Rocket trajectory with Gravity Turn

This is a simplified model for a rocket trajectory going into LEO (Low Earth Orbit) in python, taking as reference the Electron from RocketLab.

From the Figure 1, the trajectory in relation to the Earth, for simplicity as said, we will ignore the Earth´s rotation and write the equations respect to a non rotating earth.

Figure: 1. Representation of a gravity turn

Differential Equations

T direction:

$$\Large \dot{v} = -\frac{T}{m} - g \cos(\psi) + \frac{1}{2m} \rho A C_d v^2$$

N direction:

$$\Large \dot{\phi} = -\frac{g}{v} \sin(\psi) $$ From geometry: $$\Large \dot{h} = v \cos(\psi) $$ $$\Large \dot{\theta} = \frac{v \sin(\psi)}{R_e + h} $$ $$\Large \psi = \dot{\phi} - \dot{\theta} $$

Modeling enviroment

Drag Force

$$\Large D = \frac{1}{2} \rho v^2 C_D A$$

Atmospheric Density

$$\Large \rho = \rho_0 e^{-\frac{h}{H}}$$

Gravity Acceleration

$$\Large g = \frac{g_0}{\left(1 + \frac{h}{R_e}\right)^2}$$

Vehicle Parameters

LiftOff Mass (Kg)	First Stage (kN)	Second Stage (kN)	Payload (Kg)
13500	218.982	29.22	300

Results

Disclaimer

This is an early version and some values havent been modified, next updates will make this in a more realistic orbit for the Electron Rocket.