Lecture 5-6
25
Astronautics 309 Lecture 5-6
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Transcript of Lecture 5-6
Astronautics 309 Lecture 5-6
Astronautics - Winter 2015
Outline
2
The Two-Body Problem
Position as a function of
Time
What other forces?
Circular orbit
Elliptical orbit Parabolic arc
Hyperbolic arc
The EoM
Astronautics - Winter 2015
The Orbit Formula (true anomaly)
3
The position as a function of true anomaly
v? =h
rRate of change of true anomaly
Removing position from the equation:
tp = 0
✓p = 0
Astronautics - Winter 2015
Circular Orbit
4
When (e=0) the integral is simple:
relation between true anomaly and time:
as a fraction of one period:
Astronautics - Winter 2015
Elliptical Orbit
5
From the integration:
The mean anomaly:
in elliptic orbit:
The mean motion:
Astronautics - Winter 2015
Elliptical Orbit
6
Kepler’s transformation:uniform circular motion for an ellipse
Eccentric anomaly
the relation between two geometries:
Astronautics - Winter 2015
Kepler’s Equation
7
Relation between true anomaly and eccentric anomaly:
Kepler’s equation
mean anomaly eccentric anomaly
Astronautics - Winter 2015
Example
8
Astronautics - Winter 2015
Example
9
Astronautics - Winter 2015
Parabolic Trajectory
10
when (e=1), the integration is simple:
mean anomaly:
Astronautics - Winter 2015
Example
11
Astronautics - Winter 2015
Hyperbolic Trajectory
12
for e>1, well!
defining the mean anomaly:
Astronautics - Winter 2015
Hyperbolic Trajectory
13
to simplify the integration:
cosh
2 F � sinh
2 F = 1defining an auxiliary angle F:
Astronautics - Winter 2015
simplifies:
Hyperbolic Trajectory
14
cosh
2 F � sinh
2 F = 1
hyperbola equation
when:
hyperbolic mean anomaly hyperbolic eccentric anomalyrelation between true anomaly and
hyperbolic eccentric anomaly
Astronautics - Winter 2015
Summary
15
similarity between ellipse an hyperbola
Astronautics - Winter 2015
Force Model
16
Newtonian motion:
r = F(t, r, r)/m
Two-Body Gravitational Force:
r = � µ
r3r
Other forces on a satellite in the Earth’s atmosphere:
• Geopotential (the effect of non-spheric body on the gravity) • Solar Radiation Pressure (SRP) • Atmospheric Drag
Mathematically modeling the acting forces on all the particles of a dynamical system.
Astronautics - Winter 2015
Geopotential
17
Using potential to derive equations of motion:
r = �rU
U = �µ
r
Gravitational Potential:
For a small element:
Astronautics - Winter 2015
Geopotential
18
We can use an polynomial expansion to write:
Using spherical coordinate:
where Legendre Polynomial
Astronautics - Winter 2015
Geopotential
19
Legendre Polynomial
Astronautics - Winter 2015
Geopotential
20
Using normalized coefficient:
Then, the force can be written as:
Astronautics - Winter 2015
Geopotential
21
The Earth:
Astronautics - Winter 2015
Geopotential
22
The Earth:
Astronautics - Winter 2015
Solar Radiation Pressure (SRP)
23
The pressure by absorbed and reflected photons
Using solar flux � =1
A
dE
dt
momentum by a single photon : ⌫ p⌫ =E⌫
c
dp =dE
c=
�
cAdtchange of the momentum for a small element:
F =dp
dt=
�
cAForce: Pressure: P =
�
c
� = 1367 Wm�2
P� = 4.56⇥ 10�6Nm�2At 1AU:
Astronautics - Winter 2015
Solar Radiation Pressure (SRP)
24
Absorption Reflection
Astronautics - Winter 2015
Atmospheric Drag
25
Depends on relative velocity
Assume that atmosphere rotates with the Earth
Drag force model:
