Difference between revisions of "Tutorial: Basic Orbiting (Math)/fr"

Revision as of 16:23, 23 April 2014

KSP apparently greatly increases the understanding of orbital mechanics

Dans le tutoriel de mécanique orbitale de base à été introduit le concept d'orbite, de stabilisation d'orbite, d'orbite de transfert ainsi qu'une table des vitesses orbitales. Mais que faire si vous désirez suivre une orbite qui n'est pas dans cette table ? Ou si vous voulez orbiter autour d'un autre corps que Kerbin ? Et avec une période spécifique ? C'est là que vient le moment de faire un peu de maths !

Vitesse orbitale

La relation entre vitesse orbitale et accélération est donnée par la formule

${\displaystyle a={\frac {v^{2}}{r}},}$

Où ${\displaystyle a}$ est l'accélération centripète, ${\displaystyle v}$ est la vitesse horizontale (orbitale) et ${\displaystyle r}$ est le rayon de l'orbite (distance au centre du corps orbité).

Cette accélération centripète correspond à la "force" qui nous attire vers le sol, à savoir l'accélération de la pesanteur, qui dépend de la distance au centre du corps orbité et se calcule comme suit :

${\displaystyle a={\frac {\mu }{\left(R+h\right)^{2}}}}$

Où

• ${\displaystyle \mu =G\cdot M_{C}}$ est un paramètre gravitationnel du corps orbité (3530,461 km³/s² pour Kerbin).
• ${\displaystyle G}$ est la constante gravitationnelle (${\displaystyle 6,673\times 10^{-11}\ {\mbox{m}}^{3}\ {\mbox{kg}}^{-1}\ {\mbox{s}}^{-2}}$).
• ${\displaystyle M_{C}}$ est la masse du corps orbité.
• ${\displaystyle R}$ est le rayon du corps orbité (600 km pour Kerbin).
• ${\displaystyle h}$ est l'altitude de l'orbite.

Si nous orbitons autour de Kerbin, on peut obtenir cette valeur de ${\displaystyle a}$ à partir de la valeur de ${\displaystyle g_{0}=9,81m/s^{2}}$, qui est l'accélération de la pesanteur à la surface de Kerbin, par la formule

${\displaystyle a=g_{0}\cdot \left({\frac {R}{R+h}}\right)^{2}}$

Notons que le rayon de l'orbite est précisément la somme du rayon du corps et de l'altitude de l'orbite, donc ${\displaystyle r=R+h}$.

Comme nous avons deux expression de la valeur de ${\displaystyle a}$, nous pouvon égaler les deux membres et en extraire la valeur de ${\displaystyle v}$ :

${\displaystyle {\begin{aligned}g_{0}\left({\frac {R}{R+h}}\right)^{2}&={\frac {v^{2}}{r}}\\g_{0}\left({\frac {R}{R+h}}\right)^{2}&={\frac {v^{2}}{R+h}}\\g_{0}{\frac {R^{2}}{(R+h)^{2}}}&={\frac {v^{2}}{R+h}}\\g_{0}{\frac {R^{2}}{R+h}}&=v^{2}\\v^{2}&=g_{0}{\frac {R^{2}}{R+h}}\\v&={\sqrt {g_{0}{\frac {R^{2}}{R+h}}}}\\v&=R{\sqrt {\frac {g_{0}}{R+h}}}\\\end{aligned}}}$

Et en substituant ${\displaystyle R}$ et ${\displaystyle g_{0}}$ par leur valeur (pour Kerbin), on a :

${\displaystyle v=600\ 000\ \mathrm {m} \cdot {\sqrt {\frac {9,81\ \mathrm {m} /\mathrm {s} ^{2}}{600\ 000\ \mathrm {m} +h}}}}$

Période orbitale

La formule de base de la mécanique nous donne

${\displaystyle d=vt}$ ou ${\displaystyle t=d/v}$

Connaissant ${\displaystyle v}$ grâce à la formule précédente, et ${\displaystyle d}$ étant simplement la circonférence d'un cercle de rayon ${\displaystyle r}$ (rayon de l'orbite, soit ${\displaystyle R+h}$), on obtient la période ${\displaystyle T}$ (en secondes) :

${\displaystyle T=2\pi \cdot {\frac {R+h}{v}}}$

Et pour Kerbin :

${\displaystyle t=2\pi {\frac {600\ 000\ \mathrm {m} +h}{v}}}$

Orbital Equations

For this section it is assumed that the craft in question is orbiting a particular body and is a lot smaller than the body that it is orbiting. Other then in the general orbital equation and the energy equation, the orbits are assumed to be bounded orbits completely within the body's sphere of influence and no others. This section also assumes that classical physics applies in the game. All distances and radii are from the center of the body being orbited and the sea level radius of the body needs to be added to the altitude given in game to recover the values here. The section uses the standard gravitational parameter ${\displaystyle \mu =GM}$ which ${\displaystyle G}$ is Newton's Gravitational Constant and ${\displaystyle M}$ is the mass of the body that is being orbited. From here on out ${\displaystyle m}$ will refer to the mass of the craft.

When the craft is orbiting a body, the distance between the craft and the body is given by the equation

${\displaystyle r(\phi )={\frac {l^{2}}{\mu m^{2}}}{\frac {1}{1+\epsilon cos\phi }}}$

where ${\displaystyle l}$ is the angular momentum of the craft, ${\displaystyle \epsilon }$ is the eccentricity of the orbit, and ${\displaystyle \phi }$ is an angle parameter for the orbit. ${\displaystyle \phi }$ runs from ${\displaystyle 0\deg }$ to ${\displaystyle 360\deg }$, ${\displaystyle l=m\cdot r\cdot v}$ at apoapsis or periapsis, and ${\displaystyle \epsilon \geq 0}$. When ${\displaystyle \epsilon <1}$ the orbit is bounded and is an ellipse and when ${\displaystyle \epsilon \geq 1}$ the orbit is unbounded. At ${\displaystyle \phi =0\deg }$the craft is at the periapsis and at ${\displaystyle \phi =180\deg }$ the craft is at the apoapsis. At periapsis,

${\displaystyle r_{p}={\frac {l^{2}}{\mu m^{2}}}{\frac {1}{1+\epsilon }}}$

and at apoapsis,

${\displaystyle r_{a}={\frac {l^{2}}{\mu m^{2}}}{\frac {1}{1-\epsilon }}}$.

The semimajor axis is

${\displaystyle a={\frac {l^{2}}{\mu m^{2}}}{\frac {1}{1-\epsilon ^{2}}}}$

and the semiminor axis is

${\displaystyle b={\frac {l^{2}}{\mu m^{2}}}{\frac {1}{\sqrt {1-\epsilon ^{2}}}}}$.

${\displaystyle \epsilon ={\frac {\sqrt {a^{2}-b^{2}}}{a}}}$

Given the results for ${\displaystyle r_{p}}$ and ${\displaystyle r_{a}}$ above,

${\displaystyle l^{2}=2\mu m^{2}{\frac {r_{p}r_{a}}{r_{p}+r_{a}}}}$ and

${\displaystyle \epsilon ={\frac {r_{a}-r_{p}}{r_{p}+r_{a}}}}$ thus

${\displaystyle r(\phi )=2{\frac {r_{p}r_{a}}{r_{p}+r_{a}+(r_{a}-r_{p})cos\phi }}}$

The velocity for an object in elliptical orbit about a much larger object is

${\displaystyle v={\sqrt {\mu ({\frac {2}{r}}-{\frac {1}{a}})}}={\sqrt {\mu {\frac {r_{p}^{2}+r_{a}^{2}+(r_{a}^{2}-r_{p}^{2})cos\phi }{r_{p}r_{a}(r_{p}+r_{a})}}}}={\sqrt {2\mu {\frac {r_{p}+r_{a}-r}{r(r_{p}+r_{a})}}}}}$.

The energy for an object in orbit is

${\displaystyle E={\frac {\mu ^{2}m^{3}}{2l^{2}}}(\epsilon ^{2}-1)={\frac {\mu m(r_{p}+r_{a})}{4r_{p}r_{a}}}(\epsilon ^{2}-1)=-{\frac {\mu m}{r_{p}+r_{a}}}}$.

One should note that for bounded orbits, the energy is negative and for escape orbits, energy is non-negative.

The orbital period is

${\displaystyle \tau ={\sqrt {{\frac {4\pi ^{2}}{\mu }}a^{3}}}={\sqrt {\frac {\pi ^{2}(r_{p}+r_{a})^{3}}{2\mu }}}}$.

Let ${\displaystyle r_{c,1}}$ and ${\displaystyle r_{o,1}}$ be the two apsides of a given orbit and let a craft execute a burn imparting a specific ${\displaystyle \Delta v}$ all at the single point in time the craft is at ${\displaystyle r_{c,1}}$. For this problem ${\displaystyle \Delta v}$ can be either positive or negative. If it is positive then the thrust is to speed up the craft but if it is negative the ture delta v is the oppositie of that in the problem but the thrust is to slow down the craft. This problem assumes that the thust is in the same or opposite direction of travel and is not enough to reverse the craft. The new orbit will have apsides of ${\displaystyle r_{c,2}}$ and ${\displaystyle r_{o,2}}$ with ${\displaystyle r_{c,2}=r_{c,1}}$.

The speed of the craft before the burn is

${\displaystyle v_{1}={\sqrt {2\mu {\frac {r_{c,1}+r_{o,1}-r_{c,1}}{r_{c,1}(r_{c,1}+r_{o,1})}}}}={\sqrt {2\mu {\frac {r_{o,1}}{r_{c,1}(r_{c,1}+r_{o,1})}}}}}$.

The speed of the craft after the burn is

${\displaystyle v_{2}=v_{1}+\Delta v={\sqrt {2\mu {\frac {r_{c,2}+r_{o,2}-r_{c,2}}{r_{c,2}(r_{c,2}+r_{c,2})}}}}={\sqrt {2\mu {\frac {r_{o,2}}{r_{c,1}(r_{c,1}+r_{o,2})}}}}={\sqrt {2\mu {\frac {r_{o,1}}{r_{c,1}(r_{c,1}+r_{o,1})}}}}+\Delta v}$

which yields

${\displaystyle \Delta v={\sqrt {\frac {2\mu }{r_{c,1}}}}({\sqrt {\frac {r_{o,2}}{r_{c,1}+r_{o,2}}}}-{\sqrt {\frac {r_{o,1}}{r_{c,1}+r_{o,1}}}})}$

and

${\displaystyle r_{o,2}=r_{c,1}{\frac {r_{c,1}({\sqrt {2\mu {\frac {r_{o,1}}{r_{c,1}(r_{c,1}+r_{o,1})}}}}+\Delta v)^{2}}{2\mu -r_{c,1}({\sqrt {2\mu {\frac {r_{o,1}}{r_{c,1}(r_{c,1}+r_{o,1})}}}}+\Delta v)^{2}}}=r_{c,1}{\frac {r_{c,1}v_{2}^{2}}{2\mu -r_{c,1}v_{2}^{2}}}}$