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The enhancement of a cross-section due to gravitational focusing is given by $$\sigma_{\rm eff} = \pi a_J^2 \left(1 + \frac{2GM_{\odot}}{a_J\ v^2}\right),$$ where $a_J$ is the semi-major axis of Jupiter's orbit (assumed circular), $v$ is the relative velocity (at infinity) and I have ignored the mass of Jupiter. Thus, using $v=30$ km/s (as specified in ...

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I had to write my own code to calculate this, since I could not find any tools to solve two-body problem. From my simulation it seems that Jupiter orbit cross section is 6.5 AU, assuming radius of Jupiter orbit as 5 AU. Seems like a tiny difference, but surface-wise it's 1.7x increase. This assuming 20 km/s, it's smaller for faster scouts. This means one ...

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The gravitational field of the Earth has not a border, what you can leave and there is weightlessness. Instead, going more and more far away from the Earth, so will its gravity weaker and weaker. At some point, the gravity of other bodies (Moon, Sun, other planets) will be stronger than the Earths. What will happen to a body which leaves the space where ...

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There are many definitions for the "Goldilocks zone" or Habitable zone. If you want liquid water on the surface then that can happen for a wide variety of atmospheric pressures (totally unknown for the vast majority of exoplanets) and temperatures. In fact Mars is considered to be in the habitable zone using only this criteria. More strict criteria exist ...

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From the Wikipedia article Circumstellar habitable zone, which is just another name for the Goldilocks zone: Estimates for the habitable zone within the Solar System range from 0.38 to 10.0 astronomical units, though arriving at these estimates has been challenging for a variety of reasons. Numerous planetary mass objects orbit within, or close to, this ...

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This idea would be a direct reduction from Newton's laws of motion. In his Principia he states "Hence the common centre of gravity of the earth, the sun, and all the planets, is to be esteemed the centre of the world". So that would be the first evidence of the Sun moving outside of a geocentric universe. Of course he wouldn't have known about Uranus and ...

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Addressing @Allure's comment below @ JohnHoltz's excellent answer, the synodic period is simply a function of the two periods. It will return something like the average value between two successive events where the planets would line up if they orbited in the same plane, but it does not predict the exact times as pointed out in that answer. In addition to ...

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Mercury's orbit is highly eccentric: 0.21 according to Wikipedia. Therefore, the actual time between repeating occurrences will vary depending on the year. If you were to perform your calculations for many periods, the average should approach the value given by Stellarium. The theoretical synodic period, using the sidereal period of Earth and Mercury, is ...

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It's possible the book you seek is by Christopher Graney. I cannot find them now, but I believe he had put together instructions on how to build a telescope similar to what Galileo used and how it could be used to argue for and against a heliocentric universe. But his overall conclusion is that it's poorly suited for the task, and scientists of the time ...

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