Wikipedia:Reference desk/Archives/Science/2018 July 19

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July 19

Does gravity always point to *somewhere* in the Earth's core?

I know there are points on the Earth's surface where gravity points along a line that detectably misses the Earth's center of mass. But are there any where a line in the direction of gravity doesn't even go through the Earth's core? NeonMerlin 08:11, 19 July 2018 (UTC)

No, the Earth's gravitational field is nearly spherically symmetrical. Deviations from this symmetry are small. Ruslik_Zero 08:45, 19 July 2018 (UTC)

[e/c] :No. Assuming the Earth is a homogeneous density sphere, (which it is not) the gravity vector will always pass through the geocenter. However, since the Earth is actually a bit elipsoidic, (see Reference ellipsoid) there is a difference in angle between spherical and ellipsoid normals (Vertical deflection -- see also: geoid). This difference is very small, however, relative to the diameter of the Earth's core. Also, since the core is more dense than Earth's average, this will add even more "attraction" to the center. There are some gravitational anomalies (local variations of the gravity field) that distort the force vector angle, but doubtfully enough to bypass the core; there might be an artificial condition where this is not true -- perhaps standing right next to Fort Knox? —2606:A000:1126:4CA:0:98F2:CFF6:1782 (talk) 09:05, 19 July 2018 (UTC) ... since your two questions are diametrically opposed (colloquially, not literally), the answers are "yes" and "no".

The core is really big, with a radius of 3,470 km, it would have to be a huge deflection (33°), which is highly unlikely - see vertical deflection, which talks about deflections up to 50 seconds of arc. Mikenorton (talk) 09:30, 19 July 2018 (UTC)

• Guys, that's the kind of question where you need to show the numbers.

Gravity of Earth is the relevant article here. The two most important effects that explain local variations of gravity are (1) the centrifugal force due to Earth's rotation and (2) the fact Earth is not exactly round but slightly flattened at the poles. The article states that the combined effect is a 0.5% change in g between poles and equator (this is why all space rocket launching pads are near the equator). By comparison, local deviations are about 0.005% of g (5e-4 m/s2 max changes, g~10m/s2).

The exact details of how much of the variation in magnitude changes the direction of the gravity vector depending on the latitude need more calculus than I am willing to do, but fortunately very rough assumptions get us what we want. In the worst case scenario, the 0.5% change in g's magnitude comes from an additional pull perpendicular to g (this maximises the deviation). A bit of trigonometry (solve cos(theta)=1/(1 + 0.5%) for theta) tells us the deviation is less than 6° (by the way, this is higher than I would have expected).

On the other hand, Earth's inner core is seen from an angle of two times 10.8° (some more trigo with Earth radius = 6400km, core radius = 1220km). So even with all assumptions lined up in the worse case, we still get that the gravity vector will pass through the inner core. Tigraan^{Click here to contact me} 09:40, 19 July 2018 (UTC)

Indeed. The Schiehallion experiment measured deviations from the local zenith of around 11 seconds of arc. Similar (but less precise) measurements around Chimborazo measured deviations of 8 seconds of arc. Over a distance of 6360 km (radius of the Earth) these sizes of deflections result in a deviation of less than 1 km away from the geometric centre of the Earth. Gandalf61 (talk) 11:43, 19 July 2018 (UTC)

Re "(this is why all space rocket launching pads are near the equator)". The difference in gravitational force is not the main reason rockets are launched from near the equator. Rather, it's because the earth's rotation means the rocket is already traveling east at close to 1000 miles/hour even before it launches. The additional velocity means less fuel required. The hard part of getting into orbit is achieving orbital velocity, not reaching a specific height. [1] [2] — Preceding unsigned comment added by CodeTalker (talk • contribs)

Huh... I somehow thought that was the same effect, but of course it is not (speed in the geocentric frame of reference vs. centrifugal force in the terrestrial, non-inertial frame). Struck accordingly, see also discussion below. Tigraan^{Click here to contact me} 08:06, 20 July 2018 (UTC)

To dispel some very common misconceptions: spacecraft launch sites can be located anywhere on Earth. The geographic location of the launch site is selected for many reasons, not the least of which is orbital dynamics; but there are many other factors, including range safety, redundancy, and convenience - it's important to have easy access to a bunch of rocket scientists, who don't necessarily all want to live on the launch site! Here's a list of NASA's primary launch sites; but not all American orbital (and sub-orbital) launches are managed by NASA. Notably, yesterday a New Shepard 3 launched from West Texas, an operation that was largely conducted without overtly depending on NASA's money or help.

I would use caution before spouting anything about equatorial rotational inertia unless you've worked through the orbital flight dynamics equations - orbits aren't really that simple - and I can spout some counterexamples off the top of my head. For example I've been to spacecraft launch sites at Kodiak Island and Vandenberg Air Force Base: sites from which the spacecrafts can fly westbound or northbound or pretty much where-ever they want to fly. Quoting directly from NASA's website, "Kodiak Island is one of the best locations in the world for polar launch operations, providing a wide launch azimuth and unobstructed downrange flight path..."!

Nimur (talk) 18:33, 19 July 2018 (UTC)

Do they launch low-inclination orbits from Kodiak? Sagittarian Milky Way (talk) 19:51, 19 July 2018 (UTC)

With enough ΔV, you can get to any orbit! That maneuver would be called a "dog leg" on the launch trajectory, and it's expensive (you have to spend fuel to do it) - but any orbit is possible if the people buying the launch want to use their mass (and dollar) budget in that way. Nimur (talk) 22:22, 19 July 2018 (UTC)

Indeed. Likewise a launch from the equator that ends up going over the poles would require a dogleg. Not sure whether it would use more energy than was gained by the equatorial launch -- it feels like it should but I would have to do the math. --Guy Macon (talk) 23:45, 19 July 2018 (UTC)

Orbital inclination changed are ludicrously expensive in terms of energy. If they weren't, there wouldn't be separate low and high inclination launch sites. An extreme example, a LEO craft launched from KSC want to transfer to a 90 degree inclination polar orbit. Plane change delta v for circular orbits is :$\Delta {v_{i}}={2v\,\sin \left({\frac {\Delta {i}}{2}}\right)}$. Assuming a LEO orbital speed of 7 km/s, the delta v required for that plane change is 7.4 km/s again. Or almost as much as it took to launch in the first place. Of course, there are tricks to lower that, but it illustrates my point. Doglegging during launch is severely restricted by overflight considerations on most launch sites. Fgf10 (talk) 07:06, 20 July 2018 (UTC)

Maybe sort of related: shell theorem. Although the Earth isn't perfectly symmetrical, the deviation is pretty tiny given the scale of the Earth. The standard factoid is that the Earth is, to scale, smoother than a billiard ball. --47.146.63.87 (talk) 06:54, 20 July 2018 (UTC)

No. A pool ball is much smoother than the (whole) Earth would be if it were shrunk down to the size of a pool ball, though much of the Earth’s surface is indeed smoother than a pool ball. DroneB (talk) 13:11, 20 July 2018 (UTC)