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The Tindallgrams have some notes on the matter. See pages 307 to 354 of http://www.collectspace.com/resources/tindallgrams/tindallgrams02.pdf Briefly, improvements were made in a couple of areas: The model of the moon's gravity field (based on tracking data from earlier Apollo and Lunar Orbiter flights) Real-time estimation of the spacecraft's trajectory -...


7

From Wikipedia: In the more general case of elliptical orbits, there are no longer stationary points in the same sense: it becomes more of a Lagrangian “area”. The Lagrangian points constructed at each point in time, as in the circular case, form stationary elliptical orbits which are similar to the orbits of the massive bodies. A more complete ...


7

Wikipedia has a good page on the subject of artificial gravity, highlighting many (very interesting) possibilities. Rotation Linear acceleration Mass Magnetism Gravity generator/gravitomagnetism Rotation is exactly what you would expect. The spacecraft is a big rotating cylinder, just like in physics textbooks. Linear acceleration, however,...


7

Gravimetric Doppler measurements can actually confirm that Vesta has an iron core, as it eliminates the alternatives. This works in the following way: The mass of Vesta is first measured accurately, for example using the orbital period of a satellite, for instance the Dawn spacecraft. Then, based on observed geological features, its average density, the ...


6

One thing that may be tripping you up is that the d term in the gravitational formula is the distance between the centers of mass of the objects, not the altitude above Earth's surface. The other thing to keep an eye on is your units. The big G gravitational constant is ~6.67 x 10-11 m3 kg-1 s-2; if you're using that value, make sure you're consistently ...


6

Do Curiosity's reported measurements of Mars' surface gravity include centrifugal effects? Of course. That's how surface gravity is defined. The measurements in the cited paper were made with accelerometers during periods where when the vehicle was at rest to the surface. Accelerometers cannot measure centrifugal effects, any more than they can measure ...


5

The Moon's gravitational field itself, by far. The perturbations on a satellite due to the reshaping of the Earth due to the Moon and Sun's gravity fields are small effects. The primary impact of this secondary tidal effects is a slight change in $J_2$ ($C_{2,0}$). A description of how the Earth tides affect the gravitational field is in section 6.2 of IERS ...


4

For these type of files, use your same first equation, but in this form: $$V(r,\phi,\theta)=-{\frac{GM}{r}} +\sum _{n=2}^{\infty}\sum _{m=0}^{n}{\frac {P_{n}^{m}(\sin \phi )(C_{n}^{m}\cos m\theta +S_{n}^{m}\sin m\theta )}{r^{n+1}}}$$ Note that the second summatory now begins at $m=0$. Zonal harmonics are $C^0_{n}=J_n$ and $S^0_{n}=0$, $n=2...\infty$. ...


3

At first glance, this looks like a horribly complicated problem. Here is an example of a tadpole orbit around around $L_5$: Geometrically, it is very irregular. That, however, is a projection in a co-rotating frame of reference. The same orbit in the inertial frame is actually an almost perfect ellipse: In for instance the Sun-Earth $L_4$ and $L_5$ points, ...


3

This might help to get you started. It's just a 1D radial solver, but you can play with the math. I included a 3D version of the derivative function to show one way to make the acceleration a vector in NumPy. Don't forget to add the rotation of the earth. The idea about starting on top of a mountain like Mt. Kilimenjaro is worth exploring for sure. I just ...


3

There are two search channels in LIGO, the first is for CBCs (Compact Binary Coalescence). These are binary systems of very dense objects, either BH-BH, BH-NS or NS-NS mergers (BH: Black Hole, NS: Neutron Star). They produce well known signals that increase in frequency over time, an example of this is shown here: The second search channel is looking for "...


3

I think the problem here is the calculation of gravity is a bit sensitive to nuances you've over-looked. I am confident that the reported result is the 'experienced' force, as it's the sensible way to do it and fits with the values the literature expects. I'd like to show the value is predictable but that's a bit beyond a short Q/A (and I think I'd quickly ...


3

This article provides estimates of zonal harmonics down to $J_8$ derived from Doppler data in the Juno mission.* Tesseral harmonics are "statistically zero as expected for a fluid planet in equilibrium". The article also compares the zonal harmonics estimates to past estimates from the Pioneer and Voyager missions. The values are reported in Table 2, ...


2

@DavidHammen's answer says to include all accelerations, and @drjpizzle's answer recommends to look at Mars' shape and consider both the poles and equator. So here's a complete tally using $$a_G = -GM \frac{\mathbf{r}}{r^3}$$ From Geopotential_model; The_deviations of Earth's gravitational field from that of a homogeneous sphere: $$a_{J2x} = J2 \frac{\...


1

For Saturn: Data from Cassini-Huygens are given in this table from phase 76 of Reference 1 $J_2×10^6=16324.19\pm0.11$ (observed) $J_4×10^6=-939.32\pm0.98$ (observed), $-971$ (theory) $J_6×10^6=91\pm 5$ (observed) $J_8×10^6=-10$ (assumed) References 1. Saturn from Cassini-Huygens edited by Michele Dougherty, Larry Esposito, Stamatios Krimigis. (...


1

I'm going to give an opposing answer. As long as the mass is spherically symmetric, you shouldn't be able to determine its distribution, because spherically symmetric changes in the mass distribution shouldn't change the resulting gravitational field. If the Sun, the planets, and the other bodies in the solar system truly were spherically symmetric, and ...


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