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# lecture27 - Computation of drag force Consider a small...

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Computation of drag force Consider a small atmospheric mass element Δ m that hits the spacecraft’s cross-sectional area A in a time interval Δ t , Δ m = ρAv rel Δ t with ρ the atmospheric density at the satellite’s location and v rel the speed of the satellite rela- tive to the atmosphere. The impulse (change of momentum) caused by this mass element is just the mass element times the velocity, I = Δ mv rel = ρAv 2 rel Δ t. This is true if the spacecraft presents a perpen- dicular, flat, absorbing surface to the oncoming wind. In reality, there is a coefficient of drag, dependent on material properties, shape, and attitude of the spacecraft that modifies this with a coefficient of drag C D , I = 1 2 C D ρAv 2 rel Δ t. L. Healy – ENAE404 – Spring 2007 – Lecture 27 (May 3) 1

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Acceleration due to drag The force is the impulse divided by the time interval so acceleration due to drag is ¨ r drag = - 1 2 C D A m B ρv 2 rel ˆ v rel . The ballistic coefficient B = C D A/m is mostly a property of the satellite, so the acceleration ¨ r drag = - 1 2 ρBv 2 rel ˆ v rel is composed of B , mostly a property of the spacecraft and its attitude, ρ , a property of the atmosphere; depends on location and conditions, particularly be- havior of the sun, v rel , a property of its orbital motion relative to the atmosphere. L. Healy – ENAE404 – Spring 2007 – Lecture 27 (May 3) 2
Drag variables The parameters that contribute to drag are: B = C D A/m = (dimensions M - 1 L 2 ); C D = coefficient of drag, dimensionless, depends on cross-sectional shape of satel- lite. Frequently, we take C D 2 if it’s not otherwise known. It is 1.0 for a per- fectly absorbing sphere. m = Mass of satellite (dimensions [ M ]), A = cross-sectional area of satellite in direction (dimensions [ L 2 ]); for a non- spherical satellite, this will depend on attitude ρ = atmospheric density (dimensions [ ML - 3 ]), v rel velocity of satellite relative to atmo- sphere. Could be taken as relative to IJK, not really accurate because atmosphere ro- tates with earth and has fluctuations (wind). L. Healy – ENAE404 – Spring 2007 – Lecture 27 (May 3) 3

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Drag coefficients Drag coefficients vary with the shape, material and orientation of the vehicle. Some values C D B ( m 2 / kg ) Typical 2–4 0 . 001 to 0 . 1 Hubble 3.33-4 0 . 0052 to 0 . 034 For Wednesday (May 3, 2006), ISS B = 1 . 59 × 10 - 3 m 2 / kg , Hubble B = 5 . 35 × 10 - 4 m 2 / kg . Computed by reading the B* (see Lecture 4 “drag term”) in the twoline elements from space- track.org and multiplying by 12.7416. L. Healy – ENAE404 – Spring 2007 – Lecture 27 (May 3) 4
Exponential atmospheric model (static) A simple, time-independent model of atmo- spheric density may be derived from the uni- versal gas law ρ = ρ 0 e - ( z - z 0 ) /H with H = scale height, ρ 0 = density at reference altitude z 0 , z = altitude of satellite.

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lecture27 - Computation of drag force Consider a small...

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