Calculate Bending of Light: Find the Middle Point

[vaibhavtewari]

Hello,

while solving bending of light problem, where I shoot a light ray from one tower to other. How much light has fallen upto the middle point ? I was able to reduce the eq to

$$\phi=\int[1-a(\cos\theta+\frac{1}{1+\cos \theta})]^{-1/2}d\theta$$

a is very small, also if there is no gravity "a" will be zero.

Is there a way I can express this integral as a series ? the first term being $$\phi=\theta+...$$

I tried expanding the square root but the higher order integrals become ugly. I was hoping to find a neat looking series..please help

 

[Dale]

If you substitute $\theta = \delta\theta + \theta$ where $\delta\theta$ is a small quantity then you can do a Taylor series expansion about $\delta\theta=0$

$$\frac{1}{\sqrt{a (-\cos (\theta ))-\frac{a}{\cos (\theta )+1}+1}}+\delta \theta <br /> \left(\frac{a \sin (\theta )}{2 (\cos (\theta )+1)^2 \left(a (-\cos (\theta<br /> ))-\frac{a}{\cos (\theta )+1}+1\right)^{3/2}}-\frac{a \sin (\theta )}{2 \left(a<br /> (-\cos (\theta ))-\frac{a}{\cos (\theta )+1}+1\right)^{3/2}}\right)+O^2$$

Since both $a$ and $\delta\theta$ are small quantities then their product is second order, so this reduces to

$$\frac{1}{\sqrt{a (-\cos (\theta ))-\frac{a}{\cos (\theta )+1}+1}}+O^2$$

 

[vaibhavtewari]

Thankyou for your effort, $$\theta$$ itself is very small as $$\phi$$, the angle towers make wrt to center of Earth is small. My question is more mathematical..is there is a way I can get a series solution to the integral equation. I don't intend to ignore any terms, no matter what order they are.

Thank You

 

[Dale]

is there is a way I can get a series solution to the integral equation. I don't intend to ignore any terms, no matter what order they are.

What do you mean by this? A series where you don't ignore any terms? That doesn't make sense to me.

 

[vaibhavtewari]

I mean something like this,

$$\int \frac{\sin (x)}{x}dx=\int \frac{x+x^3/3+2x^5/15+...}{x}dx=x+x^3/9+2x^5/75+...$$

We can't have any analytical integration but as we see we can ha ve a series solution and if x is small we can just consider first few terms, but I have the compete series solution...series solution are useful in many ways and the problem I am trying to solve might become easier with series solution...

So I am looking for a series solution of the integral...

Thanks for putting effort and helping me out...

 

[vaibhavtewari]

I didnt mean only solution the way I have given in my example...I mean a sum over double series or any weird series looking solution...

 

[Dale]

According to Mathematica the first several terms are:

$$\frac{\theta }{\sqrt{1-\frac{3 a}{2}}}-\frac{a \theta ^3}{16 \left(1-\frac{3<br /> a}{2}\right)^{3/2}}+\frac{a (3 a+16) \theta ^5}{320 \sqrt{4-6 a} (2-3 a)^2}+\frac{a<br /> \left(-477 a^2+816 a-32\right) \theta ^7}{17920 \sqrt{4-6 a} (3<br /> a-2)^3}+O\left(\theta ^9\right)$$

but you will have to figure out the expression for arbitrary terms on your own.

 

[vaibhavtewari]

Thankyou for giving the expansion, it was helpful.