Homework Help: How is gravitational acceleration affected with distance?

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1. Mar 18, 2017

Unteroffizier

Note: I didn't really know where to put this. It isn't a specific problem, but I've been asked by my physics teacher, who decided to give me and a few others an individual physics course of sorts, to find the means of solving similar problems. It's the first problem he assigned us, since we're focusing on astronomy. I'm not studying in an English-speaking country, so my terminology might be a bit off.

1)
So, here's the problem, exactly as it was given, during my actual independent astronomy course (away from our engineering school):
If the Earth is orbiting at a distance of precisely 1AU from the Sun, how long will it take it to fall into the Sun along a straight path (if the laws of inertia do not apply)?

2) There are two solutions to this problem, one of which I cannot use. Our physics teacher will also attempt to guide us through integral calculus early so that we can simplify these problems.

Equation for solution a) T1^2/T2^2 = a1^3/a2^3 (Kepler's third law)

3) The steps of solution a)

1) Create an imaginary object, for which a = 0.5AU applies. e (eccentricity) = approaching 1. In other words, aphelium 1 goes through Earth's center, while aphelium 2 goes through the center of the Sun. The orbital path is eccentric enough to be considered a straight line.
2) Find this object's orbital period by means of 1/x^2 = 1/0.5^3
3) X/2 = approximately 0.165y
It would take Earth approximately 0.165 years to fall into the Sun under given circumstances.

Now, the problem with this result is the fact that it's not perfectly accurate, and it requires some pretty strange twists to achieve.

Steps of solution b)

1) Take gravitational acceleration at 1AU.
2) Take gravitational acceleration at r = approaching 1AU.
3) etc.. Infinite calculations. Can't solve the area under a curved line without severe simplification, which would in turn ruin result accuracy. Unsolvable within reasonable timespan.

We were told that it can be solved with the use of integral calculus by integrating. Problem is, we can't integrate. He's teaching us to do so, but asked us to find the formula for the solution of this problem with the use of integral calculus.

Since I don't know this, and he gave us no restrictions in regards to the sources of our information, I ask if anyone could help me out. He wants us to present a formula, so that he can teach us how it works later. Not sure why he doesn't give it to us himself, but it is what it is, and he probably has his reasons.

2. Mar 18, 2017

PeroK

3. Mar 18, 2017

Unteroffizier

Danke!

I appreciate your response. Looked over the equation, got lost pretty quick. Looks like it's way above my mathematical level, to be honest. However, I see the word "differential equation", so this is differential calculus? I'm supposed to find the method involving integration, because the main goal is to teach us to integrate earlier than the other, less astronomy-oriented students.

Or am I just looking at it wrong?

4. Mar 18, 2017

PeroK

Some alternative approaches are here

Solving differential equation is, essentially, a form of integration.

5. Mar 18, 2017

Unteroffizier

Ah, I see. Well then, I'll give it a closer look.

Thanks for the links! I'll put them to good use.