On Tuesday, October 13, 2020 at 6:16:04 PM UTC-5 Brent wrote: > > > On 10/13/2020 3:12 PM, Lawrence Crowell wrote: > > On Tuesday, October 13, 2020 at 4:13:05 PM UTC-5 Brent wrote: > >> >> >> On 10/13/2020 1:34 PM, Lawrence Crowell wrote: >> >> On Tuesday, October 13, 2020 at 3:28:21 PM UTC-5 Lawrence Crowell wrote: >> >>> On Tuesday, October 13, 2020 at 3:26:14 PM UTC-5 Lawrence Crowell wrote: >>> >>>> I will try to give a definitive answer. The Schwarzschild metric is >>>> >>>> ds^2 = c^2(1 – 2m/r)dt^2 – (1 – 2m/r)dr^2 – r^2(dθ^2 – sin^2θdφ^2) >>>> >>>> for m = GM/c^2. For the motion of a satellite in a circular orbit there >>>> is no radial motion so dr = 0. We set this on a plane with θ = π/2 so dθ = >>>> 0 and this reduces this to >>>> >>>> ds^2 =c^2(1 – 2m/r)dt^2 – r^2dφ^2. >>>> >>>> For circular motion dφ/dt = ω and the velocity v = ωr means this is >>>> >>>> ds^2 = [c^2(1 – 2m/r) – r^2ω^2]dt^2 >>>> >>>> and so ds^2 = [c^2(1 – 2m/r) – v^2]dt^2 the term Γ = 1/√[c^2(1 – 2m/r) >>>> – v^2] is a general Lorentz gamma factor and in flat space with m = 0 >>>> reduces the form we know. ds is an increment in the proper time on the >>>> orbiting satellite and t is a coordinate time, say on the ground of the >>>> body. >>>> >>> >> Another erratum. The coordinate time t is for a clock very far removed, >> not on the ground. On the ground that clock ticks away with a factor Γ = >> 1/√[c^2 – v^2] change. So there is a relative time difference. >> >> >> A clock on the ground is also moving with rotation of the Earth, with >> different speed at different latitudes. This is taken out of the equations >> by comparing the GPS clock to ideal clocks on a fixed (non-rotating Earth) >> and then after GPS calculates the location on the non-rotating Earth, it >> calculates what point this is on the rotating Earth. >> >> Brent >> > > This gets really complicated. I did a lot of post-Newtonian parameter work > on this back in the late 80s. A lot of it was numerical, because on the > ground there are different values of gravity, and these too can cause > drift. Gravitation, thinking of a Newtonian force, is different near a > mountain than on the top of it, and the direction can vary some from the > radius. It also fluctuates with tides! The surging in and out of a lot of > ocean water actually changes the Newtonian gravitation potential and force. > > > LC > > > And it's further complicated by the Earth being non-spherical. The > calculations find the lat/long of a WGS84 ellipsoid. But of course the > real Earth isn't exactly an WGS84 ellipsoid either and there have to be > local corrections in look-up tables. Off the coast of California where I > used to be involved in developing sea-skimming targets the WGS84 "sea > level" is about 120ft under water. > > Brent > > The world geodetic system might be thought of as a set of Bessel functions and similar polynomials. The Earth is not a perfect ellipsoid and there are other mass distributions going on with the tectonic plates and interior mass distributions. The model is subject to constant refinement.
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