L5_Astrodynamics-2 - Swedish Institute of Space Physics - Uppsala

Compass-2 

Microsatellite Technology 

L5 – Astrodynamics 2 

Jan Bergman, January 28, 2010 

Adapted after Christian Hånberg’s lecture 2008 

E-mail: jb@irfu.se 

Tel. +46(0)18-4715916 

Room: Å84116 

Swedish Institute of Space 

Physics, Uppsala, Sweden 

http://www.irfu.se

Part 1 

L5 – Astrodynamics Part 2 


• Repetition Classical Orbital Elements 

• Orbit Maneuvering ch. 6.3 

Coplanar Orbit transfers 

Orbit Plane Changes 

Orbit Rendezvous 

• Launch Windows ch. 6.4 

• Orbit Maintenance ch. 6.5 

Swedish Institute of Space Physics, Uppsala, Sweden | www..irfu.se

Classical orbital elements 


• Shape 

a: Semi-major axis 

e: Eccentricity 

• Orientation of orbital plane 

i: Inclination (vertical tilt) 

• i < 90 prograde or direct 

• i > 90 retrograde 

Ω: Right ascension of the 

ascending node 

• Line of node - equator crossings 

• Orientation of the ellipse 

 

ω: Argument of perigee 

• Angle between ascending node 

and semi-major axis 

• Where (in time) 

 

v: True anomaly 

• Angle between satellite and 

perigee in the orbit plane 


Orbit Maneuvering 


• A some point in most satellites life, one or more 

of the orbital elements have to be changed. 

This can be due to 

• orbit corrections, 

• transfer the satellite between different orbits, 

• rendezvous or intercept with another satellite. 

• To change the satellites orbit, we have to 

change the satellites velocity vector. 

 

∆v = v NEED - v CURRENT 


Orbit maneuvering 


• Any maneuver 

changing the orbit 

of the satellite 

must occur at a 

point where the 

old orbit intersects 

the new. 

• If the two orbits 

don’t intersect we 

must use a 

transfer orbit, 

intersecting both 

orbits. 

1. Initial orbit 

2. Transfer orbit 

3. Final orbit 


The Vis-viva equation 


• Total energy = kinetic + potential energy 

• Orbiting body: 

• Total energy = half the potential energy at 

the average distance (semi-major axis) 

1 2 μm 

μm 

E = mv − =− 

2 r 2a 

• Solving for v yields the Vis-viva equation 

2 1 

v = μ ⎛ 

⎜ − 

⎞ 

⎟ 

⎝r 

a⎠ 

• Vis-viva – “it’s alive” (Newton) 

• Shows that orbit speed is inversely 

proportional to the square root of the radius 

• Applications 

• Delta-v calculations, Escape, transfers, etc. 


Coplanar Orbit Transfers 


• Three main types of coplanar orbit transfers: 

Hohmann Transfer 

• The most fuel efficient transfer between 

two circular coplanar orbits. 

One-Tangent Burn 

• The quickest transfer between two 

coplanar orbits. 

Spiral Transfer 

• Using a constant low-thrust burn, which 

results in a spiral transfer. 


Hohmann Transfer 


• Represents the 

most fuel 

efficient transfer 

between two 

circular coplanar 

orbits. 

• When going 

from a smaller to 

a larger orbit, 

the velocity 

change is applied 

in the direction 

of the motion 

and vice versa 


Hohmann Transfer Derivation 


• Using the Vis-viva 

equation it can be 

shown that for a 

Hohmann transfer 

Δ V = 

μ ⎛ 2 R ' ⎞ 

1 

R ⎜ 

− 

R R' 

⎟ 

⎝ + ⎠ 

Δ V ' = 

μ ⎛ 2R 

⎞ 

1 

R' 

⎜ 

− 

R+ 

R' 

⎟ 

⎝ ⎠ 

• Home assignment! 


One-Tangent Burn 


• Represents the 

quickest transfer 

between two 

coplanar orbits. 

• Tangential burn 

anywhere in the 

orbit. 

• Allows us to 

choose transfer 

orbit size and 

time for transition. 


Spiral transfer 


• Using a constant lowthrust 

burn, which 

results in a spiral 

transfer 

• This burn can be 

achieved with light 

ion-engines. 

• Uses up more energy 

than Hohmanntransfer, 

since the 

burn is continuous. 


Orbit plane changes 


• To change the orientation of the satellite, we 

have to add a component to the velocity vector, 

perpendicular to the orbital plane 

• If the size of the orbit is constant, we have a 

simple plane change. 

• If both the size and orientation of the orbit is 

changed, we have a combined plane change. 


Orbit Rendezvous 


• Intercepting with other objects in space is a 

bit more complicated and requires a phasing 

orbit. 


Launch Windows 


• The launch window is the time frame in which we can 

launch into the desired orbital plane. 

• For a launch window to exist, the launch site must pass 

through the orbital plane. 

• If we put i = inclination and L = latitude 

No launch window exists if L > i for direct orbits, or 

L > 180 deg - i for retrograde orbits. 

• Direct orbits, is when the satellite moves eastward. 

• Retrograde orbits, is when it moves westward. 

One launch window exists if L = i or L = 180 deg - i 

Two launch windows exists if L 


Launch Windows 



Orbit maintenance 


• Some satellites require orbital adjustments 

to correct perturbed orbital elements. 

• Two particular cases are satellites with 

repeating ground tracks and 

geosynchronous equatorial satellites. 


Repeating Ground Track 


• A satellite has a repeating ground track if 

it has exactly an integer number of 

revolutions per integer number of days. 

P = (m sidereal days) / (k sidereal days) 

• m,k = integers, 

• P = period 

• Perturbations are mainly caused by the 

Earth’s oblateness and have to be 

corrected to maintain the period. 


Repeating Ground Track 



Geosynchronous orbits 


• Fix orbit, with respect to 

the Earths surface. i.e. 

rotates with the same 

speed as the Earth. 

• Altitude maintained by the 

centripetal force canceling 

out the gravitational pull. 

• Perturbations mainly 

causes by the 

nonspherical Earth and 

third body pull from the 

Sun and the moon. 

• Calculations! 


End of L5 and Astrodynamics 2 

Jan Bergman 

jb@irfu.se

L5_Astrodynamics-2 - Swedish Institute of Space Physics - Uppsala

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