mass $m_1$ ranging from $2*10^{30}kg$ to $3*10^{30}kg$ we can see that at mass $m_1=2*10^{30}kg$ and the contstant mass $m_2= 6*10^{24}kg$ or earths mass, the power lost by G-waves is 200W!!

If you notice that as $m_1$ increases with all other stuff constant then the power lost is increasing logarithamically as you can see in this graph:

you can see that at ten times the mass of sun how much the energy release by G-waves increase to 600,000W! Still not so much!

If we take two neutron stars with mass of sun at a distance of 2*10^8m away then see the graph as r increases or decreases.

This is a graph of Power radiated with respect to distance between objects. The mass is 1 Solar mass for both objects and they are neutron stars. at $2*10^8m$ is about $10^{30}W!!!$ that means in a few days or months the stars would collide!

The graph might look flat or constant near 10^30W or as r increase but dont be fooled by that and look at the Power axis and see the scale ratio. the graph looks like this over their.

again dont be fooled that the power is gone positive, keep your eyes open and look at the axis,

it shows it in this way for understanding.

And the graph will never go positive unless mass is negative!! which cant happen.

TIME FOR MULTIVARIABLE GRAPH:

Now we are going to see the graph with respect to two variables. First with $m_1$ and $m_2$;

hmmm, not such an interesting graph but it has a lot of info in it.

The vertical axis is the Power, the rest two axis are the masses of the two bodies, As you can see that as $m_1$ which is the top horizontal axis increases along with $m_2$ increasing the power radiated is exponentially increasing and then it suddenly does a strange thing and becomes a fold!

again, not to be fooled it looks like this at that fold:

But again note that these graphs axis ratios are not the same, so the graph looks different a bit.

Now we shall look at a 3D graph of our neutron star case. here the x and z axis will have equal rates of change( or axis ratio ). taking again with respect to $m_1$ and $m_2$:

basically its the same thing :(

Now taking the graph with respect to $m_1$ and $r$:

BTW: no this is not a waterfall model but it can be (Moral: different formulas make different real world models!)

Now that is interesting :)! This time I added some colour to show the heights in it.

As you can see another fold in these graphs too!! In the colour coded graph we can clearly see grey for the fold, so ill call it a well!(The water one)

The graph will be similar for neutron star case.

THIS WAS PART 1

, so $$

. we will take this as simple keplarian orbit in x-y or 2-D space.