David Sims

Time to Fall in a Plunge Orbit The moon doesn't fall on Earth because it has enough transverse velocity to remain in an orbit around Earth, and because angular momentum is conserved as a natural law. Now, if you removed all of the moon's transverse velocity when the moon reached its aphelion, leaving the moon momentarily stationary with respect to Earth, then the moon and Earth would indeed fall together. How long would it take? Two bodies having a total mass M (i.e., M=M₁+M₂) are initially at rest, separated by a distance d, in vacuum, and isolated from all forces except their mutual gravitational attraction. Find the time elapsed from the initial moment to the moment at which the separation is r, such that 0<r<d. The problem involves the conservation of energy in a gravity dominated physical system, so we begin with the Vis Viva equation, solved for the speed in orbit: v = √[GM(2/r − 1/a)] where G = 6.6743e-11 m³ kg⁻¹ sec⁻² The eccentricity of a plunge orbit is one, just as it is for a parabolic orbit. Why? Because for an elliptical orbit, e = √(1 − b²/a²) where b is the length of the semiminor axis, and a is the length of the semimajor axis. As b→0, the elliptical orbit becomes a plunge orbit, and e→1. The apoapsis separation of M₁ and M₂ is therefore twice the semimajor axis. e = 1 d = a(1+e) d = 2a a = d/2 And, since the angular momentum in a plunge orbit is zero, all of the motion therein is radial, and so v = ∂r/∂t Therefore, the Vis Viva equation can be rewritten as ∂r/∂t = √[2GM(1/r − 1/d)] which is a non-linear differential equation with variables separable. We shall integrate both sides over the entire course of the fall, from the apoapsis of the plunge orbit to contact, assuming that both M₁ and M₂ are point masses. ∫(t₀,tᵢ) ∂t = − ∫(rᵢ,d) { ∂r / √[2GM(1/r − 1/d)] } The minus sign appears because the actual motion in the plunge orbit begins at d and ends at rᵢ, and I flipped those limits. However, hereafter I'll omit the limits and treat the indefinite integral. tᵢ−t₀ = −√[d/(2GM)] ∫ ∂r/√(d/r−1) We make this substitution: u = √(d/r−1) ∂u/∂r = −½ d r⁻²/√(d/r−1) r = d/(u²+1) ∂r = (−2d) u [∂u/(u²+1)²] Plugging them in, we get t−t₀ = 2d √[d/(2GM)] ∫ ∂u/(u²+1)² Integrating by parts, t−t₀ = 2d √[d/(2GM)] { ½ ∫ [∂u/(u²+1)] + ½ u/(u²+1) } t−t₀ = d √[d/(2GM)] { u/(u²+1) + ∫ [∂u/(u²+1)] } We notice an integral-trigonometric identity. ∫ ∂u/(u²+1) = arctan u So, t−t₀ = d √[d/(2GM)] { u/(u²+1) + arctan u } Reversing the substitution, t−t₀ = d √[d/(2GM)] { √(d/r−1)/[(d/r−1)+1] + arctan √(d/r−1) } After simplification, we get as the general solution for the time to fall in a plunge orbit: tᵢ−t₀ = √[d/(2GM)] { √(rᵢd−rᵢ²) + d arccos √(rᵢ/d) } In the case where the motion in a plunge orbit begins at the apoapsis and ends in contact (assuming point masses), rᵢ = 0 tᵢ−t₀ = π √[d³/(8GM)] When determining the time to fall from r₁ to r₂, such that d>r₁>r₂>0, t₂−t₁ = (t₂−t₀) − (t₁−t₀) Let's consider what happens when we let the moon drop straight down from its apogee to first contact between the two bodies, allowing them to keep their actual size and shape. M = 6.0483e+24 kilograms d = 405,503,560 meters r₁ = 8,108,400 meters t₁−t₀ = 450871.423 seconds = 125.242062 hours Now let's run the fall again, from apogee to center-to-center contact, assuming that Earth and moon are both point masses. M = 6.0483e+24 kilograms d = 405,503,560 meters r₂ = 0 t₂−t₀ = 451416.430 seconds = 125.393453 hours The difference is about 9 minutes and 5 seconds. As another example, let's find out how much time is required for the separation to decrease from r₁=d/2 to r₂=d/3. t₁−t₀ = √[d/(2GM)] { √(r₁d−r₁²) + d arctan √(d/r₁−1) } t₁−t₀ = √[d/(2GM)] { √(d²/2−d²/4) + d arctan √(2d/d−1) } t₁−t₀ = √[d/(2GM)] { √(d²/4) + d arctan √(2−1) } t₁−t₀ = √[d/(2GM)] { d/2 + dπ/4 } t₁−t₀ = {1/2 + π/4} √[d³/(2GM)] t₂−t₀ = √[d/(2GM)] { √(r₂d−r₂²) + d arctan √(d/r₂−1) } t₂−t₀ = √[d/(2GM)] { √(d²/3−d²/9) + d arctan √(3d/d−1) } t₂−t₀ = √[d/(2GM)] { √(2d²/9) + d arctan √(3−1) } t₂−t₀ = √[d/(2GM)] { √(2d²/9) + d arctan √(2) } t₂−t₀ = {√(2/9) + arctan √(2)} √[d³/(2GM)] t₂−t₁ = {√(2/9) + arctan √(2) − 1/2 − π/4} √[d³/(2GM)] t₂−t₁ ≈ 0.141322975518 √[d³/(2GM)] As a teaser, I'll ask readers to calculate the fraction of d that is closed in half of the total amount of time required for two point masses to fall to contact.

Determining the Keplerian Elements of an Elliptical Orbit from Four Observations Presented hereafter is a method for determining a preliminary heliocentric orbit from four geocentric directions of a sun-orbiting object at four distinct times of observation. I take this method after that presented in chapter six of The Determination of Orbits by A.D. Dubyago. The initial data t₁, X⊕₁, Y⊕₁, Z⊕₁, α₁, δ₁ t₂, X⊕₂, Y⊕₂, Z⊕₂, α₂, δ₂ t₃, X⊕₃, Y⊕₃, Z⊕₃, α₃, δ₃ t₄, X⊕₄, Y⊕₄, Z⊕₄, α₄, δ₄ The times of observation, tᵢ, are given in Julian date format. The vectors [X⊕ᵢ,Y⊕ᵢ,Z⊕ᵢ] are the positions of the Earth in heliocentric ecliptic coordinates, with the components being in astronomical units. The αᵢ are the geocentric right ascensions for Ceres. The δᵢ are the geocentric declinations for Ceres. The time intervals should be about 0.5% to 1% of the object's period (estimated as 8 to 16 days for a main belt asteroid), should be near opposition with the sun, but should NOT span an apside of the object's orbit. The precision in right ascension should be 0.01 seconds or better, and the precision in declination should be 0.1 arcseconds or better. The Earth's orbital mean motion, κ = 0.01720209895 radians/day We will use a single value for the obliquity of the ecliptic to transform all four of the observation angles from celestial coordinates to ecliptic coordinates. First, we find the middle of the observation time window in units of 10000 years since 1 January 2000, and then we use that number to find the obliquity using the 10-degree polynomial fit of J. Laskar. t = (t₁ + t₄)/2 T = (t − 2451545)/3652500 The obliquity in seconds of arc is ε" = 84381.448 − 4680.93 T − 1.55 T² + 1999.25 T³ − 51.38 T⁴ − 249.67 T⁵ − 39.05 T⁶ + 7.12 T⁷ + 27.87 T⁸ + 5.79 T⁹ + 2.45 T¹⁰ The obliquity in radians is ε = (π / 648000) ε" Although Earth's axial tilt (i.e. the obliquity of the ecliptic) does change over time, it changes so slowly that the difference will almost always be negligible across the t₁ to t₄ time window. However, in the rare case when this isn't true, separate evaluations of the obliquity will have to be made for each time of observation. The geocentric positions of the sun in celestial coordinates are (for i = 1 to 4) Xᵢ = −X⊕ᵢ Yᵢ = −Y⊕ᵢ cos ε + Z⊕ᵢ sin ε Zᵢ = −Y⊕ᵢ sin ε − Z⊕ᵢ cos ε The geocentric unit vectors in the direction of the target object, in celestial coordinates, are (for i = 1 to 4) aᵢ = cos αᵢ cos δᵢ bᵢ = sin αᵢ cos δᵢ cᵢ = sin δᵢ The squares of the Sun-Earth distances at times t₁ and t₄ are R₁² = X₁² + Y₁² + Z₁² R₄² = X₄² + Y₄² + Z₄² The values of 2Rᵢ cos θᵢ at times t₁ and t₄ are 2R₁ cos θ₁ = −2 ( a₁X₁ + b₁Y₁ + c₁Z₁ ) 2R₄ cos θ₄ = −2 ( a₄X₄ + b₄Y₄ + c₄Z₄ ) where θ is the supplementary angle to the sun-Earth-object angle. We find some time differences: τ₁ = κ (t₄−t₂) τ₂ = κ (t₂−t₁) τ₃ = κ (t₄−t₁) τ₄ = κ (t₄−t₃) τ₅ = κ (t₃−t₁) We find this pair of determinants: Φ = a₂b₄−b₂a₄ φ = a₃b₄−b₃a₄ We calculate some intermediate quantities: A = ( a₁b₂ − b₁a₂ ) / Φ B = ( a₂Y₁ − b₂X₁ ) / Φ C = ( b₂X₂ − a₂Y₂ ) / Φ D = ( a₂Y₄ − b₂X₄ ) / Φ A' = ( a₁b₃ − b₁a₃ ) / φ B' = ( a₃Y₁ − b₃X₁ ) / φ C' = ( b₃X₃ − a₃Y₃ ) / φ D' = ( a₃Y₄ − b₃X₄ ) / φ And then more intermediate quantities: E = τ₁/τ₂ F = (4/3) τ₁τ₃ G = AE H = F(A−G) I = 4Aτ₁² K = E (B + C) + C + D L = F (B − C + D − K) M = 4 (Bτ₁² + τ₁τ₂C) E' = τ₄/τ₅ F' = (4/3) τ₄τ₃ G' = A'E' H' = F'(A'−G') I' = 4A'τ₄² K' = E' (B' + C') + C' + D' L' = F' (B' − C' + D' − K') M' = 4 (B'τ₄² + τ₄τ₅C') Make initial guesses for the sun-object distance r₁ at time t₁ and for the sun-object distance r₄ at time t₄. For main belt asteroids, a reasonable initial guess for both times is 2.75 AU. Then use the loop below to converge, by successive approximations, to the true sun-object distances, r, and for the Earth-object distances, ρ, distances at times t₁° and t₄° (i.e. the times for the first and fourth observations, corrected for the speed of light travel time). O = 9.999e+99 N = r₁ + r₄ while |N−O|/N > 1ᴇ-11 do ξ = (r₁ + r₄)⁻³ η = (r₄ − r₁) / (r₁ + r₄) P = G + ξH + ηξI Q = K + ξL + ηξM P' = G' + ξH' + ηξI' Q' = K' + ξL' + ηξM' ρ₁ = (Q'−Q)/(P−P') ρ₄ = Pρ₁ + Q r₁ = √[R₁² + (2R₁cos θ₁)ρ₁ + ρ₁²] r₄ = √[R₄² + (2R₄cos θ₄)ρ₄ + ρ₄²] O = N N = r₁ + r₄ endwhile The positions of the object at times t₁ and t₄ in geocentric celestial coordinates are x₁ = a₁ρ₁ − X₁ y₁ = b₁ρ₁ − Y₁ z₁ = c₁ρ₁ − Z₁ x₄ = a₄ρ₄ − X₄ y₄ = b₄ρ₄ − Y₄ z₄ = c₄ρ₄ − Z₄ The reciprocal of the speed of light ç = 0.00577551833 days/AU The sun's gravitational parameter μ = 1.32712440018ᴇ20 m³ sec⁻² The conversion factor from AU to meters is U = 1.495978707ᴇ11 The conversion factor from AU/day to m/sec is β = 1731456.8368 Correcting times of observation for planetary abberation. t₁° = t₁ − çρ₁ t₄° = t₄ − çρ₄ The nominal time associated with the forthcoming state vector is t₀ = ½ (t₁° + t₄°) The nominal heliocentric distance of the object at time t₀ is r₀ = ½ (r₁ + r₄) Find the heliocentric position vector [x',y',z'] for the object at time t₀ in celestial coordinates. x" = ½ (x₁ + x₄) y" = ½ (y₁ + y₄) z" = ½ (z₁ + z₄) r" = √[(x")²+(y")²+(z")²] x' = (r₀/r") U x" y' = (r₀/r") U y" z' = (r₀/r") U z" Find the sun-relative velocity vector for the object at time t₀ in celestial coordinates. S = √[ (x₄ − x'/U)² + (y₄ − y'/U)² + (z₄ − z'/U)² ] s = √[ (x'/U − x₁)² + (y'/U − y₁)² + (z'/U − z₁)² ] Ψ = S + s ψ = √[(x₄−x₁)² + (y₄−y₁)² + (z₄−z₁)²] Vx' = β (Ψ/ψ) (x₄−x₁) / (t₄°−t₁°) Vy' = β (Ψ/ψ) (y₄−y₁) / (t₄°−t₁°) Vz' = β (Ψ/ψ) (z₄−z₁) / (t₄°−t₁°) The object's position in heliocentric ecliptic coordinates x₀ = x' y₀ = y' cos ε + z' sin ε z₀ = −y' sin ε + z' cos ε The object's sun-relative velocity in ecliptic coordinates Vx₀ = Vx' Vy₀ = Vy' cos ε + Vz' sin ε Vz₀ = −Vy' sin ε + Vz' cos ε The object's speed relative to the sun V₀ = √[(Vx₀)² + (Vy₀)² + (Vz₀)²] The semimajor axis of the object's orbit, in AU a = (2/r₀ − V₀²/μ)⁻¹ / U The angular momentum per unit mass in the object's orbit hx = y₀ Vz₀ − z₀ Vy₀ hy = z₀ Vx₀ − x₀ Vz₀ hz = x₀ Vy₀ − y₀ Vx₀ h = √[(hx)² + (hy)² + (hz)²] The eccentricity of the object's orbit e = √[1 − h²/(aμU)] The inclination of the object's orbit i = arccos(hz/h) The longitude of the ascending node of the object's orbit Ω' = arctan(−hx/hy) if hy>0 then Ω = Ω' + π If hy<0 and hx<0 then Ω = Ω' + 2π The true anomaly at time t₀ sin θ₀ = h ( x₀ Vx₀ + y₀ Vy₀ + z₀ Vz₀ ) / (r₀μ) cos θ₀ = h²/(r₀μ) − 1 θ₀' = arctan( sin θ₀ / cos θ₀ ) If cos θ₀ < 0 then θ₀ = θ₀' + π If cos θ₀ > 0 and sin θ₀ < 0 then θ₀ = θ₀' + 2π The sum of the true anomaly at time t₀ and the argument of the perihelion of the object's orbit sin(θ₀+ω) = z₀ / (r₀ sin i) cos(θ₀+ω) = ( x₀ cos Ω + y₀ sin Ω ) / r₀ (θ₀+ω)' = arctan[ sin(θ₀+ω) / cos(θ₀+ω) ] If cos(θ₀+ω) < 0 then θ₀ = θ₀' + π If cos(θ₀+ω) > 0 and sin(θ₀+ω) < 0 then (θ₀+ω) = (θ₀+ω)' + 2π The argument of the perihelion of the object's orbit ω' = (θ₀+ω) − θ₀ if ω'<0 then ω=ω'+2π else ω=ω' The eccentric anomaly of the object at time t₀ cos u₀ = 1 − r₀/(aU) sin u₀ = (x₀ Vx₀ + y₀ Vy₀ + z₀ Vz₀) / √(aμU) u₀' = arctan( sin u₀ / cos u₀ ) If cos u₀ < 0 then u₀ = u₀' + π If cos u₀ > 0 and sin u₀ < 0 then u₀ = u₀' + 2π The mean anomaly of the object at time t₀ M₀ = u₀ − e sin u₀ The period of the object's orbit in days P = 365.256898326 a¹·⁵ The object's time of perihelion passage T = t₀ − PM₀/(2π) Example problem. Find the orbit of Ceres. Observation #1 t₁ = JD 2457204.625 X⊕₁ = +0.155228396 AU Y⊕₁ = −1.004732775 AU Z⊕₁ = +0.00003295786 AU α₁ = 20h 46m 57.02s δ₁ = −27°41′33.9″ Observation #2 t₂ = JD 2457214.625 X⊕₂ = +0.319493277 AU Y⊕₂ = −0.965116604 AU Z⊕₂ = +0.0000311269 AU α₂ = 20h 39m 57.10s δ₂ = −28°47′21.5″ Observation #3 t₃ = JD 2457224.625 X⊕₃ = +0.4747795623 AU Y⊕₃ = −0.8983801739 AU Z⊕₃ = +0.00002841127 AU α₃ = 20h 31m 22.81s δ₃ = −29°49′22.7″ Observation #4 t₄ = JD 2457234.625 X⊕₄ = +0.616702829 AU Y⊕₄ = −0.8063620175 AU Z⊕₄ = +0.00002486325 AU α₄ = 20h 22m 06.57s δ₄ = −30°41′57.3″ Converting the observation angles to radians: α₁ = 5.44084723 δ₁ = −0.483329666 α₂ = 5.41030979 δ₂ = −0.502468171 α₃ = 5.37290956 δ₃ = −0.520509058 α₄ = 5.33245865 δ₄ = −0.53580299 Continuing through the procedure, t = 2457219.63 T = 0.001553628 ε = 0.409057547 radians X₁ = −0.155228396 Y₁ = +0.921851498 Z₁ = +0.399597005 X₂ = −0.319493277 Y₂ = +0.885503087 Z₂ = +0.383841559 X₃ = −0.474779562 Y₃ = +0.824271594 Z₃ = +0.357299982 X₄ = −0.616702829 Y₄ = +0.739843884 Z₄ = +0.320703493 a₁ = +0.589463371 b₁ = −0.660726081 c₁ = −0.464730008 a₂ = +0.563195268 b₂ = −0.671477524 c₂ = −0.4815901 a₃ = +0.532276132 b₃ = −0.685093499 c₃ = −0.497321844 a₄ = +0.49965704 b₄ = −0.69978587 c₄ = −0.510531662 R₁² = 1.03358381 R₄² = 1.03054208 2R₁ cos θ₁ = 1.772595 2R₄ cos θ₄ = 1.97920299 τ₁ = 0.344041979 τ₂ = 0.17202099 τ₃ = 0.516062969 τ₄ = 0.17202099 τ₅ = 0.344041979 Φ = −0.058607619 φ = −0.030167527 A = +0.404275125 B = −7.08013785 C = +4.84883362 D = −0.043927505 A' = +1.72864027 B' = −12.7399767 C' = +3.76138574 D' = +0.951283093 E = +2 F = +0.236729767 G = +0.80855025 H = −0.095703956 I = +0.191407912 K = +0.342297675 L = −2.91537363 M = −2.20429551 E' = +0.5 F' = +0.118364883 G' = +0.864320133 H' = +0.102305152 I' = +0.204610303 K' = +0.223373365 L' = −1.86702289 M' = −0.617533885 r₁ = 2.75 (initial guess) r₄ = 2.75 (initial guess) 1st approximation ρ₁ = 1.97723208 ρ₄ = 1.9223289 r₁ = 2.90652064 r₄ = 2.92071388 2nd approximation ρ₁ = 2.001149 ρ₄ = 1.94460337 r₁ = 2.93008666 r₄ = 2.94292188 3rd approximation ρ₁ = 2.00417695 ρ₄ = 1.94741724 r₁ = 2.93307059 r₄ = 2.94572742 4th approximation ρ₁ = 2.00455348 ρ₄ = 1.94776713 r₁ = 2.93344165 r₄ = 2.94607627 5th approximation ρ₁ = 2.0046002 ρ₄ = 1.94781054 r₁ = 2.93348769 r₄ = 2.94611956 6th approximation ρ₁ = 2.00460599 ρ₄ = 1.94781593 r₁ = 2.9334934 r₄ = 2.94612492 7th approximation ρ₁ = 2.00460671 ρ₄ = 1.9478166 r₁ = 2.93349411 r₄ = 2.94612559 8th approximation ρ₁ = 2.0046068 ρ₄ = 1.94781668 r₁ = 2.9334942 r₄ = 2.94612567 9th approximation ρ₁ = 2.00460681 ρ₄ = 1.94781669 r₁ = 2.93349421 r₄ = 2.94612568 10th approximation (final, converged) ρ₁ = 2.00460681 ρ₄ = 1.94781669 r₁ = 2.93349421 r₄ = 2.94612568 HEC positions in AU at t₁ & t₄ x₁ = +1.33687069 y₁ = −2.2463475 z₁ = −1.33119794 x₄ = +1.58994315 y₄ = −2.10289848 z₄ = −1.31512559 The reciprocal of the speed of light ç = 0.00577551833 days/AU Aberration corrections to time çρ₁ = 0.011577643 days t₁° = 2457204.61 çρ₄ = 0.011249651 days t₄° = 2457234.61 Epoch of state vector t₀ = 2457219.61 JD The object's state vector in heliocentric ecliptic coordinates x₀ = +1.46520344 AU y₀ = −2.52458426 AU z₀ = −0.349479243 AU Vx₀ = +14610.4367 m/s Vy₀ = +7967.42879 m/s Vz₀ = −2442.63758 m/s The object's distance from the sun at t₀ r₀ = 2.93980995 AU Sun-relative speed V₀ = 16819.9661 m/s The sun's gravitational parameter μ = 1.32712440018ᴇ20 m³ sec⁻² The semimajor axis of the object's orbit a = 2.76694735 AU The angular momentum per unit mass in the object's orbit hx = +1.3390648ᴇ15 m²/sec hy = −2.2844842ᴇ14 m²/sec hz = +7.26435028ᴇ15 m²/sec h = 7.39026848ᴇ15 m²/sec The eccentricity of the object's orbit e = 0.076026341 The inclination of the object's orbit i = 10.5918141° The longitude of the ascending node of the object's orbit Ω = 80.3183813° The true anomaly at time t₀ θ₀ = 147.669798° The sum of the true anomaly at time t₀ and the argument of the perihelion of the object's orbit (θ₀+ω) = 220.296384° The argument of the perihelion of the object's orbit ω = 72.6265867° The eccentric anomaly of the object at time t₀ u₀ = 145.259666° The mean anomaly of the object at time t₀ M₀ = 142.777370° The period of the object's orbit P = 1681.12408 days Times of perihelion passage T₀ = 2456552.87 T₁ = 2458234.01 = T₀+P A summary of the calculation results and a comparison with JPL's numbers Orbital elements (as calculated) a = 2.76694735 AU e = 0.076026341 i = 10.5918141° Ω = 80.3183813° ω = 72.6265868° T₀ = JD 2456552.87 T₁ = JD 2458234.01 Orbital elements (From the JPL Small-Body Database) a = 2.76916515 AU e = 0.076009027 i = 10.5940672° Ω = 80.3055309° ω = 73.5976947° T = JD 2458238.75

Elliptical Transfer-Intercept Orbits Non-Hohmann Transfer Orbits that Take You Somewhere Most of the time, when someone speaks of transfer orbits, he's referring to a special case known as a Hohmann transfer orbit. Hohmann transfer orbits have a departure occurring at one of its apsides (perihelion or aphelion) and an arrival occurring at the other apside. Thus, I could describe Hohmann transfer orbits as transfer orbits having two anchored apsides. In this essay, I will treat a more general case of transfer orbits that have only one anchored apside, which turns out to be enough to close the equation set and permit the Keplerian elements of the transfer orbit to be found, as well as the changes of velocity required for transfer orbit insertion and, later, for matching velocity with the destination object. In what follows, the following example problem will be used for illustration: A spaceship is initially in Earth's orbit, but is on the opposite side of the sun from Earth. Its captain wants to enter a transfer orbit, bound for Vesta, at 12h UT on 26 June 2017. The navigator does some trial runs on a computer and discovers an elliptical transfer orbit having its aphelion at Vesta upon arrival at 4h 45m 36.036s UT on 12 June 2018. Check the navigator's work to ensure that an elliptical transfer orbit does exist for these times for departure and arrival. Show the elements of the transfer orbit and the delta-vees required for transfer orbit insertion (departure) and for matching velocity with Vesta at arrival. Spaceship initial orbit. a = 1.000002 AU e = 0.016711 i = 0.0° Ω = 0.0° ω = 103.095° T = JD 2454285.96 Vesta's orbital elements. a = 2.36126914 AU e = 0.089054753 i = 7.13518389° Ω = 103.91484282° ω = 149.85540185° T = JD 2454267.1969204 Departure time, t₁ = 12h UTC, 26 June 2017 Arrival time, t₂ = 4h 45m 36.036s UTC, 12 June 2018 It is convenient to convert t₁ and t₂ from calendar date format to Julian date format. Converting from Calendar Date to Julian Date The time zone must be Greenwich, Zulu, UT, UTC (all the same zone) Y = the four-digit year M = the month of the year (1=January... 12=December) D = the day of the month Q = the time of the day in decimal hours A = integer [ (M−14) / 12 ] B = integer { [ 1461 (Y + 4800 + A) ] / 4 } C = integer { [ 367 (M − 2 − 12A) ] / 12 } E = integer [ (Y + 4900 + A) / 100 ] F = integer [ (3E) / 4 ] t = B + C − F + D − 32075.5 + Q/24 Converting the time of departure, t₁, from calendar date to Julian date t₁ = 12h UTC, 26 June 2017 Y = 2017 M = 6 D = 26 Q = 12 A = 0 B = 2489909 C = 122 E = 69 F = 51 t₁ = JD 2457931.0 Converting the time of arrival, t₂, from calendar date to Julian date t₂ = 4h 45m 36.036s UTC, 12 June 2018 Y = 2018 M = 6 D = 12 Q = 4.76001 A = 0 B = 2490274 C = 122 E = 69 F = 51 t₂ = JD 2458281.69833375 Instead of having the initial position vectors given to us, we must calculate them by reducing the elements of the spaceship's initial orbit (around the sun) and the time of departure therefrom, t₁, in order to obtain the position vector r₁, and by reducing the elements of Vesta's orbit and the time of arrival thereto, t₂, in order to obtain the position vector r₂. For what passes below, the Sun's gravitational parameter, GM = 1.32712440018ᴇ20 m³ sec⁻² The ratio of the astronomical unit to the meter, AU = 1.495978707ᴇ11 m au⁻¹ And the Definition of the two-dimensional arctangent function atn(z) = single argument arctangent function of the argument z. Function arctan( y , x ) . if x = 0 and y greater than 0 then angle = +π/2 . if x = 0 and y = 0 then angle = 0 . if x = 0 and y less than 0 then angle = −π/2 . if x greater than 0 and y greater than 0 then angle = atn(y/x) . if x less than 0 then angle = atn(y/x) + π . if x greater than 0 and y less than 0 then angle = atn(y/x) + 2π arctan = angle Unless otherwise indicated, the coordinate system to which all unprimed vectors in this essay refer is ecliptic coordinates — heliocentric for position, and sun-relative for velocity. Reducing Keplerian orbital elements and a time to position and velocity in heliocentric ecliptic coordinates Find the period, P, in days. P = (365.256898326 days) a¹·⁵ Find the mean anomaly, m, in radians. m₀ = (t − T) / P m = 2π [ m₀ − integer(m₀) ] Find the eccentric anomaly, u, in radians. The Danby first approximation for the eccentric anomaly, u, in radians. u' = m + (e − e³/8 + e⁵/192) sin(m) + (e²/2 − e⁴/6) sin(2m) + (3e³/8 − 27e⁵/128) sin(3m) + (e⁴/3) sin(4m) The Danby's method refinement for the eccentric anomaly. u = u' REPEAT U = u F₀ = U − e sin U − m F₁ = 1 − e cos U F₂ = e sin U F₃ = e cos U D₁ = −F₀ / F₁ D₂ = −F₀ / [ F₁ + D₁F₂/2 ] D₃ = −F₀ / [ F₁ + D₁F₂/2 + D₂²F₃/6 ] u = U + D₃ UNTIL |u−U| is less than 1ᴇ-14 The loop, just above, converges u to the correct value of the eccentric anomaly. Usually. However, when e is near one and the orbiting object is near the periapsis of its orbit, there is a chance that this loop will fail to converge. In such cases, a different root-finding method will be needed. Find the canonical position vector of the object in its orbit at time t. x''' = a (cos u − e) y''' = a sin u √(1−e²) z''' = 0 Find the true anomaly, θ. We'll use it below when we find the velocity. θ = arctan( y''' , x''' ) Rotate the triple-prime position vector by the argument of the perihelion, ω. x'' = x''' cos ω − y''' sin ω y'' = x''' sin ω + y''' cos ω z'' = z''' = 0 Rotate the double-prime position vector by the inclination, i. x' = x'' y' = y'' cos i z' = y'' sin i Rotate the single-prime position vector by the longitude of the ascending node, Ω. x = x' cos Ω − y' sin Ω y = x' sin Ω + y' cos Ω z = z' The unprimed position vector [x,y,z] is the position in heliocentric ecliptic coordinates. Find the canonical (triple-prime) heliocentric velocity vector. k = √{ GM / [ a AU (1 − e²) ] } k is a speed in meters per second. Vx''' = −k sin θ Vy''' = k (e + cos θ) Vz''' = 0 Rotate the triple-prime velocity vector by the argument of the perihelion, ω. Vx'' = Vx''' cos ω − Vy''' sin ω Vy'' = Vx''' sin ω + Vy''' cos ω Vz'' = Vz''' = 0 Rotate the double-prime velocity vector by the inclination, i. Vx' = Vx'' Vy' = Vy'' cos i Vz' = Vy'' sin i Rotate the single-prime velocity vector by the longitude of the ascending node, Ω. Vx = Vx' cos Ω − Vy' sin Ω Vy = Vx' sin Ω + Vy' cos Ω Vz = Vz' The unprimed velocity vector [Vx,Vy,Vz] is the sun-relative velocity in ecliptic coordinates. Calculate the position and velocity of the spaceship in its initial orbit at the time of departure P = 365.257994 m₀ = 9.97935722 m = 6.15348288 u' = 6.15128508 u = 6.15128508 x''' = +0.974604719 y''' = −0.131499998 θ = 6.14906877 x'' = −0.092732158 y'' = +0.979054316 x' = −0.092732158 y' = +0.979054316 z' = 0 xi = −0.092732158 yi = +0.979054316 zi = 0 k = 29788.8217 Vx''' = +3983.20734 Vy''' = +30019.1146 Vx'' = −30140.9504 Vy'' = −2921.69307 Vx' = −30140.9504 Vy' = −2921.69307 Vz' = 0 Vxi = −30140.9504 Vyi = −2921.69307 Vzi = 0 Calculate the position and velocity of Vesta at the time of arrival P = 1325.30752 m₀ = 3.02910935 m = 0.182899417 u' = 0.200646945 u = 0.200648459 x''' = +2.10361404 y''' = +0.468742457 θ = 0.219245394 x'' = −2.0545179 y'' = +0.651051227 x' = −2.0545179 y' = +0.646009389 z' = +0.080867606 xf = −0.13298229 yf = −2.14957848 zf = +0.080867606 k = 19460.2928 Vx''' = −4232.48025 Vy''' = +20727.481 Vx'' = −6748.92645 Vy'' = −20049.7967 Vx' = −6748.92645 Vy' = −19894.5281 Vz' = −2490.40168 Vxf = +20933.6861 Vyf = −1766.64767 Vzf = −2490.40168 We will refer to a "hypothetical" transfer orbit until we have assured ourselves that it satisfies the condition that the calculated transit time be equal, or very nearly equal, to the required transit time. The required transit time is the amount of time that the destination object (Vesta, in our example) takes to go from where it is at t₁ to where it is at t₂. This time difference is, of course, t₂−t₁. The calculated transit time is the amount of time, Δt, that the spaceship takes to travel, along the hypothetical transfer orbit, from where it is at t₁ to the intersection of the hypothetical transfer orbit with the orbit of the destination object. In general, Δt will differ substantially from t₂−t₁. It is necessary that t₁ and t₂ be chosen such that Δt is nearly equal to t₂−t₁. Once we know that to be the case, we can drop the word "hypothetical," for we will have determined that the transfer orbit does, indeed, exist. The determination of an elliptical transfer-intercept orbit from a position and time of departure and from a position and time of arrival At time t₁ a spaceship in free orbit around the sun (i.e. there is no planet nearby) has this state vector: xi , yi , zi , Vxi , Vyi , Vzi At time t₂ (such that t₂>t₁) an asteroid in free orbit around the sun (i.e. there is no significant perturbing third mass) has this state vector: xf , yf , zf , Vxf , Vyf , Vzf We want to find out whether or not there exists a transfer orbit between the position elements of those two state vectors, such that x₁ = xi y₁ = yi z₁ = zi x₂ = xf y₂ = yf z₂ = zf Where the subscript 1 denotes "pertaining to the transfer orbit at transfer orbit insertion," and the subscript 2 denotes "pertaining to the transfer orbit when the spaceship crosses the destination object's orbit." (Whether the destination object is actually there at t₂ is a question that we will answer presently.) r₁ = √[ x₁² + y₁² + z₁² ] r₂ = √[ x₂² + y₂² + z₂² ] d = √[ (x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)² ] We define the integer variable β and permit it to have only the values 1 and 2. If β=1, an apside (perihelion or aphelion) of the transfer orbit occurs at departure. If β=2, an apside (perihelion or aphelion) of the transfer orbit occurs at arrival. β = either 1 or 2 φ = 3 − β N = (−1)ᵠ The variables β and φ will usually be subscripts. The variable N is a sign toggle factor. m : mean anomaly u : eccentric anomaly θ : true anomaly If the apside at the apsidal endpoint of the intended trajectory is the perihelion, then mᵦ = uᵦ = θᵦ = 0 If the apside at the apsidal endpoint of the intended trajectory is the aphelion, then mᵦ = uᵦ = θᵦ = π radians The eccentricity of a conic section, having the sun at a focus, which includes the point of departure and the point of arrival, is found by solving, simultaneously, The polar equation which relates the heliocentric distance with the true anomaly, cos θ₂ − cos θ₁ = { [a (1−e²) / r₂ − 1] − [a (1−e²) / r₁ − 1] } / e The law of cosines, d² = r₁² + r₂² − 2 r₁ r₂ cos(θ₂−θ₁) We can eliminate the semimajor axis by recalling that rᵦ is equal to either a(1+e), the transfer orbit aphelion, or to a(1−e), the transfer orbit perihelion. In general, the semimajor axis can always be eliminated because one or the other endpoints of the intended trajectory occurs at one or the other apside of the transfer orbit. This is why you don't need a third point on the transfer orbit to determine its elements. After some algebra, we get the eccentricity of the hypothetical transfer orbit. e = 2 (cos θᵦ) rᵦ (rᵦ−rᵩ) / (rᵩ² − rᵦ² − d²) The semimajor axis of the hypothetical transfer orbit is found from a = rᵦ / (1 − e cos θᵦ) The true anomaly in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory is found as follows: θᵩ = θᵦ + N arccos{(rᵦ² + rᵩ² − d²) / (2rᵦrᵩ)} The eccentric anomaly in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory is found as follows: sin uᵩ = (rᵩ/a) sin θᵩ / √(1−e²) cos uᵩ = (rᵩ/a) cos θᵩ + e uᵩ = arctan(sin uᵩ , cos uᵩ) The mean anomaly in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory is found as follows: mᵩ = uᵩ − e sin uᵩ The period of the hypothetical transfer orbit is P = (365.256898326 days) a¹·⁵ The mean motion in the hypothetical transfer orbit is μ = 2π/P For short path trajectories (for which the arc of true anomaly going from departure to arrival is less than π radians), the calculated transit time in the hypothetical transfer orbit is Δt = (N/μ) [mᵩ − π sin(θᵦ/2)] Here's the test that determines whether a transfer orbit exists between heliocentric position r₁ at time t₁ and heliocentric position r₂ at time t₂. It is necessary that Δt ≈ t₂ − t₁ And the match should be a close one, ideally a small fraction of a second. In general, this will not be the case. If the difference in the required and the calculated transit times is unacceptably large, then the spaceship pilot will have to choose either a different departure time, or a different arrival time, or both, and try again. The procedure being demonstrated here finds elliptical transfer orbits of the short path, by which it is meant that the arc of true anomaly along the intended trajectory, from departure to arrival, is strictly less than π radians. One of the transfer orbit's apsides will occur at either the position of departure or at the position of arrival, but the other apside will not occur at all within the intended trajectory. To be complete about things, we will calculate the time of perihelion passage in the transfer orbit, whether or not the spaceship is ever there. T = tᵦ − mᵦ/μ The inclination of the transfer orbit is found from the cross product of the heliocentric position vectors of departure and arrival, r₁ x r₂. Xn' = y₁ z₂ − z₁ y₂ Yn' = z₁ x₂ − x₁ z₂ Zn' = x₁ y₂ − y₁ x₂ Rn' = √[ (Xn')² + (Yn')² + (Zn')² ] Xn = Xn' / Rn' Yn = Yn' / Rn' Zn = Zn' / Rn' The vector Rn is a unit normal to the transfer orbit in the direction of the orbit's angular momentum. i = arccos(Zn) Having found the components of the vector normal to the transfer orbit (in the direction of the angular momentum), we now use it to find the velocity in the transfer orbit at the apsidal endpoint of the intended trajectory. Vxᵦ'' = Yn zᵦ − Zn yᵦ Vyᵦ'' = Zn xᵦ − Xn zᵦ Vzᵦ'' = Xn yᵦ − Yn xᵦ Vᵦ'' = √[ (Vxᵦ'')² + (Vyᵦ'')² + (Vzᵦ'')² ] Vxᵦ' = Vxᵦ'' / Vᵦ'' Vyᵦ' = Vyᵦ'' / Vᵦ'' Vzᵦ' = Vzᵦ'' / Vᵦ'' Vᵦ = √[ (GM/AU) (2/rᵦ − 1/a) ] Vxᵦ = Vᵦ Vxᵦ' Vyᵦ = Vᵦ Vyᵦ' Vzᵦ = Vᵦ Vzᵦ' Now we find the angular momentum per unit mass in the transfer orbit. hx = AU (yᵦ Vzᵦ − zᵦ Vyᵦ) hy = AU (zᵦ Vxᵦ − xᵦ Vzᵦ) hz = AU (xᵦ Vyᵦ − yᵦ Vxᵦ) From here, we find the longitude of the ascending node of the transfer orbit. Ω = arctan( hx , −hy ) Notice that hx is proportional to sin Ω, while −hy is proportional to cos Ω. We find the argument of the perihelion of the transfer orbit as follows: cos ω'' = (xᵦ cos Ω + yᵦ sin Ω) / rᵦ If sin i = 0 then sin ω'' = (yᵦ cos Ω − xᵦ sin Ω) / rᵦ If sin i ≠ 0 then sin ω'' = zᵦ / (rᵦ sin i) ω'' = arctan( sin ω'' , cos ω'' ) ω' = ω'' − θᵦ If ω' ≥ 0 then ω = ω' If ω' < 0 then ω = ω' + 2π The two known points on the hypothetical transfer orbit are x₁ = −0.092732158 AU y₁ = +0.979054316 AU z₁ = 0 x₂ = −0.13298229 AU y₂ = −2.14957848 AU z₂ = +0.080867606 AU The sides of the sun-departure-arrival triangle are r₁ = 0.98343612 AU r₂ = 2.15520567 AU d = 3.129936551 AU We are given that the apoapsis of the hypothetical transfer orbit occurs at the arrival position, so β = 2 φ = 1 N = −1 m₂ = u₂ = θ₂ = π radians The eccentricity and the semimajor axis of the hypothetical transfer orbit are e = 0.37484849 a = 1.56759505 AU The true, eccentric, and mean anomalies in the hypothetical transfer orbit at the non-apsidal endpoint of the intended trajectory (i.e. the departure position) are θ₁ = 0.16062918 radians u₁ = 0.10844236 radians m₁ = 0.067872532 radians The period and mean motion of the hypothetical transfer orbit are P = 716.884602 days μ = 0.00876457005 radians/day The calculated and required transit times in the hypothetical transfer orbit, from departure to arrival, are Δt = 350.698335 days t₂−t₁ = 350.69833375 days Subject to roundoff error, the difference is about one-tenth of a second. That's close enough. The transfer orbit exists. The time of perihelion passage in the transfer orbit (although the spaceship is never there) is T = JD 2457923.256033 = 18h 8m 41s UTC on 18 June 2017 Find the unit normal vector to the plane of the transfer orbit Xn' = +0.079173779 Yn' = +0.0074990276 Zn' = +0.329531936 Rn' = 0.338992654 Xn = +0.233556030 Yn = +0.0221215048 Zn = +0.972091673 The transfer orbit's inclination to the ecliptic i = 13.56812324° Find the velocity in the transfer orbit at the apsidal endpoint of the intended trajectory Vx₂'' = +2.091376254 Vy₂'' = −0.148158093 Vz₂'' = −0.499105248 V₂'' = 2.155205676 Vx₂' = +0.970383605 Vy₂' = −0.068744295 Vz₂' = −0.231581261 V₂ = 16041.367805 m/s Vx₂ = +15566.280326 m/s Vy₂ = −1102.752513 m/s Vz₂ = −3714.880181 m/s The angular momentum per unit mass hx = +1.207943483ᴇ15 m²/s hy = +1.144116363ᴇ14 m²/s hz = +5.027623568ᴇ15 m²/s The transfer orbit's longitude of the ascending node Ω = 95.41068849° Find the transfer orbit's argument of the perihelion cos ω'' = −0.987126840 sin ω'' = +0.159939380 ω'' = 2.980963411 radians ω' = −0.160629243 radians ω = 350.79662233° Keplerian elements of the transfer orbit a = 1.56759505 AU e = 0.37484849 i = 13.56812324° Ω = 95.41068849° ω = 350.79662233° T = JD 2457923.256033 Now that you have the elements of the transfer orbit, you can calculate the changes-of-velocity needed for transfer orbit insertion (departure) and for matching velocity with the target asteroid at arrival. Calculation of the delta-vees for departure and for arrival When we reduce the elements of the transfer orbit with the time of departure, t₁, we find that the velocity of the spaceship in the transfer orbit is Vx₁ = −34166.4329 m/s Vy₁ = −1690.83202 m/s Vz₁ = +8247.34992 m/s The velocity of the spaceship in its initial orbit at t₁ was Vxi = −30140.9504 m/s Vyi = −2921.69307 m/s Vzi = 0.0 m/s So the change of velocity required at departure for transfer orbit insertion is ΔVx₁ = −4025.4825 m/s ΔVy₁ = +1230.8611 m/s ΔVz₁ = +8247.3499 m/s ΔV₁ = 9259.4983 m/s The velocity of Vesta, when it is intercepted by the spaceship, is Vxf = +20933.6861 m/s Vyf = −1766.64767 m/s Vzf = −2490.40168 m/s When we reduce the elements of the transfer orbit with the time of arrival, t₂, we find that the velocity of the spaceship in the transfer orbit is Vx₂ = +15566.2801 m/s Vy₂ = −1102.75259 m/s Vz₂ = −3714.88014 m/s So the change of velocity required of the spaceship at arrival to match velocity with Vesta is ΔVx₂ = +5367.4060 m/s ΔVy₂ = −663.8951 m/s ΔVz₂ = +1224.4785 m/s ΔV₂ = 5545.1917 m/s Remember that all of the unprimed vectors in this tutorial are referred to ecliptic coordinates. If you want them in celestial coordinates (so that you can use a star chart to show you the right ascension and declination in which to point the nose of your spaceship when you apply thrust), you'll still have that to do. A check on the accuracy of the method by a numerical evolution of the state vector As calculated from the Keplerian elements of the transfer orbit, at time t₁=JD 2457931.0, the spaceship's heliocentric state vector is x₁ = −0.092732158 AU y₁ = +0.979054316 AU z₁ = 0 Vx₁ = −34166.4329 m/s Vy₁ = −1690.83202 m/s Vz₁ = +8247.34992 m/s As calculated from the Keplerian elements of the transfer orbit, at time t₂ = JD 2458281.69833375, the spaceship's heliocentric state vector is x₂ = −0.13298229 AU y₂ = −2.14957848 AU z₂ = +0.080867606 AU Vx₂ = +15566.2801 m/s Vy₂ = −1102.75259 m/s Vz₂ = −3714.88014 m/s The transit time of the spaceship in the transfer orbit is t₂−t₁ = 30300336.036 sec If we take the state vector at t₁ and numerically walk it forward in time by 1 second intervals for 30300336 seconds, we get this result. x = −0.132983124 AU y = −2.149579643 AU z = +0.080867833 AU Vx = +15566.27354 m/s Vy = −1102.76889 m/s Vz = −3714.87818 m/s