Flight dynamics (spacecraft)

The thrust of a rocket engine, in the general case of operation in an atmosphere, is approximated by:[3]

${\displaystyle F_{n}={\dot {m}}\;v_{e}={\dot {m}}\;v_{e-opt}+A_{e}(p_{e}-p_{amb})}$

where:
${\displaystyle {\dot {m}}}$	=  exhaust gas mass flow
${\displaystyle v_{e}}$	=  effective exhaust velocity (sometimes otherwise denoted as c in publications)
${\displaystyle v_{e-opt}}$	=  effective jet velocity when Pamb = Pe
${\displaystyle A_{e}}$	=  flow area at nozzle exit plane (or the plane where the jet leaves the nozzle if separated flow)
${\displaystyle p_{e}}$	=  static pressure at nozzle exit plane
${\displaystyle p_{amb}}$	=  ambient (or atmospheric) pressure

The effective exhaust velocity of the rocket propellant is proportional to the vacuum specific impulse and affected by the atmospheric pressure: [4]

${\displaystyle {v_{e}}=(I_{sp-vac}-(A_{e}\,p_{amb})/{\dot {m}})\cdot g_{0}}$

where:

${\displaystyle I_{sp-vac}}$ has units of seconds

${\displaystyle g_{0}}$ is the acceleration at the surface of the Earth

The specific impulse relates the delta-v capacity to the quantity of propellant consumed according to the Tsiolkovsky rocket equation:[5]

${\displaystyle \Delta v\ =v_{e}\ln {\frac {m_{0}}{m_{1}}}}$

where:

${\displaystyle m_{0}}$ is the initial total mass, including propellant, in kg (or lb)

${\displaystyle m_{1}}$ is the final total mass in kg (or lb)

${\displaystyle v_{e}}$ is the effective exhaust velocity in m/s (or ft/s)

${\displaystyle \Delta v\ }$ is the delta-v in m/s (or ft/s)

Aerodynamic forces, present near a body with significant atmosphere such as Earth, Mars or Venus, are analyzed as: lift, defined as the force component perpendicular to the direction of flight (not necessarily upward to balance gravity, as for an airplane); and drag, the component parallel to, and in the opposite direction of flight. Lift and drag are modeled as the products of a coefficient times dynamic pressure acting on a reference area:[6]

${\displaystyle \mathbf {L} =C_{L}qA_{\text{ref}}}$

${\displaystyle \mathbf {D} =C_{D}qA_{\text{ref}}}$

where:

• C_L is roughly linear with α, the angle of attack between the vehicle axis and the direction of flight (up to a limiting value), and is 0 at α = 0 for an axisymmetric body;
• C_D varies with α²;
• C_L and C_D vary with Reynolds number and Mach number;
• q, the dynamic pressure, is equal to 1/2 ρv², where ρ is atmospheric density, modeled for Earth as a function of altitude in the International Standard Atmosphere (using an assumed temperature distribution, hydrostatic pressure variation, and the ideal gas law); and
• A_ref is a characteristic area of the vehicle, such as cross-sectional area at the maximum diameter.

The gravitational force that a celestial body exerts on a space vehicle is modeled with the body and vehicle taken as point masses; the bodies (Earth, Moon, etc.) are simplified as spheres; and the mass of the vehicle is much smaller than the mass of the body so that its effect on the gravitational acceleration can be neglected. Therefore the gravitational force is calculated by:

${\displaystyle \mathbf {W} =m\cdot g_{0}\left({\frac {r_{0}}{r}}\right)^{2}}$

where:

${\displaystyle W}$ is the gravitational force (weight);

${\displaystyle m}$ is the space vehicle's mass; and

${\displaystyle r}$ is the radial distance of the vehicle to the planet's center; and

${\displaystyle r_{0}}$ is the radial distance from the planet's surface to its center; and

${\displaystyle g_{0}}$ is the gravitational acceleration at the surface of the planet

For a circular orbit, r_a = r_p = a, and eccentricity is 0. Circular velocity at a given radius is:

${\displaystyle v_{c}={\sqrt {\frac {\mu }{r}}}\,}$

For an elliptical orbit, e is greater than 0 but less than 1. The periapsis velocity is:

${\displaystyle v_{p}={\sqrt {\frac {\mu (1+e)}{a(1-e)}}}\,}$

and the apoapsis velocity is:

${\displaystyle v_{a}={\sqrt {\frac {\mu (1-e)}{a(1+e)}}}\,}$

The limiting condition is a parabolic escape orbit, when e = 1 and r_a becomes infinite. Escape velocity at periapsis is then

${\displaystyle v_{e}={\sqrt {\frac {2\mu }{r_{p}}}}\,}$

The specific angular momentum of any conic orbit, h, is constant, and is equal to the product of radius and velocity at periapsis. At any other point in the orbit, it is equal to:[11]

${\displaystyle h=rv\cos \varphi ,\,}$

where φ is the flight path angle measured from the local horizontal (perpendicular to r.) This allows the calculation of φ at any point in the orbit, knowing radius and velocity:

${\displaystyle \varphi =\arccos \left({\frac {r_{p}v_{p}}{rv}}\right)\,}$

Note that flight path angle is a constant 0 degrees (90 degrees from local vertical) for a circular orbit.

It can be shown that the angular momentum equation given above also relates the rate of change in true anomaly to r, v, and φ, thus the true anomaly can be found as a function of time since periapsis passage by integration:[12]

${\displaystyle \nu =r_{p}v_{p}\int _{t_{p}}^{t}{\frac {1}{r^{2}}}\,dt}$

Conversely, the time required to reach a given anomaly is:

${\displaystyle t={\frac {1}{r_{p}v_{p}}}\int _{0}^{\nu }r^{2}\,d\nu }$

Once in orbit, a spacecraft may fire rocket engines to make in-plane changes to a different altitude or type of orbit, or to change its orbital plane. These maneuvers require changes in the craft's velocity, and the classical rocket equation is used to calculate the propellant requirements for a given delta-v. A delta-v budget will add up all the propellant requirements, or determine the total delta-v available from a given amount of propellant, for the mission. Most on-orbit maneuvers can be modeled as impulsive, that is as a near-instantaneous change in velocity, with minimal loss of accuracy.

An elliptical orbit is most easily converted to a circular orbit at the periapsis or apoapsis by applying a single engine burn with a delta v equal to the difference between the desired orbit's circular velocity and the current orbit's periapsis or apoapsis velocity:

To circularize at periapsis, a retrograde burn is made:

${\displaystyle \Delta v\ =v_{c}-v_{p}\,}$

To circularize at apoapsis, a posigrade burn is made:

${\displaystyle \Delta v\ =v_{c}-v_{a}\,}$

Hohmann transfer orbit, 2, from an orbit (1) to a higher orbit (3)

A Hohmann transfer orbit is the simplest maneuver which can be used to move a spacecraft from one altitude to another. Two burns are required: the first to send the craft into the elliptical transfer orbit, and a second to circularize the target orbit.

To raise a circular orbit at ${\displaystyle v_{1}}$, the first posigrade burn raises velocity to the transfer orbit's periapsis velocity:

${\displaystyle \Delta v_{1}\ =v_{p}-v_{1}}$

The second posigrade burn, made at apoapsis, raises velocity to the target orbit's velocity:

${\displaystyle \Delta v_{2}\ =v_{2}-v_{a}}$

A maneuver to lower the orbit is the mirror image of the raise maneuver; both burns are made retrograde.

A bi-elliptic transfer from a low circular starting orbit (dark blue) to a higher circular orbit (red)

A slightly more complicated altitude change maneuver is the bi-elliptic transfer, which consists of two half-elliptic orbits; the first, posigrade burn sends the spacecraft into an arbitrarily high apoapsis chosen at some point ${\displaystyle r_{b}}$ away from the central body. At this point a second burn modifies the periapsis to match the radius of the final desired orbit, where a third, retrograde burn is performed to inject the spacecraft into the desired orbit.[13] While this takes a longer transfer time, a bi-elliptic transfer can require less total propellant than the Hohmann transfer when the ratio of initial and target orbit radii is 12 or greater.[14][15]

Burn 1 (posigrade):

${\displaystyle \Delta v_{1}\ ={v_{p}}_{1}-v_{1}}$

Burn 2 (posigrade or retrograde), to match periapsis to the target orbit's altitude:

${\displaystyle \Delta v_{2}\ ={v_{a}}_{2}-{v_{a}}_{1}}$

Burn 3 (retrograde):

${\displaystyle \Delta v_{3}\ =v_{2}-{v_{p}}_{2}}$

Plane change maneuvers can be performed alone or in conjunction with other orbit adjustments. For a pure rotation plane change maneuver, consisting only of a change in the inclination of the orbit, the specific angular momentum, h, of the initial and final orbits are equal in magnitude but not in direction. Therefore, the change in specific angular momentum can be written as:

${\displaystyle \Delta h=2h\sin \left({\frac {|\Delta i|}{2}}\right)}$

where h is the specific angular momentum before the plane change, and Δi is the desired change in the inclination angle. From this it can be shown[16] that the required delta-v is:

${\displaystyle \Delta v={\frac {2h\sin {\frac {|\Delta i|}{2}}}{r}}}$

From the definition of h, this can also be written as:

${\displaystyle \Delta v=2v\cos \varphi \sin \left({\frac {\left|\Delta i\right|}{2}}\right)}$

where v is the magnitude of velocity before plane change and φ is the flight path angle. Using the small-angle approximation, this becomes:

${\displaystyle \Delta v=v\cos(\varphi )\left|\Delta i\right|}$

The total delta-v for a combined maneuver can be calculated by a vector addition of the pure rotation delta-v and the delta-v for the other planned orbital change.

This must be timed so that the Moon will be in position to capture the vehicle, and might be modeled to a first approximation as a Hohmann transfer. However, the rocket burn duration is usually long enough, and occurs during a sufficient change in flight path angle, that this is not very accurate. It must be modeled as a non-impulsive maneuver, requiring integration by finite element analysis of the accelerations due to propulsive thrust and gravity to obtain velocity and flight path angle:[18]

${\displaystyle {\dot {v}}={\frac {F\cos \alpha }{m}}-g\cos \theta \,}$

${\displaystyle {\dot {\theta }}={\frac {F\sin \alpha }{mv}}+\left({\frac {g}{v}}-{\frac {v}{r}}\right)\sin \theta ,\,}$

${\displaystyle v=\int _{t_{0}}^{t}{\dot {v}}\,dt}$

${\displaystyle \theta =\int _{t_{0}}^{t}{\dot {\theta }}\,dt}$

where:

F is the engine thrust;

α is the angle of attack;

m is the vehicle's mass;

r is the radial distance to the planet's center; and

g is the gravitational acceleration, which varies with the inverse square of the radial distance:

${\displaystyle g=g_{0}\left({\frac {r_{0}}{r}}\right)^{2}\,}$[18]

Altitude ${\displaystyle h}$, downrange distance ${\displaystyle s}$, and radial distance ${\displaystyle r}$ from the center of the Earth are then computed as:[18]

${\displaystyle h=\int _{t_{0}}^{t}v\cos \theta \,dt}$

${\displaystyle r=r_{0}+h\,}$

${\displaystyle s=r_{0}\int _{t_{0}}^{t}{\frac {v}{r}}\sin \theta \,dt}$

A simple lunar trajectory stays in one plane, resulting in lunar flyby or orbit within a small range of inclination to the Moon's equator. This also permits a "free return", in which the spacecraft would return to the appropriate position for reentry into the Earth's atmosphere if it were not injected into lunar orbit. Relatively small velocity changes are usually required to correct for trajectory errors. Such a trajectory was used for the Apollo 8, Apollo 10, Apollo 11, and Apollo 12 manned lunar missions.

Greater flexibility in lunar orbital or landing site coverage (at greater angles of lunar inclination) can be obtained by performing a plane change maneuver mid-flight; however, this takes away the free-return option, as the new plane would take the spacecraft's emergency return trajectory away from the Earth's atmospheric re-entry point, and leave the spacecraft in a high Earth orbit. This type of trajectory was used for the last five Apollo missions (13 through 17).

In the Apollo program, the retrograde lunar orbit insertion burn was performed at an altitude of approximately 110 kilometers (59 nautical miles) on the far side of the Moon. This became the pericynthion of the initial orbits, with an apocynthion on the order of 300 kilometers (160 nautical miles). The delta v was approximately 1,000 meters per second (3,300 ft/s). Two orbits later, the orbit was circularized at 110 kilometers (59 nautical miles). [19] For each mission, the flight dynamics officer prepares 10 lunar orbit insertion solutions so the one can be chosen with the optimum (minimum) fuel burn and best meets the mission requirements; this is uploaded to the spacecraft computer and must be executed and monitored by the astronauts on the lunar far side, while they are out of radio contact with Earth.[19]

Body	Eccentricity[21]	Mean distance 106 km[22]	Orbital speed km/sec[22]	Orbital period years[22]	Mass Earth = 1[22]	${\displaystyle \mu }$ km3/sec2[22]
Sun	---	---	---	---	333,432	1.327x1011
Mercury	.2056	57.9	47.87	.241	.056	2.232x104
Venus	.0068	108.1	35.04	.615	.817	3.257x105
Earth	.0167	149.5	29.79	1.000	1.000	3.986x105
Mars	.0934	227.8	24.14	1.881	.108	4.305x104
Jupiter	.0484	778	13.06	11.86	318.0	1.268x108
Saturn	.0541	1426	9.65	29.46	95.2	3.795x107
Uranus	.0472	2868	6.80	84.01	14.6	5.820x106
Neptune	.0086	4494	5.49	164.8	17.3	6.896x106

For the purpose of preliminary mission analysis and feasibility studies, certain simplified assumptions may be made to enable delta-v calculation with very small error:[23]

• All the planets' orbits except Mercury have very small eccentricity, and therefore may be assumed to be circular at a constant orbital speed and mean distance from the Sun.
• All the planets' orbits (except Mercury) are nearly coplanar, with very small inclination to the ecliptic (3.39 degrees or less; Mercury's inclination is 7.00 degrees).
• The perturbating effects of the other planets' gravity is negligible.
• The spacecraft will spend most of its flight time under only the gravitational influence of the Sun, except for brief periods when it is in the sphere of influence of the departure and destination planets.

Since interplanetary spacecraft spend a large period of time in heliocentric orbit between the planets, which are at relatively large distances away from each other, the patched-conic approximation is much more accurate for interplanetary trajectories than for translunar trajectories.[23] The patch point between the hyperbolic trajectory relative to the departure planet and the heliocentric transfer orbit occurs at the planet's sphere of influence radius relative to the Sun, as defined above in Orbital flight. Given the Sun's mass ratio of 333,432 times that of Earth and distance of 149,500,000 kilometers (80,700,000 nautical miles), the Earth's sphere of influence radius is 924,000 kilometers (499,000 nautical miles) (roughly 1,000,000 kilometers).[24]

The transfer orbit required to carry the spacecraft from the departure planet's orbit to the destination planet is chosen among several options:

• A Hohmann transfer orbit requires the least possible propellant and delta-v; this is half of an elliptical orbit with aphelion and perihelion tangential to both planets' orbits, with the longest outbound flight time equal to half the period of the ellipse. This is known as a conjunction-class mission.[25][26] There is no "free return" option, because if the spacecraft does not enter orbit around the destination planet and instead completes the transfer orbit, the departure planet will not be in its original position. Using another Hohmann transfer to return requires a significant loiter time at the destination planet, resulting in a very long total round-trip mission time.[27] Science fiction writer Arthur C. Clarke wrote in his 1951 book The Exploration of Space that an Earth-to-Mars round trip would require 259 days outbound and another 259 days inbound, with a 425-day stay at Mars.
• Increasing the departure apsis speed (and thus the semi-major axis) results in a trajectory which crosses the destination planet's orbit non-tangentially before reaching the opposite apsis, increasing delta-v but cutting the outbound transit time below the maximum.[27]
• A gravity assist maneuver, sometimes known as a "slingshot maneuver" or Crocco mission after its 1956 proposer Gaetano Crocco, results in an opposition-class mission with a much shorter dwell time at the destination.[28][26] This is accomplished by swinging past another planet, using its gravity to alter the orbit. A round trip to Mars, for example, can be significantly shortened from the 943 days required for the conjunction mission, to under a year, by swinging past Venus on return to the Earth.

The required hyperbolic excess velocity v_∞ (sometimes called characteristic velocity) is the difference between the transfer orbit's departure speed and the departure planet's heliocentric orbital speed. Once this is determined, the injection velocity relative to the departure planet at periapsis is:[29]

${\displaystyle v_{p}={\sqrt {{\frac {2\mu }{r_{p}}}+v_{\infty }^{2}}}\,}$

The excess velocity vector for a hyperbola is displaced from the periapsis tangent by a characteristic angle, therefore the periapsis injection burn must lead the planetary departure point by the same angle:[30]

${\displaystyle \delta =\arcsin {\frac {1}{e}}\,}$

The geometric equation for eccentricity of an ellipse cannot be used for a hyperbola. But the eccentricity can be calculated from dynamics formulations as:[31]

${\displaystyle e={\sqrt {1+{\frac {2\varepsilon h^{2}}{\mu ^{2}}}}},}$

where h is the specific angular momentum as given above in the Orbital flight section, calculated at the periapsis:[30]

${\displaystyle h=r_{p}v_{p},\,}$

and ε is the specific energy:[30]

${\displaystyle \varepsilon ={\frac {v^{2}}{2}}-{\frac {\mu }{r}}\,}$

Also, the equations for r and v given in Orbital flight depend on the semi-major axis, and thus are unusable for an escape trajectory. But setting radius at periapsis equal to the r equation at zero anomaly gives an alternate expression for the semi-latus rectum:

${\displaystyle p=r_{p}(1+e),\,}$

which gives a more general equation for radius versus anomaly which is usable at any eccentricity:

${\displaystyle r={\frac {r_{p}(1+e)}{1+e\cos \nu }}\,}$

Substituting the alternate expression for p also gives an alternate expression for a (which is defined for a hyperbola, but no longer represents the semi-major axis). This gives an equation for velocity versus radius which is likewise usable at any eccentricity:

${\displaystyle v={\sqrt {\mu \left({\frac {2}{r}}-{\frac {(1-e^{2})}{r_{p}(1+e)}}\right)}}\,}$

The equations for flight path angle and anomaly versus time given in Orbital flight are also usable for hyperbolic trajectories.

There is a great deal of variation with time of the velocity change required for a mission, because of the constantly varying relative positions of the planets. Therefore, optimum launch windows are often chosen from the results of porkchop plots that show contours of characteristic energy (v_∞²) plotted versus departure and arrival time.