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Coriolis force: $$F_c=m v \wedge \omega$$ We need the velocity component perpendicular to the axis of rotation, which is $$v_c = v sin(\theta)$$ Now we integrate the acceleration twice with respect to time in order to get displacement: $$x = \int\left(\int{v sin(\theta)\omega dt}\right)$$ Noting that $$\theta = \frac{v.t}{R}$$ where $R$ = radius of ...

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inner radius of habitable zone (AU) $= \sqrt{\frac{L}{1.1}}$ outer radius of habitable zone (AU) $= \sqrt{\frac{L}{0.53}}$ where $L$ is absolute luminosity of the star. See Calculating the Habitable Zone for more information.

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$\theta$ is the angle made between the positive x-axis and the spacecraft's velocity vector (all at closest approach). Don't you need $\theta$ to be the angle of the spacecraft's velocity relative to Mars? Try subtracting the Mars velocity vector from the spacecraft's velocity vector, calculating $\theta$ for the resultant vector and using that angle ...

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I interpreted the question as asking whether it's possible to show that a sphere is the minimum energy shape of an object being acted on by its own gravity. I attempted to do this using spherical harmonics, but got stuck part of the way there; I'm posting this anyways just in case someone can figure out how to complete the last bit. The gravitational ...

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One perspective (heh) involves the following relation among position vectors: $$\vec{r}_{A\rightarrow C} = \vec{r}_{A\rightarrow B} + \vec{r}_{B\rightarrow C}.$$ These position vectors can be for anything; object $A$ could be a house, object $B$ an ant, and object $C$ a leaf on the river. Here's a diagram to help: So if you want to know the position of ...

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Would the asteroids count? Or maybe the rings around Uranus or Neptune? I don't think the rings on Saturn would fit in your requirements since they are too broad.

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There is a reasonable chance that yes, planets can form before the star "ignites" (which I take to mean the fusion of hydrogen into helium, not the very brief phase of deuterium burning which certainly will take place before planets can form). Planets form in a disk of circumstellar material around their parent protostars. The "core-accretion" model of ...

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I'm having trouble understanding what you're trying to say, but assuming you're just looking for period resonances of the form $a:b$ where $1\leq a,b\leq 5$, the following 4 lines of Mathematica code using your example list of extrasolar periods should suffice: A = {0.44, 0.8, 0.9, 0.9, 1.2, 2.0, 3.0}; n = Length[A]; d = 0.05; ResonanceMatrix = ...

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The planets appear to have formed by accretion of dust and gas around initially-small nuclei in orbit around the sun. As orbital radius (distance from the sun) increases, more material is available (in a uniform dust/gas disc) to accrete, so you get bigger planets further out; bigger planets sweep up more of their neighbors, so they tend to be spaced further ...

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As you have said, the general solution for the Kepler problem is an elliptic orbit. The shape of an ellipse is determined by its semi-major axis and the eccentricity. The semi-major axis is determined entirely by the energy, but the eccentricity depends also on the angular momentum. Both energy and angular momentum are conserved in the two-body problem, so ...

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In a two body system the orbit can have any eccentricity you want. The key point is that the eccentricity cannot change. In a three or many body system the bodies perturb each others orbits and the eccentricity can change with time. For example this graph shows the eccentricities of the rocky planets as a function of time. These changes are mainly caused by ...

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It's because of Conservation of Angular Momentum. When all these systems are in the process of creation, all motions not existing in the same horizontal plane cancel each other out which confines them into that plane. The system continues to exist that way due to conservation of Angular Momentum.

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it is because of the way the solar system formed from a cloud of gas. the molecules of gas (mostly hydrogen) were pulled together by gravity which formed a spinning disk of gas around our sun, thus forming the planets all on the same field.

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