After the outer ring falls, the government drafts a whopping 250,000 citizens into a massive army to take the outer ring back. Attack on Titan brings this to us early on, as a way of showing just how utterly fucked humanity is against the Titans.The grains in the Mars calculation also made a smaller collision angle with the surface on rebound. Major findings include a comparison of earth and Mars grain trajectories that show Mars length scales to be longer and to fall with a higher terminal grain velocity. These calculations show the importance of a lifting force on the grain to initiate motion in both earth and Mars calculations. These modified equations were then solved to estimate grain motion for Mars. The numerical solutions for earth were used and empirically adjusted to correlate with existing experimental data for Mars. To study this phenomenon, both experimental investigation in an atmospheric wind tunnel and numerical solutions of the equations of motion of a single grain under Martian surface conditions.
Mariner 9 spacecraft images showing evidence of variable surface features and surface erosion resulting from atmospheric wind on Mars have caused a renewed interest in the eolian mechanics of saltating grains. We also compute the deviations from the vertical for short falls and compare them to standard eastwards and southwards deviation expressions obtained with other modelizations of the gravity of the rotating Earth, in particular, for a rotating spherical body. We find that when the flattening of the Earth is taken into account, gravity tunnels have no exit: an object dropped from a point of the surface (other than on the equator or a pole) never reaches the surface again, unless the flattening has very specific (and unnatural) values.
This study follows a procedure presented several times in this journal and elsewhere, i.e., the trajectory is first computed in the geocentric (non-rotating) frame, and it is then analysed in the terrestrial (rotating) frame.
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It turns out that the gravitational force acting on an object located inside the spheroid has a simple form (it is harmonic), so that it is straightforward to compute analytically the free fall trajectories. The Earth is described as a McLaurin spheroid, an exact equilibrium shape for a rotating homogeneous deformable body that provides a fair description of the actual shape of the Earth. In this article, we determine the shape of gravity tunnels by taking into account the polar flattening of the Earth resulting from its rotation. It is well known that because of the rotation of the Earth, such tunnels are not straight lines but instead they emerge westward of the antipodal point, when the Earth is approximated as a rotating sphere. The results can be scaled to other planets, stars, or even globular clusters.Ī "gravity tunnel" is the name given to a fictitious deep shaft drilled inside the Earth so that objects dropped from the surface of the Earth would free fall without ever touching the walls. Analytic solutions for the trajectories in a homogeneous sphere are derived and compared to the numeric results. Trajectories through objects of low central condensation show a focussing effect that decreases as the central condensation increases. Numerical integration of the equations of motion are then used to calculate the angular distance you can travel along the surface and the traversal time as a function of the parabolic periapsis distance for each model. Numerical integrations of the equations describing polytropes were used to generate the initial models. The n = 3 results are also scaled to the Sun.
We use models of the interior of the Earth with three values of the polytropic index (n) to calculate interior orbits that travel between surface points. We describe how parabolic Keplerian orbits outside the object merge onto quasi-Keplerian orbits inside the object. Projectiles move along an orbital tunnel faster than they would along a brachistochrone connecting the same points but the body force components cancel. If you use a brachistochrone for the tunnel, the body forces in the tunnel become overwhelmingly large for small angular distances traveled. We describe how orbital tunnels could be used to transport payloads through the Earth.
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