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A circle is a degenerate ellipse, and you can also think of a circle as having two foci (on top of one another) as the eccentricity approaches 1. The foci of conic sections in general originate from the approach in which the curves are defined - using a focus (point) and directrix (line). This approach leads to rational parametric expressions for the conic ...

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The equation of an ellipse whose semi-major and semi-minor axes are parallel to the x and y axes is given by: $$(\frac{x-h}{a})^2+(\frac{y-k}{b})^2=1$$ (where $a$,$b$ are the lengths of its semi-major and semi-minor axes respectively.) A focus, $c$ is defined by $c^2=a^2-b^2$, and therefore there can be two foci at a distance of $(a^2-b^2)^{1/2}$ on both ...

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The foci are simply points that define the ellipse by the relation $c^2 = a^2 - b^2$, where $c$ equals the length of each one of the foci to the center and $a$ is the length of a focus to the end of the ellipse. For a circle, $a$ = $b$. Given any two foci, a point on the ellipse is a point that is equal I he sum of he lengths of the foci.

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If you plot the path of the moon with respect to the star you should get a cycloid pattern averaging around the path of the planet. That's what our moon does. The planet and the moon are doing a gyrating dance about their center of mass, with the moon doing the most motion (because it has the smallest mass), while the planet/moon duo orbit the star. I've ...

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What local forces cause the damage? Tidal forces. Tidal forces can only be neglected in extremely small regions, and you have an extended body. More details follow. You suppose an extended body, bound together, interacting gravitationally with a black hole. With the center of mass staying outside the photon sphere, moving on a more than barely unbounded ...

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First, let us look at the current rate at which the moon slows down. I have a few different sources, and they don't all give me the same answer. First, there is this claim that Earth slows down at a rate of about 0.005 seconds per year per year. A year has approximately $365.25 \cdot 24 \cdot 3600 = 3.15\cdot 10^7 \mathrm{sec}$, so 0.005 seconds change ...

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The Moon rotates around the Earth slower than the rotation of the Earth itself. That's why, from a fix point on the Earth, the Moon appears to be moving. The Moon creates the tide on Earth. So the tide "follows" the Moon. However as the Earth rotates faster than the Moon it will tend to carry the tide with itself "forward". The Moon pulls the tide toward ...

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Two reasons this effect doesn't occur. No observer sees any part of the spaceship/sun crossing the event horizon. The distant observer sees it get closer and closer (due to time dilation) but not actually reach it. The observer in the spaceship can't even detect that they have reached the event horizon. No observer sees any part of the ship cross the event ...

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The answers are yes and no. Special relativity does make ellipses precess, but it only accounts for 7" out of 43" per century of Mercury's anomalous precession. I wonder if Einstein and/or Sommerfeld knew that. To first order, incorporating special relativity results in a small inverse cube correction to the gravitational force, which is well known to ...

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