Estimating convergence of GRACE twin-satellites due to gravitational mass

[jacmacg]

Homework Statement

Hello noble physicists,

I am struggling to solve a problem with any sort of confidence whatsoever, so to you I turn in the hopes of guidance.

The problem refers to GRACE twin satellite convergence due to gravitational anomalies.

I’d like to estimate the convergence of the two satellites due to the gravitational mass of Mt. Everest, and subsequently estimate the size of the smallest mountain detectable by GRACE.

Information:
Twin trapezoid satellites: 3.1 m × 0.8 m × (1.9–0.7) m each
460kg each

500km orbital height
220km apart
Can measure their separation to within 10 microns
16 orbits/day

It is assumed Everest is a pyramid
Length: 8.8km
Width: 8.8km
Height: 8.8km
Rock Density: 3000kgm^3

Homework Equations

?

I think:

F = Gm/R^2
a = F/m
s = at^2
s = Vi t + 1/2at^2

The Attempt at a Solution

Time spent above Everest = 1.2 seconds

Circumference of Earth = 40075km

40 075 000m x 16 orbits = 6.412x10^8 m
Speed = 6.412x10^8 m / 86 400 secs = 7421m/s

Width of Everest = 8800m
Time spend over Everest = 8800/7421 = 1.2 secs


Mass of Everest = 6.815x10^5 kg

V = 1/3 (base area) height
V = 227.157m^3

Mass = Vol x Density = 6.815x10^5 kg



Radius of satellite from centre of Earth = 6.88x10^6m

Radius of Earth + Orbit height
6.38x10^6 + 500000m = 6.88x10^6m


Now I’m unsure of which steps to take next to calculate the convergence of the two satellites, I hope someone would be able to offer some guidance.

Many thanks in advance.

 

[mfb]

Time spent above Everest = 1.2 seconds

The time of influence is longer - of the order of the flight time for the orbital height (550km).

You could assume that one satellite passes 110km "left" of the mountain (relative to the orbits) and another one passes 110km "right" of it. Calculate the sidewards acceleration for both and integrate over a fly-by (or use some approximation for the integral). Everest can be approximated as point-mass. This gives the change in relative velocity during a central fly-by.

 

[jacmacg]

Thanks for your help,

though could you explain how to get the correct time of influence?

 

[mfb]

Well, the range of gravity is infinite, but the minimal distance is about 550km - if the satellites are >>550km away, the remaining influence does not matter any more. There is no precise number for the time of influence. If you are just interested in an order of magnitude approximation, you can take the maximal force and use 550km flight distance as range of influence with that maximal force.

 

[jacmacg]

thanks

 

[jacmacg]

I think the problem has been over complicated.

Here is a massively simplicated solution which seems to present a reasonable answer.

Though if there was one thing I were not too sure sure about it would be whether to include in the time of influence calculations the actual distance between the satelites - 220km - or half of that as they are equal distances below...


G 6.67E-11 ms^-2
Mass of EARTH 5.98E+24 kg
Radius Earth 6400000 m
Orbital Height 500000 m
Orbital Radius 6900000 m
Satellite
Separation 220000 m

MASS OF EVEREST
Vol = 1/3(base area) height
Everest Base Width 8800 m
Everest Height 8800 m
Everest Density 3.00E+03 kgm^-3
Volume Everest 2.27E+11 m^3
M = V D
Mass of Everest 6.81472E+14 kg


SATELLITE SPEED
V^2 = GM/r (r = orbital radius = 6900km)
Satellite Velocity 7603.0695555589 ms^-1 V^2 = GM/r


GRAVITATIONAL ACCELERATION
DUE TO EVEREST
F = (G.Meverest)/ r^2 (r = 500000)
Acceleration 1.818167296E-07 ms^-2


TIME OF INFLUENCE
T = D/S = 220000/Satellite Velocity
Time 28.93568162 secs


CONVERGENCE OF SATELLITES
s =( Vi t) + 0.5(a t^2)
s 220000.000076115 m
s - 220km 0.0000761152 m
x2 Satellites 0.00015223 m


Convergence in Microns 152.23 microns

OR

38.06 microns using 110km equal separation.

 

[mfb]

That is a temporary distance modification during a single fly-by if those satellites follow each other - after they passed Mount Everest (and ignoring other mountains), their distance is the same as before.

 

[hydro13]

Hi I'm working on a similar question and am trying to understand the benefit of interpreting the convergence of acting across the two satallites at 220km spacing or two 110km equally spaced. Depending on interpretation the convergance is drastically different.

Based on the fact that GRACE can only detect convergance of 10 microns or greater - the interpretation of the convergance gives differing impressions of the detectable mass