A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns....

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A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire de Météorologie Dynamique, CNRS 14/12/2010 1

Transcript of A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns....

Page 1: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns.

Application to Mars.

François Forget & Michel Capderou

Laboratoire de Météorologie Dynamique, CNRS

14/12/2010

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Page 2: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

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Atmospheric drag and velocity change Δv during an orbital pass through the atmosphere

Page 3: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Coupling an orbitography numerical model and an atmospheric model. Here:

Ixion orbitography(Capderou 2005)

http://climserv.ipsl.polytechnique.fr/ixion.html

Mars Climate database(Forget et al. 2006)

“Mars aerobraking simulator”

Page 4: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Example : Delta V during a 200 sols (1314 orbits) “drag compensated” aerobraking campaign

- MGS like shape- Inclination = 75°- Excentricity = 0.4- Initial conditions:• Ls=0°• Periapsis at 0°N, z=100km

Can we calculate Delta V without having to run complex models for quick look estimation and mission strategy design ?

Page 5: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Previous analytical work: very complex equations by King-Hele (1987) on the “contractions of orbit”

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Page 6: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Calculation of aerobraking Δv over one orbit

• Drag force on spacecraft: • Change of velocity a time t:

• Integration over one orbit (period T ) :

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Page 7: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

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v2

ρ

Page 8: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Calculation of aerobraking Δv over one orbit (2)

• Assuming that around perihelion the atmosphere is relatively isothermal (scale height H) :

• Integration of one orbit (period T ) :

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• Playing with the Keplerian equations (with simplification and development around periaspis) give the variation of altitude as function of time:

Page 9: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Density at perisapsis

Relative velocity at perisapsis

Scale height at periapsis (RT/g)

Page 10: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Validation : Delta V during a 200 sols (1314 orbits) “drag compensated” aerobraking campaign

- MGS like shape- Inclination = 75°- Excentricity = 0.4- Initial conditions:• Ls=0°• Periapsis at 0°N, z=100km

Page 11: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Validation : Delta V during a 200 sols (1314 orbits) “drag compensated” aerobraking campaign

- MGS like shape- Inclination = 75°- Excentricity = 0.4- Initial conditions:• Ls=0°• Periapsis at 0°N, z=100km

Page 12: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Statistics of errors (simple equation – reference model) over one Mars year simulations

Orbit Inclination:

Page 13: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Statistics of errors (simple equation – reference model) over one Mars year simulations

Page 14: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Impact of the relative wind

Spacecraft velocity Vspc(Galilean frame)

Atmospheric wind Va(axial component)

Atmospheric motion Vt(transverse component, galilean frame)

• One must take into account the motion of the atmosphere in addition to the spacecraft velocity Vspc.

• Notice that the transverse component Vt is always negligible, unlike the axial component Va, because V0

2 ~ (Vspc + Va)2 >> Vt2

Page 15: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Estimation of the relative wind• Case 1 : spacecraft velocity and atmospheric wind known from models:

• Case 2: neglect planet rotation and winds (use only spacecraft velocity in galilean frame)

• Case 3: atmosphere rotating like the solid planet

Page 16: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Polar orbits

The error resulting from the assumption on winds depends on the orbit inclination

Page 17: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Mean zonal wind predicted by the LMD Global Climate model

Page 18: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

Conclusions and perpectives• Velocity change Δv during an orbital pass through the

atmosphere can be computed using a simple equation with error < ~5% (better than error on density…)

• e.g. :

• This should be useful for preliminary mission design by non specialists, simple scaling,to develop autonomous aerobraking system, etc…

• Conversely, estimation of the velocity change provide an estimation of density at periapsis in real time.

• This equation should be valid on other planets

Page 19: A simple analytical equation to calculate the atmospheric drag during aerobraking campaigns. Application to Mars. François Forget & Michel Capderou Laboratoire.

• Thank you

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