gemseo.problems.ode.orbital_dynamics module#
The 2-body astrodynamics problem.
Predict the motion and position of a massive object orbiting a fixed mass in an elliptic trajectory with a given eccentricity. This problem is treated here as a classical central force problem.
Consider the frame defined by one particle. The position \((x, y)\) and the velocity \((v_x, v_y)\) of the other particle as a function of time can be described by the following set of equations:
with \(r = \sqrt{x(t)^2 + y(t)^2}\).
We use the initial conditions:
where \(e\) is the eccentricity of the particle trajectory.
The Jacobian of the right-hand side of this ODE is:
- class OrbitalDynamics(eccentricity=0.5, times=(0.0, 0.5))[source]#
Bases:
ODEProblemProblem describing a particle following an elliptical orbit.
The particle is supposed to represent a massive object immersed in a gravitational field generated by a single, fixed, massive object.
The dynamic is parametrized by its eccentricity, a strictly positive parameter, that is supposed to be between 0 (included), and 1 (excluded) for elliptic orbits. All parameters of the system are rescaled so that the length of the semi-major axis of the trajectory is normalized to 1.
By Kepler's law, the orbit of the particle is planar and follows a conic section. Thus, the reference system is oriented so that the trajectory lies in the x-y plane, and the periapsis is placed on the axis y=0. The horizontal coordinate of the periapsis can be computed as x = 1 - eccentricity. The time interval is scaled so that the mass reaches its periapsis in t=0. At the periapsis, the velocity of the massive particle is orthogonal to the line of the apsides, and its magnitude is equal to \(sqrt{frac{1 + eccentricity}{1 - eccentricity}}\). We suppose that the orbit follows a counter-clockwise path.
The state of the ODE consists in four variables:
\(x\), the horizontal coordinate,
\(y\), the vertical coordinate,
\(v_x\), the horizontal component of the velocity,
\(v_y\), the vertical component of the velocity.
The system follows the dynamic below:
\(\frac{dx}{dt} = v_x\),
\(\frac{dy}{dt} = v_y\),
\(\frac{dv_x}{dt} = - x / (\sqrt(x^2 + y^2))^3\),
\(\frac{dv_y}{dt} = - y / (\sqrt(x^2 + y^2))^3\).
- Parameters:
eccentricity (float) --
The eccentricity of the particle trajectory.
By default it is set to 0.5.
times (ArrayLike) --
The initial and final time.
By default it is set to (0.0, 0.5).