ARO 3191
Space Environments
S. Nanning
Last revised: 2021-04-02
Gravitational Fields
Gravitational Fields
Gravitation (or gravity) is one of the four fundamental forces
Weakest, but is capable of acting over large distances
Classical mechanics definition will suffice for our applications, as opposed to
General Relativity
Gravitational Force is applied instantaneously
Function of Position and Velocity
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Gravitational Fields
Law of Gravitation
First presented by Newton in The Principia (1687)
𝐹𝑔 =
Where
𝐺𝑚 𝑀
𝐺𝑚 𝑀
ȓ
=
r
𝑟2
𝑟3
Fg = Force of gravitation
G = Universal gravitational constant, 6.674x10-11 m3/kgS
M = mass of Primary body, Kg
M = mass of secondary body, Kg
R = position vector of m wrt M, meters
ȓ = unit vector r/r
The negative sign shows the force is attractive
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Gravitational Fields
Acceleration of Gravity is the Gravitational Force per Unit Mass
gEarth = 𝑟 2
𝐺𝑀
𝑚 𝑠 2 = 9.82 m/s2
𝑒𝑎𝑟𝑡ℎ
𝐺𝑀
This can be generalized to g0 = 𝑟 2 where r0 is taken at the surface of a
celestial body
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Gravitational Potential
Higher order Gravitational Potential
Needed to determine the precise gravitational Force from a body without a
Homogeneous mass distribution
Recall that a surface spherical harmonic function is a function of θ & λ and
describes a function to the surface of a sphere
Variation of Gravitational Potential has (n-m) nodes in latitude over 0 <= θ <= π and
2m nodes in Longitude over 0 <= λ <= 2π
Bottom line: it has bumps, both as surface and Potential
Three categories of Harmonic functions:
Zonal
Sectoral
Tesseral
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Gravitational Potential
Zonal functions are defined as zero order (m = 0)
Constant longitude
Vary in Latitude
Specific potential creates n=nodes in parallels of latitude
The cloud bands of Jupiter are a good example of this
Sectoral functions have both degree and order being equal (n = m)
Variations in potential has nodes at 2m parallels of longitude
None at latitude
Surface divided into vertical sectors, similar to a Mercator projection map
Tesseral functions are defined as terms where degree and order are not equal and
are non-zero
Variation in potential is divided up into n-m parallels of latitude and 2m lines
of longitude
Checkerboard pattern
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Gravitational Models
Gravitational models of a celestial body include:
Geometric representation of the body
Its gravitational field
Positions of geographic points
Dynamic phenomena such as tides and crustal motion
Several gravitational models available for Earth
World Geodetic System 84 (WGS84)
Earth Gravitational Models (EGM84, 96, 2008)
[Table 7.1]
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Gravitational Models
Geopotential modeling is made complex by the presence of the Sun and Moon
Causes Earth deformation
Influences the geopotential being modeled
We simplify this in one of three ways
Tide Free
Mean Tide
Removes influence of direct/indirect effects of the Sun and Moon
Removes time effects and models an average of Sun/Moon influence
Zero Tide
Removes time independent direct effects, but retains time independent indirect effects
E.g.: tides and tidal deformation are removed but elastic impacts to Earth’s shape are retained
While the models have detailed higher order terms, when considering the geopotential on
a spacecraft, where for (a/r)n the term r >> a (lower effects over distance/altitude) the
higher order terms become negligible
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Geoids of Earth
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Geoids of Earth
Defines the surface of constant gravity potential
Closely approximates global mean sea level
Provides a reference surface to measure variations in local gravity
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Geoids of Earth
A reference ellipsoid maps the geopotential model onto the surface geoid
Difference between the two are known as Geoid Heights
Positive if the geoid is above the ellipsoid
Ellipsoidal Height h
Orthometric height H
Distance of topography above reference Geoid
Geoidal height N
Distance of topography above reference ellipsoid
Distance of Geoid above reference Ellipsoid
In general, h = H + N
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Gravity anomaly
Difference in gravity between geoid and reference ellipsoid
Δg = g – γ
Also known as Free-Air gravity anomalies
Free-Air Gravity anomaly (aka Free-Air Anomaly)
Determined by gravity acceleration measurements on the topographical
surface
Result of correction applied to the measurement made
ΔgF = gp – γ(Φ) + δgF
Where
gp = gravity measured at the topographic surface, mGal
mGal = mean value of theoretical gravity
γ = gravity on free ellipsoid, mGal
ΔgF = Free air anomaly, mGal
δgF = Free air anomaly correction, mGal
Φ = latitude
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Gravity anomaly
The Free Air Correction follows from
ℎ
𝑟
δgF =2h =2 γ(Φ)
Leading to
ΔgF = gp – γ(Φ) (1-2 )
The Free Air correction accounts for the decrease in gravity with distance
from Earth’s center
ℎ
𝑟
We end up with a ‘bumpy’ gravitational field
Prime benefit for orbital assets is gravitational stability ‘buckets’ in Earth’s gravity
field
Higher orbit stability with reduced maneuvering fuel allocations
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Tides
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Tidal Potential
Gradient gravitational potential that arises from multiple body interactions
For Earth, this is the Sun and Moon
Affects both liquids and solid mass of a quasi-elastic body such as Earth
A body of mass M produces a tidal potential given by
Ut=-GM/ρ = -GM/(R(t)2+ a2 – 2R(t)acosψ)1/2
Where
a = radius of disturbed body
G = Gravitational constant, m3/kgS2
M = mass of disturbed body
R(t) = distance between the two bodies, m
Ut = Tidal potential
Ψ = angle between R(t) and a
ρ = distance between disturbing body and position defined by a, Ψ
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Body Tides
Earth responds quasi-elastically to tidal potentials from the Sun and Moon
These deformations are small
May be represented by linearized functions
Love numbers
K, h, and l dimensionless parameters that reflect the rigidity of a body
Each spherical harmonic of deformation degree n and order m has a Love
number set
Kn,m; hn,m; ln,m
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Body Tides
Love number kn,m relates the gravitational potential for deformation
How susceptible it is to being pulled/pushed
For a rigid body, k would be zero
Love number hn,m relates to the radial potential for displacement
Love number ln,m relates to transverse of horizontal displacements
How susceptible it is to rotations
How susceptible it is to sideways displacements
The magnitude of Love numbers depends on the rigidity and mass distribution
of the body in question
Influence of Order, m, is small and is generally ignored
Love numbers kn, hn, ln
Magnitude of Degree 2 Love numbers is significantly grater than other Degrees
[Table 7.7]
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Ocean Tide Models
Water deforms more easily due to tidal potentials
that solid Earth
Amplitudes as high as 1 meter in open ocean
When Sun and Moon’s Tidal potentials interfere
constructively we have Spring Tides
Destructive interference gives us Neap Tides
Smallest tidal amplitudes
During a synodic month there are 2 Spring and 2 Neap
tides
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Ocean Tide Models
Two primary Ocean Tide models
CNES-FES2014 (Finite-element solution)
Provides tidal atlas of elevations on a 1/16 degree grid
NASA Global Tide Models GOT99.2 and GOT00.2
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Precession of Satellite orbits
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Precession of Satellite orbits
Departure of the gravitational potential from the
classic Newtonian definition can give rise to Orbit
Perturbations
Can be modeled in terms of secular, long-period, and
short period components
Secular are functions of powers of time
Long term are functions of Argument of Periapsis
Short Term are functions of True Anomaly
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Precession of Satellite orbits
To examine the secular precession rates, we first have to introduce a
coefficient for the zero-tide baseline C2,0
For Earth, C2,0 = 0.484167x10-3
Three orders of magnitude greater than any other geopotential term
Three orders of magnitude smaller that Newtonian term
First order Secular effects are limited to
Right Ascension of the Ascending Node, Ω
Argument of Periapsis, ω
Mean Anomaly, M
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Precession of Satellite orbits
The secular precession rates for a body with dominant C2,0 coeff are:
𝛺𝑠 =
3√5 𝑎2 𝐶2,0
n cos i
2 𝑎2 (1−𝑒 2 )𝟐
𝜔𝑠 =
𝐶2,0 𝑎𝑒2
3 5
4 𝑎2 1−𝑒 2
𝑀𝑠 = 𝑛 −
2
𝑛 4 − 5𝑠𝑖𝑛2 𝑖
𝐶2,0 𝑎𝑒2
3 5
𝑛 2 − 3𝑠𝑖𝑛2 𝑖
4 𝑎2 1−𝑒 2 3/2
Where:
a = orbit semi major axis, m
ae = Semimajor axis of Reference ellipsoid, m
C2,0 = -0.481167x103
e = eccentricity of orbit
i = orbital inclination
n = 2π/τ, Rad/s
τ = 2π 𝑎3 /GM
Note that the precession rate decreases as altitude increase
Semimajor axis in denominator, mean motion in numerator
Once you get the precession rate of the ascending node equal to that of the Sun, (0.9856 deg/day) you
achieve a Sun Synchronous Orbit
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Gravity Gradient effects
We’ve been looking at gravitational effects on an idealized point-mass
body
Not how it is in the real world
Gravitational force is non-linear
Force at differing points on a distributed mass body are dependent on their
distance from the gravitating body
We get a gravity-gradient torque on the disturbed body
In the case of the Moon, this torque forces one side to always face Earth
Gravitational Capture or Tidal Locking
The gravity gradient torque on a spacecraft can be an attitude disturbance
or be used to stabilize orientation
We’ve already discussed this idea in the form of Hysteresis Rods
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Gravity Gradient effects
The vertical gravity gradient is a function of gravitational acceleration
∆𝑟 𝑔 = −2 𝑟 3 = -2 𝑟
Where
𝐺𝑀
𝑔
g = acceleration of gravity, m/s2
GM = Gravitational parameter, m3/s2
r = radius from center of mass, m
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Gravity Gradient effects
For a distributed body in a circular orbit, the force on mass, m, displaced
by some radius, r, from the mass center of a spacecraft is given by:
𝛿𝐹 = −3𝑚𝑔0 𝑟 = -3mn2δr
𝛿𝑟
0
Where
g0 = Accl of gravity @ SC center of mass, m/s2
m = mass, kg
n = orbital mean motion, rad/s
r0 = orbit Semimajor axis, m
δF = Force on mass m due to vertical displacement, N
δr = radial displacement
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