Editing Spacecraft flight dynamics (section)

==Interplanetary flight==
In order to completely leave one planet's gravitational field to reach another, a [[hyperbola|hyperbolic]] trajectory relative to the departure planet is necessary, with excess velocity added to (or subtracted from) the departure planet's orbital velocity around the Sun. The desired heliocentric transfer orbit to a [[superior planet]] will have its [[perihelion]] at the departure planet, requiring the hyperbolic excess velocity to be applied in the posigrade direction, when the spacecraft is away from the Sun. To an [[inferior planet]] destination, [[aphelion]] will be at the departure planet, and the excess velocity is applied in the retrograde direction when the spacecraft is toward the Sun. For accurate mission calculations, the orbital elements of the planets must be obtained from an [[ephemeris]],{{sfnp|Bate| Mueller| White| 1971| p=359}} such as [[Jet Propulsion Laboratory Development Ephemeris|that published by NASA's Jet Propulsion Laboratory]].

===Simplifying assumptions===
{| {{Table|class=floatright}}
|-
! Body
! Eccentricity<ref name=ssd-mean>{{cite web | title=Keplerian elements for 1800 A.D. to 2050 A.D. | publisher=JPL Solar System Dynamics | url=http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt | archive-url=https://web.archive.org/web/20090723033252/http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt | archive-date=2009-07-23  | access-date=17 December 2009}}</ref>
! Mean<br />distance<br />(10<sup>6</sup> km){{sfnp|Bate| Mueller| White|1971| p=361}}
! Orbital<br />speed<br />(km/sec){{sfnp|Bate| Mueller| White|1971| p=361}}
! Orbital<br />period<br />(years){{sfnp|Bate| Mueller| White|1971| p=361}}
! Mass<br />Earth = 1{{sfnp|Bate| Mueller| White|1971| p=361}}
! <math>\mu</math><br/>(km<sup>3</sup>/sec<sup>2</sup>){{sfnp|Bate| Mueller| White|1971| p=361}}
|-
| Sun || --- || --- || --- || ---  || 333,432 || {{val|1.327e11}}
|-
| Mercury || .2056 || 57.9 || 47.87 || .241 || .056 || {{val|2.232e4}}
|-
| Venus || .0068 || 108.1 || 35.04 ||  .615 || .817 || {{val|3.257e5}}
|-
| Earth || .0167 || 149.5 || 29.79 || 1.000 || 1.000 || {{val|3.986e5}}
|-
| Mars || .0934 || 227.8 || 24.14 || 1.881 || .108 || {{val|4.305e4}}
|-
| Jupiter || .0484 || 778 || 13.06 || 11.86 || 318.0 || {{val|1.268e8}}
|-
| Saturn || .0541 || 1426 || 9.65 || 29.46 || 95.2 || {{val|3.795e7}}
|-
| Uranus || .0472 || 2868 || 6.80 || 84.01 ||  14.6 || {{val|5.820e6}}
|-
| Neptune || .0086  || 4494  || 5.49 || 164.8 || 17.3 || {{val|6.896e6}}
|}
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:{{sfnp|Bate| Mueller| White| 1971| pp=359, 362}}
*All the planets' orbits except [[Mercury (planet)|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 are 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 (astrodynamics)|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.{{sfnp|Bate| Mueller| White| 1971| pp=359, 362}} 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|Orbital flight]]. Given the Sun's mass ratio of 333,432 times that of Earth and distance of {{convert|149,500,000|km|nmi|sp=us|abbr=off}}, the Earth's sphere of influence radius is {{convert|924,000|km|nmi|sp=us|abbr=off}} (roughly 1,000,000 kilometers).{{sfnp|Bate| Mueller| White|1971|p=368}}

===Heliocentric transfer orbit===
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 [[apsis|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 (astronomy)|conjunction]]-class mission.{{sfnp|Mattfeld| Stromgren|Shyface|Komar|2015 |p=3}}{{sfn|Drake|Baker|Hoffman|Landau|2017}} 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.{{sfnp|Bate| Mueller| White|1971 |pp=362–363}} 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.{{sfnp|Bate| Mueller| White|1971 |pp=362–363}}
* A [[gravity assist]] maneuver, sometimes known as a "slingshot maneuver" or ''Crocco mission'' after its 1956 proposer [[Gaetano Crocco]], results in an [[astronomical opposition|opposition]]-class mission with a much shorter dwell time at the destination.{{sfnp|Mattfeld| Stromgren|Shyface|Komar|2015 |pp=3–4}}{{sfn|Drake|Baker|Hoffman|Landau|2017}} 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.

===Hyperbolic departure===
The required hyperbolic excess velocity ''v''<sub>∞</sub> (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:{{sfnp|Bate| Mueller| White| 1971| p=369}}
<math display="block">v_p = \sqrt{\frac{2\mu}{r_p} + v_\infty^2}\,</math>

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:{{sfnp|Bate| Mueller| White| 1971| p=371}}
<math display="block">\delta = \arcsin\frac 1 e</math>

The geometric equation for eccentricity of an ellipse cannot be used for a hyperbola. But the eccentricity can be calculated from dynamics formulations as:{{sfnp|Bate| Mueller| White| 1971| p=372}}
<math display="block">e = \sqrt{1+\frac{2\varepsilon h^2}{\mu^2}},</math>
where {{mvar|h}} is the specific angular momentum as given above in the [[#Flight path angle|Orbital flight]] section, calculated at the periapsis:{{sfnp|Bate| Mueller| White| 1971| p=371}}
<math display="block">h = r_p v_p,</math>
and ''ε'' is the specific energy:{{sfnp|Bate| Mueller| White| 1971| p=371}}
<math display="block">\varepsilon = \frac{v^2}2 - \frac \mu r\,</math>

Also, the equations for r and v given in [[#Orbital flight|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:
<math display="block">p = r_p(1 + e),\,</math>
which gives a more general equation for radius versus anomaly which is usable at any eccentricity:
<math display="block">r = \frac{r_p(1 + e)}{1+e\cos\nu}\,</math>

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:
<math display="block">v = \sqrt{\mu\left (\frac{2}{r}-\frac{1-e^2}{r_p(1+e)}\right)}\,</math>

The equations for flight path angle and anomaly versus time given in [[#Flight path angle|Orbital flight]] are also usable for hyperbolic trajectories.

===Launch windows===
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 plot]]s that show contours of characteristic energy (''v''<sub>∞</sub><sup>2</sup>) plotted versus departure and arrival time.