Equations of motion for an orbit

• fasterthanjoao
In summary, the conversation is about trying to integrate equation 3.33 to t_0 without success. The equation involves the substitution of a=\Omega_{0}, b=1-\Omega_{0}, and a=y^{3/2}. The resulting equation is H_{0}\int{}dt=\frac{2}{3}\int \frac{dy}{\sqrt{by^2 +a}}. The final form should be independent of y and should simplify to H_0t_0=\frac{2}{3}. The speaker also mentions using Mathematica to solve the problem, but prefers working through it manually.
fasterthanjoao
Only reason I'm posting here is that i'll get more views than in the cosmology thread, I'm afraid..

(Basically, I'm working through a couple of different models and after some work I'm a bit stuck: http://trond.hjorteland.com/thesis/node21.html

I'm basically trying to integrate equation 3.33 to $$t_0$$ - without much success. Now, cosmologists will know that $$a(t)=(\frac{t}{t_0})^\frac{2}{3}$$ which I feel should be substituted, getting rid of the $$a^\frac{1}{2}da$$? Any comments on this appreciated.

$$a=\Omega_{0} , \ b=1-\Omega_{0}, \ a=y^{3/2}$$

You get the equation

$$H_{0}\int{}dt=\frac{2}{3}\int \frac{dy}{\sqrt{by^2 +a}}$$

Can you take it from here ?

Seems to me like it's at least definitely a log, I think roughly coming out to something like :

$$H_0t=\frac{2}{3}\log(2by+2\sqrt{a+by^2})$$

Which I'm doesn't seem right, and infact I'm almost certain the final form should be independent of y, since in a limit it should simplify to

$$H_0t_0=\frac{2}{3}$$

So it seems I need to at least have a definite integral, ill just say y from 0 to 1 since I've already normalised for y to be 1 at present.

Last edited:
Integrating from 0 to 1 seems to be correct now, from Mathematica I see that it's an inverse hyperbolic sine (through I suppose there's an equivalent form in Log). Could obviously have used Mathematica at the beginning, but I'd like to be able to work through it since it's bugging me now!

If you can shed light on how this is done, that would be great. Thanks.

Think I have it ok now, was just a little rusty on my integration. Thanks for the help dextercioby.

1. What are the equations of motion for an orbit?

The equations of motion for an orbit are a set of mathematical formulas that describe the motion of a body around a central object, such as a planet orbiting a star. These equations include the law of universal gravitation, which states that the force between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them.

2. How are these equations derived?

The equations of motion for an orbit are derived from Newton's laws of motion, particularly the second law which states that the net force on an object is equal to its mass multiplied by its acceleration. By combining this with the law of universal gravitation, we can derive the equations that govern the motion of objects in orbit.

3. What factors affect the equations of motion for an orbit?

The equations of motion for an orbit are affected by several factors, including the masses of the two objects, the distance between them, and the initial velocity of the orbiting body. Other factors such as atmospheric drag and the gravitational pull of other objects can also have an impact on the equations.

4. Can the equations of motion for an orbit be used for any type of orbit?

Yes, the equations of motion for an orbit can be used for any type of orbit, including circular, elliptical, and parabolic orbits. However, the specific equations and solutions may differ depending on the type of orbit and the assumptions made in the calculations.

5. How accurate are the equations of motion for an orbit?

The equations of motion for an orbit are highly accurate when applied to ideal conditions, such as in a vacuum. However, in real-world scenarios, there may be external factors that can affect the accuracy of these equations, such as atmospheric drag, perturbations from other objects, and slight variations in the gravitational pull of the central object.

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