How long it takes the Earth to fall halfway to the Sun--ellipse method

In summary: As you will no doubt have realized, it is much easier to determine the time to fall to the sun rather than the time to fall halfway to the sun. But you can ball-park it by taking an ellipse with an eccentricity close to 1, which represents the orbit of the Earth with very little angular momentum relative to the centre of the sun. The area swept out by the Earth in the time it takes to reach the half way point is most of the area of one half of the ellipse. The remaining area of that half ellipse is a tiny fraction of the area swept out in the first half of the fall. Since the orbiting body sweeps out equal areas in equal times, you can see that
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How long does it take for the Earth to fall half the distance to the Sun, without calculus?
There's a classic physics problem that is:

If Earth is orbiting the Sun at 1 au from and is suddenly stopped. How long does it take to fall into the Sun (neglecting the size of the Sun/Earth)?

I know that a clever way to solve this problem is by using degenerate ellipses and an object "falling" into the Sun is the equivalent of the object orbiting the Sun with very high eccentricity. Halving the semi-major axis of the orbit and using Kepler's Law gives: T=1AU/(4√2). However, I want to apply to strategy to another similar problem. This time:

If Earth is orbiting the Sun at 1 au from and is suddenly stopped. How long does it take for the Earth to travel half the distance to the Sun(neglecting the size of the Sun/Earth)?

If we were to use the result for T from the first problem and then simply apply Kepler's 2nd Law by looking at ratios of areas, we should be able to find this new time it takes to fall halfway to the Sun. However, I am having a problem visualizing what area of the ellipse the Earth covers when it travels half the distance to the Sun. I know about the calculus solution which is really bashy, so I would like some assistance with figuring out this more elegant method.
 
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Use Kepler's third law to find the period of an elliptical orbit where the semi-major axis is the half the radius of the Earth's current orbit.

When stopped, the Earth is at the farthest point from the sun. Since the Earth would have no tangential speed, the orbit is the extreme elliptical orbit where the semi-minor axis is 0. In other words, and eccentricity of the ellipse is 1, putting the focii at the extrema of the ellipse. So the other extrema (also the other focus) is the centre of the sun (for practical purposes).

AM
 
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As you will no doubt have realized, it is much easier to determine the time to fall to the sun rather than the time to fall halfway to the sun. But you can ball-park it by taking an ellipse with an eccentricity close to 1, which represents the orbit of the Earth with very little angular momentum relative to the centre of the sun. The area swept out by the Earth in the time it takes to reach the half way point is most of the area of one half of the ellipse. The remaining area of that half ellipse is a tiny fraction of the area swept out in the first half of the fall. Since the orbiting body sweeps out equal areas in equal times, you can see that it takes a small fraction of that time to fall the rest of the way. This means that the time it takes to fall half way is very close to the time it takes to fall the entire way.

You can verify this by figuring out the speed of the Earth at the half-way point using change in potential energy = - change in kinetic energy.

AM
 
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1. How is the "halfway to the Sun" point determined using the ellipse method?

The halfway point to the Sun is determined by finding the semi-major axis of the Earth's elliptical orbit around the Sun. This is the longest distance from the center of the ellipse to its edge, and it is divided by two to find the halfway point.

2. How long does it take for the Earth to reach the halfway point using the ellipse method?

The time it takes for the Earth to reach the halfway point using the ellipse method depends on the specific location of the Earth in its orbit. On average, it takes about 3 months for the Earth to reach the halfway point.

3. What factors affect the time it takes for the Earth to fall halfway to the Sun using the ellipse method?

The main factor that affects the time it takes for the Earth to reach the halfway point using the ellipse method is the eccentricity of the Earth's orbit. The more elongated the orbit, the longer it will take for the Earth to reach the halfway point. Other factors such as gravitational pull from other celestial bodies and changes in the Earth's orbit can also affect the time.

4. Is the ellipse method an accurate way to measure the Earth's distance to the Sun?

Yes, the ellipse method is a very accurate way to measure the Earth's distance to the Sun. It takes into account the Earth's elliptical orbit, which is not a perfect circle, and provides a more precise measurement than simply using the average distance between the Earth and the Sun.

5. Why is the ellipse method used to measure the Earth's distance to the Sun instead of other methods?

The ellipse method is used because it takes into account the Earth's elliptical orbit, which is the actual path that the Earth follows around the Sun. Other methods, such as using the average distance between the Earth and the Sun, do not accurately represent the Earth's true distance at any given point in its orbit.

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