We describe how orbital tunnels could be used to transport payloads through the Earth. If you use a brachistochrone for the tunnel, the body forces in the tunnel become overwhelmingly large for small angular distances traveled. Projectiles move along an orbital tunnel faster than they would along a brachistochrone connecting the same points but the body force components cancel. We describe how parabolic Keplerian orbits outside the object merge onto quasi-Keplerian orbits inside the object. 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. The n = 3 results are also scaled to the Sun. Numerical integrations of the equations describing polytropes were used to generate the initial models. 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. Trajectories through objects of low central condensation show a focussing effect that decreases as the central condensation increases. Analytic solutions for the trajectories in a homogeneous sphere are derived and compared to the numeric results. The results can be scaled to other planets, stars, or even globular clusters.
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June 2019
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June 01 2019
Orbits through polytropes
Amalia Gjerløv;
Amalia Gjerløv
NASA, Goddard Space Flight Center
, Greenbelt, Maryland 20771
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W. Dean Pesnell
W. Dean Pesnell
a)
NASA, Goddard Space Flight Center
, Greenbelt, Maryland 20771
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a)
Electronic mail: [email protected]
Am. J. Phys. 87, 452–464 (2019)
Article history
Received:
April 05 2017
Accepted:
February 16 2019
Citation
Amalia Gjerløv, W. Dean Pesnell; Orbits through polytropes. Am. J. Phys. 1 June 2019; 87 (6): 452–464. https://doi.org/10.1119/1.5093295
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