Confirming Einstein’s Theory of General Relativity With Calculus, Part 7b: Predicting Precession II

In this series, I’m discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury’s orbit and thus confirm Einstein’s theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago.

We have shown that the motion of a planet around the Sun, expressed in polar coordinates $(r,\theta)$ with the Sun at the origin, under general relativity is

$u(\theta) \approx \displaystyle \frac{1 + \epsilon \cos \theta}{\alpha} + \frac{ \delta\epsilon}{\alpha^2} \theta \sin \theta$ ,

where $u = \displaystyle \frac{1}{r}$ , $\displaystyle \frac{1}{\alpha} = \frac{GMm^2}{\ell^2}$ , $\delta = \displaystyle \frac{3GM}{c^2}$ , $G$ is the gravitational constant of the universe, $m$ is the mass of the planet, $M$ is the mass of the Sun, $\ell$ is the constant angular momentum of the planet, and $c$ is the speed of light.

We will now simplify this expression, using the facts that $\delta$ is very small and $\alpha$ is quite large, so that $\delta/\alpha$ is very small indeed. We will use the two approximations

$\cos x \approx 1 \qquad \hbox{and} \qquad \sin x \approx x \qquad \hbox{if} \qquad x \approx 0$ ;

these approximations can be obtained by linearization or else using the first term of the Taylor series expansions of $\cos x$ and $\sin x$ about $x = 0$ .

We will also need the trig identity

$\cos(\theta_1 - \theta_2) = \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2$ .

With these tools, we can now simplify $u(\theta)$ :

$u(\theta) \approx \displaystyle \frac{1 + \epsilon \cos \theta}{\alpha} + \frac{ \delta\epsilon}{\alpha^2} \theta \sin \theta$

$= \displaystyle \frac{1}{\alpha} \left[1 + \epsilon \cos \theta + \frac{ \delta\epsilon}{\alpha} \theta \sin \theta \right]$

$= \displaystyle \frac{1}{\alpha} \left[1 + \epsilon \left(\cos \theta + \frac{ \delta}{\alpha} \theta \sin \theta \right) \right]$

$= \displaystyle \frac{1}{\alpha} \left[1 + \epsilon \left(\cos \theta \cdot 1 + \sin \theta \cdot \frac{ \delta \theta}{\alpha} \right) \right]$

$\approx \displaystyle \frac{1}{\alpha} \left[1 + \epsilon \left(\cos \theta \cdot \cos \frac{\delta \theta}{\alpha} + \sin \theta \cdot \sin \frac{ \delta \theta}{\alpha} \right) \right]$

$\approx \displaystyle \frac{1}{\alpha} \left[1 + \epsilon \cos \left( \theta - \frac{\delta \theta}{\alpha} \right) \right]$ .

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Confirming Einstein’s Theory of General Relativity With Calculus, Part 7b: Predicting Precession II

Published by John Quintanilla

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Published by John Quintanilla

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