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# lecture06 - Computation of orbital elements from Cartesian...

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Computation of orbital elements from Cartesian Ref.: Curtis Sec. 4.4, Vallado Section 2.5 We know about orbital elements, but how do we find them? If we’re given Cartesian state vector (position vector r and velocity vector ˙ r ), we can compute them using the conserved quantities. L. Healy – ENAE404 – Spring 2007 – Lecture 6 (Feb. 13) 1

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IJK to Classical Elements To compute the classical elements a, e, i, Ω , ω, θ ( t ) from the Cartesian state vectors r , ˙ r , we will need to know how to take dot and cross prod- ucts in Cartesian coordinates. Start with angular momentum vector h = r × ˙ r semilatus rectum p = h 2 eccentricity vector e = ˙ r × h μ - ˆ r node direction ˆ n = K × h . ˆ K is the unit vector along the polar axis. L. Healy – ENAE404 – Spring 2007 – Lecture 6 (Feb. 13) 2
Semimajor axis a and eccentricity e Eccentricity can be computed as magnitude of the eccentricity vector, e = | e | . The semimajor axis can be computed in one of two ways: a = 2 r - v 2 μ - 1 = p 1 - e 2 . Now we need to find the angular elements i , Ω, ω , θ . L. Healy – ENAE404 – Spring 2007 – Lecture 6 (Feb. 13) 3

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Finding angular elements After finding the unit vectors compute angles, RAAN: Ω = arctan(ˆ n · ˆ J , ˆ n · ˆ I ) or cos Ω = n I and if n J < 0, the angle is in the lower halfplane π < Ω < 2 π . Inclination: angular momentum direction gives inclination, since it’s always perpen- dicular to the orbital plane: cos i = ˆ h · ˆ K No need for arctangent here, 0 i 180 . Argument of perigee is the angle from the ascending node to the eccentricity vector, cos ω = ˆ n · ˆ e If the third component of e is negative, e K < 0, then the argument of perigee is below the equatorial plane, π ω 2 π . L. Healy – ENAE404 – Spring 2007 – Lecture 6 (Feb. 13) 4
Finding angular elements cont’d. True anomaly cos θ = ˆ r · ˆ e = 1 e p r - 1 If r · ˙ r < 0, then the true anomaly is below the major axis, π θ < 2 π , because (recall r ˙ r = r · ˙ r ) from perigee to apogee 0 < θ < π , dis- tance r is positive and increasing, so r · ˙ r = r ˙ r > 0 from apogee to perigee π θ < 2 π , dis- tance r is decreasing, so r · ˙ r = r ˙ r < 0. L. Healy – ENAE404 – Spring 2007 – Lecture 6 (Feb. 13) 5

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Coordinate systems We have computed the orbital elements from the Cartesian IJK state vector. What about going the other direction? It is possible to compute the inverse transformation, but we shall need to discuss coordinate systems and transformations first. So far we have used the coordinate system IJK but have skirted what coordinate systems are about and how to get from one to another (transformation). This requires some atten- tion, because coordinates are an essential part of everything we do in astrodynamics. There are many different kinds of coordinate systems; each one is “natural” for some pur- poses but not for others. Fortunately, they can be categorized, so that understanding what they are and what the transformation rules can be determined without (too much!) confusion.
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lecture06 - Computation of orbital elements from Cartesian...

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