Surface Force: Air Drag

1. Overview

1. Functions

AirDrag class inherits SurfaceForce base class and calculates air drag disturbance force and torque.

air_drag.cpp, air_drag.hpp : The AirDrag class is defined.
surface_force.cpp, surface_force.hpp : The base class SurfaceForce is defined.
- Note: SurfaceForce class inherits SimpleDisturbance class, and SimpleDisturbance class inherits Disturbance class. So, please refer them if users want to understand the structure deeply.
disturbance.ini : Initialization file

3. How to use

Make an instance of the AirDrag class in InitializeInstances function in disturbances.cpp
- Create an instance by using the initialization function InitAirDrag
Set the parameters in the disturbance.ini
- Select ENABLE for calculation and logging
- Select the following conditions of air drag calculation
  - Surface Temperature degC
  - Atmosphere Temperature degC
  - Molecular weight of the thermosphere g/mol

2. Explanation of Algorithm

1. `CalcCoefficients`

1. overview

CalcCoefficients calculates the normal and in-plane coefficients for SurfaceForce calculation. The air drag force acting on a surface is expressed as the following equation

\[ \begin{align} \boldsymbol{F}=-C_{n}\boldsymbol{n}+C_{t}\boldsymbol{t}\\ C_{n}=\frac{1}{2}\rho A v^2 C_{n}^{\prime}\\ C_{t}=\frac{1}{2}\rho A v^2 C_{t}^{\prime} \end{align} \]

This function mainly calculates the common part of the coefficient calculation. $C_{n}^{\prime}$ and $C_{t}^{\prime}$ are calculated in CalCnCt function, and they will be used in this function.

2. inputs and outputs

input
- $\boldsymbol{v}$: Relative velocity vector between the spacecraft and the atmosphere [m/s]
- $\rho$: air density [kg/m3]
output
- coefficients $C_{n}$ and $C_{t}$

2. `CalCnCt`

1. overview

CalCnCt calculates $C_{n}^{\prime}$ and $C_{t}^{\prime}$.

2. inputs and outputs

input variables
- $\boldsymbol{v}$: Relative velocity vector between the spacecraft and the atmosphere [m/s]
  - Currently, we assume that this value equals spacecraft velocity in the body-fixed frame.
input parameters
- $\sigma_{d}$: Diffuse coefficients for air drag
  - Ini file provide specularity for air drag $\sigma_{s}$, and the diffuse coefficient is derived as $\sigma_{d}=1-\sigma_{s}$.
  - Note: There is no absorption term for air drag. Thus total reflectivity is set as 1.
- $T_{w}$: Temperature of the surface [K]
- $T_{m}$: Temperature of the atmosphere [K]
- $M$: Molecular weight of the thermosphere [g/mol]
  - In the default ini file, we use $M=18$, and it is a little bit smaller than the molecular weight of atmosphere $M=29$. Structure of the Thermosphere provides information on the molecular weight of the thermosphere.
outputs
- $C_{n}^{\prime}$ and $C_{t}^{\prime}$

3. algorithm

$C_{n}^{\prime}$ and $C_{t}^{\prime}$ are calculated as following equations

\[ \begin{align} C_{n}^{\prime} &= \frac{2-\sigma_{d}}{\sqrt{\pi}}\frac{\Pi(S_{n})}{S^{2}}+\frac{\sigma_{d}}{2}\frac{\chi(S_{n})}{S^{2}}\sqrt{\frac{T_{w}}{T_{m}}}\\ C_{t}^{\prime} &=\frac{\sigma_{d}}{\sqrt{\pi}}\frac{\chi(S_{n})}{S^{2}}S_{t} \end{align} \]

$S, S_{n}, S_{t}$ are defined as follows
- $k=1.38064852E-23$ is the Boltzmann constant
- $\theta$ is the angle between the normal vector and the velocity vector
- $\cos{\theta}$ and $\sin{\theta}$ are calculated in SurfaceForce base class.

\[ \begin{align} S &= \sqrt{\frac{Mv^{2}}{2kT_{w}}}\\ S_{n} &= S\cos{\theta}\\ S_{t} &= S\sin{\theta}\\ \end{align} \]

$\Pi(x)$ and $\chi(x)$ are defined as follows
- where erf is the Gauss error function.
- These functions are defined as CalcFunctionPi and CalcFunctionChi.

\[ \begin{align} \Pi(x) &= x e^{-x^{2}}+\sqrt{\pi}(x^2+0.5)(1+erf(x))\\ \chi(x) &= e^{-x^{2}}+\sqrt{\pi}x(1+erf(x)) \end{align} \]

4. note

Please see the reference document for more information on detailed calculations.

The calculated magnitude of the air drag force is compared with other calculation results in three cases.

2. conditions for the verification

See the bottom table.

3. results

The calculation result is completely the same as the other calculation.

parameters/results	Case 1	Case 2	Case 3
$\sigma_{d}$	0.8	0.6	0.4
$\theta$ rad	0.202	0.202	0.202
$v$ m/s	7420	7420	7420
Out-plane force (S2E)	2.30297	2.68680	3.07062
Out-plane force (reference)	2.30297	2.68680	3.07062
Out-plane force (S2E)	0.31514	0.23636	0.15757
Out-plane force (reference)	0.31514	0.23636	0.15757