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The hill's equations (after George William Hill (1838 to 1914)) describe track changes to a satellite within the co-rotating reference system. They can be used to calculate which further course (path and speed) a satellite will take when changing its speed.
They are the solution of the coupled system of equations:
x
¨
+
2
ω
z
˙
=
b
x
{\ displaystyle {\ ddot {x}} + 2 \ omega {\ dot {z}} = b_ {x}}
y
¨
+
ω
2
y
=
b
y
{\ displaystyle {\ ddot {y}} + \ omega ^ {2} y = b_ {y}}
z
¨
-
2
ω
x
˙
-
3
ω
2
z
=
b
z
{\ displaystyle {\ ddot {z}} - 2 \ omega {\ dot {x}} - 3 \ omega ^ {2} z = b_ {z}}
Orbital equations
x
(
ω
,
t
)
=
(
x
0
-
2
z
˙
0
ω
)
+
2
z
˙
0
ω
cos
ω
t
+
(
6th
z
0
+
4th
x
˙
0
ω
)
sin
ω
t
-
(
6th
z
0
+
3
x
˙
0
ω
)
ω
t
{\ displaystyle x (\ omega, t) = \ left ({x_ {0} -2 {\ frac {{\ dot {z}} _ {0}} {\ omega}}} \ right) +2 {\ frac {{\ dot {z}} _ {0}} {\ omega}} \ cos \ omega t + \ left ({6z_ {0} +4 {\ frac {{\ dot {x}} _ {0}} {\ omega}}} \ right) \ sin \ omega t- \ left ({6z_ {0} +3 {\ frac {{\ dot {x}} _ {0}} {\ omega}}} \ right) \ omega t}
z
(
ω
,
t
)
=
(
4th
z
0
+
2
x
˙
0
ω
)
+
z
˙
0
ω
sin
ω
t
-
(
3
z
0
+
2
x
˙
0
ω
)
cos
ω
t
{\ displaystyle z (\ omega, t) = \ left ({4z_ {0} +2 {\ frac {{\ dot {x}} _ {0}} {\ omega}}} \ right) + {\ frac {{\ dot {z}} _ {0}} {\ omega}} \ sin \ omega t- \ left ({3z_ {0} +2 {\ frac {{\ dot {x}} _ {0}} {\ omega}}} \ right) \ cos \ omega t}
Velocity equations
x
˙
(
ω
,
t
)
=
-
3
x
˙
0
-
6th
ω
z
0
-
2
z
˙
0
sin
ω
t
+
(
6th
ω
z
0
+
4th
x
˙
0
)
cos
ω
t
{\ displaystyle {\ dot {x}} (\ omega, t) = - 3 {\ dot {x}} _ {0} -6 \ omega z_ {0} -2 {\ dot {z}} _ {0 } \ sin \ omega t + \ left ({6 \ omega z_ {0} +4 {\ dot {x}} _ {0}} \ right) \ cos \ omega t}
z
˙
(
ω
,
t
)
=
(
3
ω
z
0
+
2
x
˙
0
)
sin
ω
t
+
z
˙
0
cos
ω
t
{\ displaystyle {\ dot {z}} (\ omega, t) = \ left ({3 \ omega z_ {0} +2 {\ dot {x}} _ {0}} \ right) \ sin \ omega t + {\ dot {z}} _ {0} \ cos \ omega t}
Examples
Radial maneuver
Change in orbit of a satellite with radial change in speed
A radial maneuver results in an ellipse with a ratio of 1: 2 .
Initial conditions :
Position:
Speed:
(
x
;
z
)
=
(
0
;
0
)
{\ displaystyle (x; z) = (0; 0)}
(
x
˙
;
z
˙
)
=
(
0
;
Δ
v
)
{\ displaystyle ({\ dot {x}}; {\ dot {z}}) = (0; \ Delta v)}
Orbital equations :
x
=
2
Δ
v
ω
(
cos
ω
t
-
1
)
{\ displaystyle x = 2 {\ frac {\ Delta v} {\ omega}} \ left ({\ cos \ omega t-1} \ right)}
z
=
Δ
v
ω
sin
ω
t
{\ displaystyle z = {\ frac {\ Delta v} {\ omega}} \ sin \ omega t}
Tangential maneuver
Change in orbit of a satellite with tangential change in speed
A tangential maneuver results in a cycloidal path.
Initial conditions :
Position:
Speed:
(
x
;
z
)
=
(
0
;
0
)
{\ displaystyle (x; z) = (0; 0)}
(
x
˙
;
z
˙
)
=
(
Δ
v
;
0
)
{\ displaystyle ({\ dot {x}}; {\ dot {z}}) = (\ Delta v; 0)}
Orbital equations :
x
=
4th
Δ
v
ω
sin
ω
t
-
3
Δ
v
⋅
t
{\ displaystyle x = 4 {\ frac {\ Delta v} {\ omega}} \ sin \ omega t-3 \ Delta v \ cdot t}
z
=
2
Δ
v
ω
(
1
-
cos
ω
t
)
{\ displaystyle z = 2 {\ frac {\ Delta v} {\ omega}} \ left ({1- \ cos \ omega t} \ right)}
x
˙
1
=
-
3
x
˙
0
+
4th
x
˙
0
cos
ω
t
{\ displaystyle {\ dot {x}} _ {1} = - 3 {\ dot {x}} _ {0} +4 {\ dot {x}} _ {0} \ cos \ omega t}
After half a revolution, the satellite moves seven times
Δ
v
{\ displaystyle \ Delta v}
in the opposite direction in the co-rotating reference system :
x
˙
1
(
t
=
T
2
)
=
-
3
Δ
v
-
4th
Δ
v
=
-
7th
Δ
v
{\ displaystyle {\ dot {x}} _ {1} \ left ({t = {\ frac {T} {2}}} \ right) = - 3 \ Delta v-4 \ Delta v = -7 \ Delta v}
Hohmann maneuvers
Execution of the Hohmann crossing with two maneuvers
Two tangential maneuvers are performed at the Hohmann transition .
See also: Hill's differential equation (three-body problem)
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