Orbital and Rocket Delta-v

Orbital and Rocket Delta-v

Orbital and Rocket Delta-v: the Rocket Equation, Staging, Two-Body Orbits, Transfers, Patched Conics and J2

This page gives the model behind the tool, its conventions and what each result means. Everything here is the standard first-course material of astronautics: a spacecraft is a point mass attracted by one spherical body, a burn changes the velocity instantly, and a rocket obeys the ideal rocket equation. The results are planning values for studies and teaching. They are not a trajectory: finite burn times, the ascent through the atmosphere, the eccentric and inclined orbits of the real planets, third bodies and navigation all change the numbers of a real mission, usually by a few per cent and sometimes by much more (the ascent losses).

Units and constants

Distances are in km, speeds in km/s, times in seconds, angles in degrees and specific impulse in seconds in both unit systems; with U.S. units selected the result tables add statute miles and ft/s, and masses and thrusts are entered and shown in pounds. Altitudes are measured from the equatorial radius R of the body, radii r from its centre. The standard acceleration g₀ = 9.80665 m/s² is exact and only converts specific impulse to exhaust velocity, c = I_sp g₀; it is not the local gravity. The gravitational parameters GM are those of the JPL planetary ephemeris DE440 as listed on the JPL Solar System Dynamics page of astrodynamic parameters (for Mars and the giant planets the listed value is that of the planet with its moons). Radii, J₂, rotation periods and the sizes and periods of the planets' orbits are those of the NASA planetary fact sheets; the first results table gives the date of the sheet used. The Moon's orbit radius is its semimajor axis about Earth. A custom body takes entered values.

The rocket equation

A rocket that expels propellant at the effective exhaust velocity c gains Δv = c ln(m₀/m_f) = I_sp g₀ ln(m₀/m_f), with m₀ the mass before the burn and m_f the mass after it (the form given on the NASA Glenn page on the ideal rocket equation). The single-burn section solves it for whichever of the four quantities is chosen as the unknown and reports the mass ratio, the propellant m₀ − m_f and the propellant fraction 1 − m_f/m₀. The equation holds in free space; gravity and drag during a burn are accounted for separately as losses.

Stages

The stage table is read in firing order. For stage i, m₀ is its own dry mass and propellant plus everything above it (the later stages and the payload) and m_f is that less its propellant; the empty stage is then dropped. Each burn contributes I_sp g₀ ln(m₀/m_f) and the ideal velocity of the vehicle is the sum. With a thrust F entered, the flow rate is ṁ = F/(I_sp g₀), the burn time is the propellant mass over ṁ and the thrust-to-weight ratio at ignition is F/(m₀ g₀) (Earth weight, the usual convention; a first stage needs more than 1 to leave the pad). With a burn time entered instead, the thrust follows. The structural coefficient of a stage is ε = dry/(dry + propellant), and the payload fraction is the payload over the lift-off mass.

Parallel boosters. Boosters that burn beside stage 1 from lift-off are treated as one burn with the combined thrust and combined flow: its effective specific impulse is I_sp,eff = (F_b + F₁)/((ṁ_b + ṁ₁) g₀), the flow-weighted mean of the two. The burn ends when the boosters are empty; their dry mass is dropped and stage 1 burns the propellant it has left as a second burn. Stage 1 needs a thrust or a burn time so that its flow during the parallel burn is known, and the boosters must burn out first.

What the ascent needs. To reach a circular orbit of radius r the vehicle must supply the circular speed √(GM/r), plus what is lost to gravity while the thrust is not horizontal and to drag and steering, less the eastward speed the launch site already has, ω R cos(latitude), for a launch due east (which gives an orbit inclined by the latitude). The two losses are entered: they depend on the thrust-to-weight history and the trajectory, and only a trajectory simulation gives them. Figures between about 1.5 and 2 km/s in total are commonly quoted for launches to low Earth orbit; that range is a rule of thumb stated here from general knowledge and not taken from one of the references, so treat the defaults (1.5 and 0.2 km/s) as placeholders. The tool prints the ideal velocity less the requirement as a reserve or a shortfall.

Optimum split. With R_i = m₀/m_f of stage i, the payload ratio of the stage (mass above it over its ignition mass) is π_i = (1 − ε_i R_i)/(R_i (1 − ε_i)) and the payload fraction of the vehicle is the product of the π_i. Maximising it subject to Σ c_i ln R_i = Δv with a Lagrange multiplier η gives R_i = (η c_i − 1)/(η c_i ε_i); the sum rises steadily with η, so η is found by bisection. If a stage would come out with R_i below 1 it is left out and the others are solved again. For n identical stages the answer is the equal split, R = exp(Δv/(n c)), with payload fraction πⁿ, which tends to exp(−Δv/(c (1 − ε))) for many stages; a stage must have ε R below 1 to carry anything. The comparison uses the entered stages' own c and ε and the vehicle's own ideal velocity, and assumes that a stage of any size could be built at the same structural coefficient, which real hardware does not allow; it shows the direction in which the split could move, not a design.

Orbits about one body

Circular speed v_c = √(GM/r); period T = 2π √(a³/GM); escape speed √(2 GM/r) = √2 v_c; on any orbit the vis-viva equation v² = GM (2/r − 1/a) holds, with a the semimajor axis (negative on a hyperbola). An ellipse with apsis radii r_p and r_a has a = (r_p + r_a)/2 and e = (r_a − r_p)/(r_a + r_p). The stationary orbit is the circular equatorial orbit whose period is the sidereal rotation period, r = (GM (T/2π)²)^1/3; for slow rotators such as Venus it lies outside the sphere of influence and does not exist.

Hohmann transfer. Two tangential burns join two coplanar circular orbits through half of the ellipse that touches both: Δv₁ = |v_t(r₁) − v_c1|, Δv₂ = |v_c2 − v_t(r₂)| with the transfer speeds from vis-viva at a_t = (r₁ + r₂)/2, and the transfer takes π √(a_t³/GM). Inward transfers cost the same.

Bi-elliptic transfer. Three burns: out to an apoapsis r_b beyond both orbits, a burn there to raise the periapsis to the final orbit, and a braking burn on arrival. NASA's Orbital Flight Handbook (SP-33 Part 2, p. VI-48) gives the two radius ratios that settle the comparison: below 11.939 the Hohmann transfer is always the cheaper; above 15.582 the three-burn transfer is cheaper for any r_b beyond the outer orbit; between them it is cheaper only for a large enough r_b. The tool solves both ratios numerically and reproduces them. The saving is small and the transfer time long.

Plane change. Turning a velocity v through Δi without changing its size costs 2 v sin(Δi/2), so plane changes are cheapest where the speed is lowest. A burn that changes speed and plane together costs √(v_a² + v_b² − 2 v_a v_b cos Δi), always less than the two done separately. The tool shares the plane change between the two Hohmann burns and finds the split with the lowest total by a golden-section search; for a transfer to a much higher orbit a degree or two belongs to the first burn and the rest to the second.

Phasing. To close an angle φ on a target ahead in the same circular orbit in k revolutions the chaser enters an ellipse of period T (1 − φ/(360° k)) and returns to the burn point; two equal and opposite burns. A target ahead needs a shorter period, so the ellipse dips inside the orbit, and the tool warns when it would dip below the surface.

De-orbit. A retrograde burn at the orbit lowers the opposite apsis to the entered perigee: Δv = v_c − v_a of the ellipse between the two radii.

Departure. Leaving the body with the excess speed v_∞ needs the speed √(v_∞² + 2 GM/r) at the parking-orbit radius; C3 = v_∞² is the usual measure of launch energy. The sphere of influence of a body of gravitational parameter GM orbiting a parent at the distance a is Laplace's r = a (GM/GM_parent)^2/5.

Interplanetary Hohmann transfer by patched conics

The two planets move on circular, coplanar orbits about the Sun at their mean distances. The heliocentric Hohmann ellipse between them fixes the speed relative to each planet at the ends of the transfer, which is taken as the hyperbolic excess speed of a planet-centred hyperbola whose periapsis is the parking orbit (departure) or the capture orbit (arrival): Δv = √(v_∞² + 2 GM/r_p) − √(GM/r_p). The time of flight is half the period of the transfer ellipse; the synodic period 1/|1/T₁ − 1/T₂| is the interval between launch opportunities; and the phase angle at departure, 180° less the angle the target sweeps during the flight, says where the target must be relative to the departure planet (positive = ahead). Real windows differ in date, energy and duration because the orbits are eccentric and inclined.

J₂: nodal regression and the sun-synchronous orbit

The equatorial bulge of the body makes the orbit plane turn about the polar axis at the mean rate dΩ/dt = −(3/2) J₂ n (R/p)² cos i and the line of apsides turn in the plane at (3/4) J₂ n (R/p)² (4 − 5 sin² i), with n = √(GM/a³) and p = a (1 − e²) (the forms printed in Rickman's NASA course, charts 104 and 105). These are first-order secular rates; a numerical integration of the perturbed equations of motion agrees with the nodal rate within half a per cent for the default orbit. An orbit is sun-synchronous when its node moves eastward 360° per year, which needs an inclination a little over 90°: cos i = −(360°/year)/((3/2) J₂ n (R/p)²). The year is the tropical year for Earth and the sidereal orbit period for the other planets. The ground track of a circular orbit shifts westward each revolution by the angle the body turns in one period, corrected for the nodal rate; a repeating track needs the number of revolutions per rotation to be a ratio of whole numbers.

The velocity budget

The manoeuvres are listed in the order flown, each with a margin in per cent. Starting from the mass at the end of the mission, each row gives m_before = m_after exp(Δv (1 + margin)/(I_sp g₀)); with one specific impulse this is the same as one burn of the summed velocity. The mass at the start is compared with the payload of the launch vehicle.

The default case

The loaded numbers are illustrative and describe no particular vehicle: a two-stage launcher of 551.5 t at lift-off places 22 t in a 400 km orbit from latitude 28.5° (ideal velocity 9.136 km/s against 8.960 km/s required with the entered losses), and the payload then makes a Hohmann transfer to geostationary orbit: 2.398 + 1.457 = 3.854 km/s coplanar in 5.29 h, 4.196 km/s with the 28.5° plane change at the best split. The budget table charges 2,398 and 1,824 m/s (the plane change taken with the apogee burn) and a small allowance, with margins, to a 5,400 kg spacecraft at 320 s: 21,387 kg at the start, inside the 22,000 kg payload.

Checks against published values

The test script compares the library with an independent numerical solution (integration of the two-body and J₂ equations of motion and of the thrust equation, constrained optimisation for the staging split) and with printed values: the two bi-elliptic ratios of SP-33; the AU-18 Space Primer's 5.59 and 7.91 km/s at one Earth radius altitude, 7.8 km/s and 87.5 minutes at 150 km and the 98.75° inclination and 101.6 minute period of a 450 nautical mile sun-synchronous orbit; Rickman's geostationary altitude of 35,786 km and sun-synchronous 98.2° at a = 7,083 km; and the escape speeds, orbital speeds and synodic periods of the NASA fact sheets. Two printed values are not reproduced and are noted in the test: AU-18 gives 11.2 km/s as the escape speed "from a low Earth orbit of 161 km", which is the surface value (11.04 km/s at 161 km), and the NASA Glenn page rounds a mass ratio of 9.19 to 10.