User:Rat526/sandbox

Overview:
Apse line rotation is a coplanar, elliptical orbital transfer where the apse line is rotated around the focus of an ellipse to create a different elliptical orbit around the same focus. The orbital transfer is accomplished with a single impulse maneuver, and the two orbits must intersect. A Hohmann transfer is not possible for this orbital maneuver. η is angle of rotation around the focus or the difference between the true anomalies of the point of intersection, measured from the periapse of each orbit.

Basic Elliptical Orbit:
An elliptical orbit has several geometric features essential to apse line rotation. The apse line is the major axis of an ellipse connecting the focus, F, and the unoccupied empty focus, F’. The periapsis is the point of intersection of the apse line and ellipse closest to the focus, and the distance between from the focus to the periapsis is denoted by rP. Alternatively, the apse line and ellipse intersection furthest from the focus is the apoapsis where the distance is rA. The true anomaly, θ, is an angle defining a position on the ellipse measure from the periapsis, rotating counterclockwise about the focus. The apse line is also the eccentricity vector, whose scalar value, e, is a function of the position or radius on the orbit, r, and gravitational parameter, µ, true anomaly, θ, and relative angular momentum, h. Relative angular momentum is a function of the transverse velocity of the body in orbit (component of velocity perpendicular to the outward radial from the focus to the body in orbit).

$$h=rv_\perp$$

The eccentricity, relative angular momentum, and gravitational parameter are constants for a specific orbit. These variables are typically arranged in the format of the orbit equation, which defines the path of the body in orbit around a body positioned at the focus (1).

r= h^2/μ  1/(1+e cos⁡θ )

Apse Line Rotation:
When examining apse line rotation, two elliptical orbits are superimposed sharing a common focus. The size and the shape of the orbit may also change in the transfer, meaning that the eccentricity and angular moments are not necessarily constant between the two orbits. The true anomalies at the intersection of the two orbits, point I, are termed θ1 and θ2. The rotation angle, η, is the difference between the two true anomalies.

$$\eta= \theta_1-\theta_2$$

Calculation Details:
An apse line rotation orbital transfer can be investigated from two sets of circumstances:

(1) The apse line rotation and the eccentricity and angular momentum of each orbit are used to find the true anomaly of each orbit, i.e. given a current and desired orbit, the impulse and where it needs to occur are determined.

[θ_1,θ_2,Δv ]=f(η,r_(P_1 ),r_(A_1 ),r_(P_2 ),r_(A_2 ) )=f(η,h_1,e_1,h_2,e_2 )

(2) The true anomaly of an impulse maneuver is known and the rotation and eccentricity of the second orbit are to be determined, i.e. from a known orbit, a known impulse is applied at a known location to find the shape of a new orbit.

[η,e_2,r_(P_2 ),r_(A_2 ) ]=f(θ_1,r_(P_1 ),r_(A_1 ),Δv)=f(θ_1,h_1,e_1,Δv)

Find the true anomaly and impulse (1):
If not provided, the eccentricities and relative angular moment can be determine from the radii of perigee and apogee as follows.

$$e=(r_A-r_P)/(r_A+r_P )$$

h=√(r_P μ(1+e))

The radius of the point I is equal on each orbit, set the orbit equation for each orbit equal.

r_I )_1=r_I )_2

〖h_1〗^2/μ 1/(1+e_1  cos⁡〖θ_1 〗 )=  〖h_2〗^2/μ  1/(1+e_2  cos⁡〖θ_2 〗 )

Rearrange the orbit equation to get an equation in terms of θ1, knowing that θ2=θ1-η.

〖〖h_2〗^2 e〗_1 cos⁡〖θ_1 〗-〖〖h_1〗^2 e〗_2  cos⁡〖θ_2 〗= 〖h_1〗^2-〖h_2〗^2 〖〖h_2〗^2 e〗_1 cos⁡〖θ_1 〗-〖〖h_1〗^2 e〗_2  cos⁡(θ_1-η)= 〖h_1〗^2-〖h_2〗^2

Apply the trig identity cos⁡〖θ_1-η= cos⁡〖θ_1 〗 cos⁡η+sin⁡〖θ_1 〗  sin⁡η 〗. 〖〖h_2〗^2 e〗_1 cos⁡〖θ_1 〗-〖〖h_1〗^2 e〗_2 (cos⁡〖θ_1 〗  cos⁡η+sin⁡〖θ_1 〗  sin⁡η )= 〖h_1〗^2-〖h_2〗^2 〖〖h_2〗^2 e〗_1 cos⁡〖θ_1 〗-〖〖h_1〗^2 e〗_2  cos⁡〖θ_1 〗  cos⁡η-〖〖h_1〗^2 e〗_2  sin⁡〖θ_1 〗  sin⁡η= 〖h_1〗^2-〖h_2〗^2 (〖〖h_2〗^2 e〗_1-〖〖h_1〗^2 e〗_2 cos⁡η )  cos⁡〖θ_1 〗-〖〖〖h_1〗^2 e〗_2  sin⁡η sin〗⁡〖θ_1 〗  = 〖h_1〗^2-〖h_2〗^2

Rearranged and simplified to 		a cos⁡〖θ_1 〗+〖b sin〗⁡〖θ_1 〗 = c Where					a= 〖〖h_2〗^2 e〗_1-〖〖h_1〗^2 e〗_2  cos⁡η b= -〖〖h_1〗^2 e〗_2 sin⁡η c= 〖h_1〗^2-〖h_2〗^2

Below is the solution for θ1 using the relative angular momentums, eccentricities, and apse line rotation. Two solutions exist for both intersections of the orbits, points I and J.

θ_1=ϕ±cos^(-1)⁡(c/a cos⁡ϕ ) ϕ=tan^(-1)⁡〖b/a〗

With a solution for θ1, the orbit equation can be solved for the radius at the intersection.

r_I )_1=〖h_1〗^2/μ 1/(1+e_1  cos⁡〖θ_1 〗 )

Then solve for the velocity components and flight path angle for each orbit, v_(⊥_1 )=h_1/r v_(r_1 )=μ/h_1 e_1  sin⁡〖θ_1 〗 γ_1=tan^(-1)⁡〖v_(r_1 )/v_(⊥_1 ) 〗 v_(⊥_2 )=h_2/r v_(r_2 )=μ/h_2 e_2  sin⁡(θ_1-η) γ_2=tan^(-1)⁡〖v_(r_2 )/v_(⊥_2 ) 〗 And the speed of the object in each orbit. v_1=√(〖v_(r_1 )〗^2+〖v_(⊥_1 )〗^2 ) v_2=√(〖v_(r_2 )〗^2+〖v_(⊥_2 )〗^2 ) Finally, calculate the impulse. Δv=√(〖v_1〗^2+〖v_2〗^2-2v_1 v_2 cos⁡(γ_2-γ_1 ) )

Find the new orbit (2):
The eccentricity and relative angular momentum of the initial orbit can again be found using the perigee and apogee radii. Then consider general expressions for velocity components and relative angular momentum. v_r=μ/h e sin⁡θ			v_⊥=h/r h_2=r(v_⊥+Δv_⊥ )= h_1+r Δv_⊥ v_(r_2 )=v_(r_1 )+∆v_r

Set expressions for vr2 equal. v_(r_1 )+∆v_r=μ/h_2 e_2  sin⁡〖θ_2 〗 Plug in expressions for h2 and vr1. μ/h_1 e_1  sin⁡〖θ_1 〗+∆v_r=μ/(h_1+r Δv_⊥ ) e_2  sin⁡〖θ_2 〗

Solve for sin(θ2). sin⁡〖θ_2 〗=1/e_2  (μe_1  sin⁡〖θ_1 〗+h_1 ∆v_r )(h_1+r Δv_⊥ )/(μh_1 )

As in case 1, consider setting the orbit equation for each orbit equal. r_I )_1=r_I )_2 〖h_1〗^2/μ 1/(1+e_1  cos⁡〖θ_1 〗 )=  〖h_2〗^2/μ  1/(1+e_2  cos⁡〖θ_2 〗 )

Solve for cos(θ2). cos⁡〖θ_2 〗=1/e_2  (〖h_2〗^2+〖h_2〗^2 e_1  cos⁡〖θ_1 〗-〖h_1〗^2)/〖h_1〗^2

Plug in expressions for h2. cos⁡〖θ_2 〗=1/e_2  ((h_1+r Δv_⊥ )^2+(h_1+r Δv_⊥ )^2 e_1  cos⁡〖θ_1 〗-〖h_1〗^2)/〖h_1〗^2

Expand and simplify. cos⁡〖θ_2 〗=1/e_2  ((〖2h〗_1+r Δv_⊥ )  rΔv_⊥+(h_1+r Δv_⊥ )^2 e_1  cos⁡〖θ_1 〗)/〖h_1〗^2

Apply the trig identity tan⁡θ=sin⁡θ/cos⁡θ and simplify. tan⁡〖θ_2=h_1/μ〗 (μe_1  sin⁡〖θ_1 〗+h_1 ∆v_r )(h_1+r Δv_⊥ )/((〖2h〗_1+r Δv_⊥ )  rΔv_⊥+(h_1+r Δv_⊥ )^2 e_1  cos⁡〖θ_1 〗 )

Further simplify knowing that h_1 v_(r_1 )=μe_1 sin⁡〖θ_1 〗 and h_1=〖rv〗_(⊥_1 ). tan⁡〖θ_2=〖v_(⊥_1 )〗^2/(μ/r)〗 (v_(r_1 )+∆v_r )(v_(⊥_1 )+Δv_⊥ )/((2v_(⊥_1 )+Δv_⊥ )  Δv_⊥+(v_(⊥_1 )+Δv_⊥ )^2 e_1  cos⁡〖θ_1 〗 )

This final equation determines the true anomaly for the new orbit with only inputs of the impulse and the object’s true anomaly and speed when it was applied. The second orbit is fully defined by rearranging the equations for sin⁡〖θ_2 〗 or cos⁡〖θ_2 〗 to solve for e2. e_2=1/sin⁡〖θ_2 〗  (μe_1  sin⁡〖θ_1 〗+h_1 ∆v_r )(h_1+r Δv_⊥ )/(μh_1 )=1/cos⁡〖θ_2 〗   ((〖2h〗_1+r Δv_⊥ )  rΔv_⊥+(h_1+r Δv_⊥ )^2 e_1  cos⁡〖θ_1 〗)/〖h_1〗^2

Then the radii of perigee and apogee can be calculated for orbit 2. r_(P_2 )=〖h_2〗^2/μ 1/(1+e_2 ) r_(A_2 )=〖h_2〗^2/μ 1/(1-e_2 )

Impulse at Periapsis
If the impulse is applied at the periapsis where the true anomaly is zero, then the initial radial velocity will also be zero. Furthermore if the impulse is all in the radial direction (Δv_⊥=0) then the tan⁡〖θ_2 〗 equation simplifies to solve for the apse line rotation, η. tan⁡〖η=-(rΔv_(⊥_1 ))/(μe_1 ) Δv_r 〗

Unchanged Eccentricity and Angular Momentum
If angular moment remains unchanged between the orbits, solving the true anomaly for case 1 simplifies to θ1 = φ because the c term reduces to zero. θ_1= ϕ=tan^(-1)⁡〖b/a〗