Editing Orbital mechanics (section)

===The patched conic approximation===
{{Main|Patched conic approximation}}
The [[Hohmann transfer orbit]] alone is a poor approximation for interplanetary trajectories because it neglects the planets' own gravity.  Planetary gravity dominates the behavior of the spacecraft in the vicinity of a planet and in most cases Hohmann severely overestimates delta-v, and produces highly inaccurate prescriptions for burn timings. A relatively simple way to get a [[orders of approximation|first-order approximation]] of delta-v is based on the 'Patched Conic Approximation' technique. One must choose the one dominant gravitating body in each region of space through which the trajectory will pass, and to model only that body's effects in that region.  For instance, on a trajectory from the Earth to Mars, one would begin by considering only the Earth's gravity until the trajectory reaches a distance where the Earth's gravity no longer dominates that of the Sun.  The spacecraft would be given [[escape velocity]] to send it on its way to interplanetary space.  Next, one would consider only the Sun's gravity until the trajectory reaches the neighborhood of Mars.  During this stage, the transfer orbit model is appropriate.  Finally, only Mars's gravity is considered during the final portion of the trajectory where Mars's gravity dominates the spacecraft's behavior.  The spacecraft would approach Mars on a hyperbolic orbit, and a final retrograde burn would slow the spacecraft enough to be captured by Mars. [[Friedrich Zander]] was one of the first to apply the patched-conics approach for astrodynamics purposes, when proposing the use of intermediary bodies' gravity for interplanetary travels, in what is known today as a [[gravity assist]].<ref>{{cite journal |last1=Negri |first1=Rodolfo Batista |last2=Prado |first2=Antônio Fernando Bertachini de Alme |title=A historical review of the theory of gravity-assists in the pre-spaceflight era |journal=Journal of the Brazilian Society of Mechanical Sciences and Engineering |date=August 2020 |volume=42 |issue=8 |pages=406 |doi=10.1007/s40430-020-02489-x|s2cid=220510617 |url=http://urlib.net/8JMKD3MGP3W34R/42T4NAH }}</ref>

The size of the "neighborhoods" (or [[sphere of influence (astrodynamics)|spheres of influence]]) vary with radius <math>r_{SOI}</math>:
:<math>r_{SOI} = a_p\left(\frac{m_p}{m_s}\right)^{2/5}</math>
where <math>a_p</math> is the [[semimajor axis]] of the planet's orbit relative to the [[Sun]]; <math>m_p</math> and <math>m_s</math> are the [[mass]]es of the planet and Sun, respectively.

This simplification is sufficient to compute rough estimates of fuel requirements, and rough time-of-flight estimates, but it is not generally accurate enough to guide a spacecraft to its destination.  For that, numerical methods are required.
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====Derivation of formulas====
For a spacecraft to travel from one planet to another, the patched conic approximation breaks the trip into three phases:
#Escape from the origin planet
#Interplanetary transfer
#Capture by the destination planet

We begin by considering phase 2.  In this phase, the gravity of the planets is neglected, and only the Sun's gravity is considered.  Therefore, this phase can be treated as a standard Hohmann transfer orbit.

The [[specific orbital energy]] of an orbit is <math>-\mu/2a</math>, where <math>a</math> is the [[semimajor axis]] and <math>\mu</math> is the [[standard gravitational parameter]] of the gravitating body.  For phase 2, we are interested in the following orbits around the Sun:

{| border=1
|-
! Orbit              !! Major axis                 !! Total specific orbital energy
|-
| At origin          || <math> 2r_0 \,</math>      || <math>-\frac{\mu}{2r_0}</math>
|-
| Transfer           || <math> r_0 + r_1 \,</math> || <math>-\frac{\mu}{r_0+r_1}</math>
|-
| At destination     || <math> 2r_1 \,</math>      || <math>-\frac{\mu}{2r_1}</math>
|}

*<math>-\mu/2r_0</math> for the origin planet
*<math>-\mu/2r_1</math> for the destination planet
*<math>-\mu/(r_0+r_1)</math> for the Hohmann transfer orbit

We must first get from the origin planet's orbit to the Hohmann transfer orbit, which requires a delta-v sufficient to make up the difference in orbital energy between the two orbits involves a change in specific kinetic energy of

A useful concept is the speed of a circular orbit speed at a given distance from a given body, which we will refer to as <math>C_b(r)</math>, where <math>b</math> is the body and <math>r</math> is the distance.  (We may omit <math>b</math> and <math>r</math> for clarity.)  This speed is usually easy to determine, simplifies the formulas, and tells us a lot about the gravitational conditions of a location in space.  The [[specific orbital energy]] for a body in a circular orbit is <math>-\frac{1}{2C}</math>, and the [[specific kinetic energy]] is <math>\frac{1}{2C}</math>.
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