Editing Spacecraft flight dynamics (section)

==Orbital flight==
{{Further|Orbital mechanics}}

Orbital mechanics are used to calculate flight in orbit about a central body. For sufficiently high orbits (generally at least {{convert|100|nmi|km|disp=flip|sp=us|abbr=off}} in the case of Earth), aerodynamic force may be assumed to be negligible for relatively short term missions (though a small amount of drag may be present which results in decay of orbital energy over longer periods of time.) When the central body's mass is much larger than the spacecraft, and other bodies are sufficiently far away, the solution of orbital trajectories can be treated as a two-body problem.{{sfnp |Perry | 1967 | p=11:151}}

This can be shown to result in the trajectory being ideally a [[conic section]] (circle, ellipse, parabola or hyperbola){{sfnp| Bate| Mueller | White | 1971 | pp=11-40}} with the central body located at one focus. Orbital trajectories are either circles or ellipses; the parabolic trajectory represents first escape of the vehicle from the central body's gravitational field. Hyperbolic trajectories are escape trajectories with excess velocity, and will be covered under [[#Interplanetary flight|Interplanetary flight]] below.

Elliptical orbits are characterized by three elements.{{sfnp |Perry | 1967 | p=11:151}} The semi-major axis ''a'' is the average of the radius at [[apsis|apoapsis and periapsis]]:
<math display="block">a = \frac{r_a + r_p} 2 </math>

The [[orbital eccentricity|eccentricity]] ''e'' can then be calculated for an ellipse, knowing the apses:
<math display="block">e = \frac{r_a} a - 1 </math>

The [[orbital period|time period for a complete orbit]] is dependent only on the semi-major axis, and is independent of eccentricity:{{sfnp | Bate| Mueller | White | 1971 | p=33 }}
<math display="block"> T = 2 \pi \sqrt{\frac{a^3} \mu}</math>
where <math>\mu</math> is the [[standard gravitational parameter]] of the central body.

[[File:Orbit1.svg|thumb|200px|right|The angular [[orbital elements]] of a spacecraft orbiting a central body, defining orientation of the orbit in relation to its fundamental reference plane]]
The orientation of the orbit in space is specified by three angles:
*The ''inclination'' ''i'', of the orbital plane with the fundamental plane (this is usually a planet or moon's equatorial plane, or in the case of a solar orbit, the Earth's orbital plane around the Sun, known as the [[ecliptic]].) Positive inclination is northward, while negative inclination is southward.
*The ''longitude of the ascending node'' Ω, measured in the fundamental plane counter-clockwise looking southward, from a reference direction (usually the [[March equinox|vernal equinox]]) to the line where the spacecraft crosses this plane from south to north. (If inclination is zero, this angle is undefined and taken as 0.)
*The ''argument of periapsis'' ''ω'', measured in the orbital plane counter-clockwise looking southward, from the ascending node to the periapsis. If the inclination is 0, there is no ascending node, so ''ω'' is measured from the reference direction. For a circular orbit, there is no periapsis, so ''ω'' is taken as 0.

The orbital plane is ideally constant, but is usually subject to small perturbations caused by planetary oblateness and the presence of other bodies.

The spacecraft's position in orbit is specified by the ''true anomaly,'' <math>\nu</math>, an angle measured from the periapsis, or for a circular orbit, from the ascending node or reference direction. The ''semi-latus rectum'', or radius at 90 degrees from periapsis, is:{{sfnp | Bate| Mueller | White | 1971 | p=24}}
<math display="block">p = a(1-e^2)\,</math>

The radius at any position in flight is:
<math display="block">r = \frac p {1+e\cos\nu}</math>
and the velocity at that position is:
<math display="block">v = \sqrt{\mu\left(\frac 2 r - \frac 1 a\right)}</math>

===Types of orbit===
====Circular====
For a circular orbit, ''r''<sub>''a''</sub> = ''r''<sub>''p''</sub> = ''a'', and eccentricity is 0. Circular velocity at a given radius is:
<math display="block">v_c = \sqrt{\frac\mu r}</math>

====Elliptical====
For an elliptical orbit, ''e'' is greater than 0 but less than 1. The periapsis velocity is:
<math display="block">v_p = \sqrt{\frac{\mu(1+e)}{a(1-e)}}</math>
and the apoapsis velocity is:
<math display="block">v_a = \sqrt{\frac{\mu(1-e)}{a(1+e)}}\,</math>

The limiting condition is a '''parabolic escape orbit''', when ''e'' = 1 and ''r''<sub>''a''</sub> becomes infinite. Escape velocity at periapsis is then
<math display="block">v_e = \sqrt{\frac{2\mu}{r_p}}</math>

===Flight path angle===
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:{{sfnp| Bate| Mueller| White| 1971| p=18}}
<math display="block">h = r v\cos\varphi,</math>
where ''φ'' is the flight path angle measured from the local horizontal (perpendicular to&nbsp;''r''.) This allows the calculation of ''φ'' at any point in the orbit, knowing radius and velocity:
<math display="block">\varphi = \arccos\left(\frac{r_p v_p}{r v}\right)</math>

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

===True anomaly as a function of time===
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:{{sfnp| Bate| Mueller| White |1971| pp=31-32}}
<math display="block">\nu = r_p v_p \int_{t_p}^t \frac 1 {r^2} \, dt</math>

Conversely, the time required to reach a given anomaly is:
<math display="block">t = \frac 1 {r_p v_p} \int_0^\nu r^2 \, d\nu</math>

===Orbital maneuvers===
{{see also|Orbital maneuver}}
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|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 [[orbital maneuver#Impulsive maneuvers|impulsive]], that is as a near-instantaneous change in velocity, with minimal loss of accuracy.

====In-plane changes====

=====Orbit circularization=====
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:
<math display="block">\Delta v\ = v_c - v_p</math>

To circularize at apoapsis, a posigrade burn is made:
<math display="block">\Delta v\ = v_c - v_a</math>

=====Altitude change by Hohmann transfer=====
[[File:Hohmann transfer orbit.svg|thumb|upright|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 <math>v_1</math>, the first posigrade burn raises velocity to the transfer orbit's periapsis velocity:
<math display="block">\Delta v_1\ = v_p - v_1</math>
The second posigrade burn, made at apoapsis, raises velocity to the target orbit's velocity:
<math display="block">\Delta v_2\ = v_2 - v_a</math>

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

=====Altitude change by bi-elliptic transfer=====
[[File:Bi-elliptic transfer.svg|thumb|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 <math>r_b</math> 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.<ref name="Curtis">{{Cite book | last = Curtis | first = Howard | title = Orbital Mechanics for Engineering Students | page = 264 | publisher = [[Elsevier]] | year = 2005 | isbn = 0-7506-6169-0 | url = https://books.google.com/books?id=6aO9aGNBAgIC}}</ref> 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.<ref>{{cite journal
| last1       = Gobetz
| first1      = F. W.
| last2      = Doll
| first2     = J. R.
| date       = May 1969
| title      = A Survey of Impulsive Trajectories
| journal    = AIAA Journal
| publisher  = [[American Institute of Aeronautics and Astronautics]]
| volume     = 7
| issue      = 5
| pages      = 801–834
| doi        = 10.2514/3.5231| bibcode= 1969AIAAJ...7..801D
}}</ref><ref>{{Cite book
 | first = Pedro R.
 | last = Escobal
 | title = Methods of Astrodynamics
 | location = New York
 | publisher = [[John Wiley & Sons]]
 | year = 1968
 | isbn = 978-0-471-24528-5
}}</ref>

Burn 1 (posigrade):
<math display="block">\Delta v_1\ = {v_p}_1 - v_1</math>
Burn 2 (posigrade or retrograde), to match periapsis to the target orbit's altitude:
<math display="block">\Delta v_2\ = {v_a}_2 - {v_a}_1</math>
Burn 3 (retrograde):
<math display="block">\Delta v_3\ = v_2 - {v_p}_2</math>

====Change of plane====
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:
<math display="block">\Delta h = 2h\sin\left(\frac {|\Delta i|}{2} \right)</math>
where ''h'' is the specific angular momentum before the plane change, and Δ''i'' is the desired change in the inclination angle. From this it can be shown{{sfnp|Hintz|2015|p=112}} that the required delta-''v'' is:
<math display="block">\Delta v = \frac {2h\sin\frac {|\Delta i|}{2}}{r}</math>

From the definition of ''h'', this can also be written as:
<math display="block">\Delta v = 2v\cos \varphi\sin\left(\frac {\left|\Delta i\right|} 2 \right)</math>
where ''v'' is the magnitude of velocity before plane change and ''φ'' is the flight path angle. Using the [[small-angle approximation]], this becomes:
<math display="block">\Delta v = v \cos(\varphi) \left|\Delta i\right|</math>

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.