theoretical_background.md

Trajectory simulation

Ascent model

$$m_{tot}a_z = g\rho_{air}V - g(m_{gross} + m_{gas}) - \frac{1}{2}C_D\rho_{air}\dot{z}^2A_b$$ where:
$m_{tot} = m_{gross} + m_{gas} + m_{added}$
$m_{added} = \frac{1}{2}\rho_{air}V$ - the mass of the air moved by the balloon during the ascent
The rise in volume is the main responsible for the rise in the balloon vertical velocity, commonly called ascending rate, because in the equation of motion the acceleration is observed to depend on volume.

$$V = \frac{m_{gas}}{M_{gas}} R \frac{T_{gas}}{p_{air}}$$

Since our approximation addresses the volume change totally to the adiabatic expansion:

$$p \cdot V^{\gamma} = constant$$

Other related stuff

$C_D$ - so-called drag coefficient $\rightarrow$ yet to be figured out (or guesstimated ;) )
sources: https://www.researchgate.net/publication/253549900_Modeling_the_ascent_of_sounding_balloons_Derivation_of_the_vertical_air_motion
$A_b$ - balloon reference area ($m^2$)

Altitude variation of the pressure:

Theoretically valid only up to $\sim$ 20km

$$p = p_0 \cdot \left(1 - \frac{L \cdot h}{T_0} \right)^{\frac{g \cdot M}{R_0 \cdot L}} = p_0 \cdot \left(1 - \frac{g \cdot h}{c_p \cdot T_0} \right)^{\frac{c_p \cdot M}{R_0}} \approx p_0 \cdot exp \left(-\frac{g \cdot h \cdot M}{T_0 \cdot R_0} \right)$$ Check out the following link if you are interested in what the heck those parameters represent (actually they are straightforward...):
https://en.wikipedia.org/wiki/Atmospheric_pressure?fbclid=IwAR2keT2FBSF0ExzFWVDx_fpQHDhbOnQdD49PGO7yijUqn8QY_0d6BH12uc0

Altitude variation of the density:

Only up to $\sim$ 20 km? It maybe won`t be enough. We should discuss it next time!

$$\rho = \frac{p \cdot M}{R \cdot T}$$ where:
$p$ is described previously
$T = T_0 - L \cdot h$

$$\rho = \frac{p_0 \cdot M}{R \cdot T_0} \left( 1 - \frac{L \cdot h}{T_0} \right)^{\frac{g \cdot M}{R \cdot L} - 1}$$

Source: https://en.wikipedia.org/wiki/Density_of_air

Descent model

After the balloon bursts, the payload starts to drop slowed down by a parachute. The system acts as it is not subjected to inertial acceleration: after a brief transient of time weight is perfectly balanced by aerodynamic drag; then, the payload will fall down at an approximately constant speed called terminal velocity

$$v_t = \sqrt{\frac{2 \cdot m \cdot g}{C_D \cdot \rho_{air} \cdot A_p}}$$

Laterla displacement

"As the wind blows..." and also probably the hardest to figure out...