# For using the acceleration due to drag formula, which coordinate frame should the satellite's position and velocity vectors be in?

I am working on simulating the space environment for a VLEO satellite. I am working on the atmosphere model where I am using density values from NRLMSISE00 model and plugging them into the drag acceleration formula.

$$a_{drag} = -\frac{1}{2} \rho \left(\frac{C_D A}{m} \right) v^2_{rel} \frac{\vec{v}_{rel}}{v_{rel}}$$

I had a question regarding the acceleration due to drag formula, which coordinate frame should the satellite's position and velocity vectors be in? I am guessing they have to be in ECI frame and the acceleration also is output in the ECI frame?

First things first: A real force is the same vector in all frames of reference. All that's needed is a transformation. It's only fictitious forces that change drastically between frames. For example, the fictitious centrifugal and Coriolis forces are nonzero in a rotating frame but vanish in an inertial frame.

Drag is a real force. This means one can calculate it in the frame that makes the most sense regarding calculation, and then if needed transform it to the frame in which numerical integration is performed. The same goes for the component of acceleration due to the drag force. Simply transform the resultant acceleration to the integration frame, if needed. I'm assuming you are integrating in a pseudo-inertial frame such as ECI. Integrating in ECEF is a royal pain.

Your atmosphere model (NRL MSISE00) is based on geodetic latitude, longitude, and altitude, plus time of day and day of year. This means you must transform your integrated ECI position to ECEF to calculate geodetic latitude, longitude, and altitude. It's not that much harder to transform ECI velocity to ECEF. Simply perform the transformation, and then subtract $$\vec\Omega_\text{ECEF}\times\vec r_\text{ECEF}$$, where $$\vec\Omega_\text{ECEF}$$ is the Earth's rotational angular velocity with respect to inertial, expressed in ECEF coordinates and $$\vec r_\text{ECEF}$$ is the position vector transformed to ECEF. For the purpose of calculating drag, $$\vec\Omega_\text{ECEF}$$ is easy: It's two pi radians per sidereal day in magnitude, pointing in the $$\hat z_\text{ECEF}$$ direction. You can get more sophisticated than this, but since drag is not well known, there's no point in doing so.

If you aren'y modeling winds, the ECEF vehicle velocity is $$\vec v_\text{rel}$$ as the atmosphere rotates more or less with the solid Earth (i.e., $$\vec v_\text{wind} = 0$$ in this simplified model.) If you are modeling winds, most models provide the winds as u* and v components, toward the local geodetic east and north, respectively. (Some wind models also have a geodetic up/down component as well.) These oftentimes come in the form of NetCDF files. There are multiple libraries in many languages that deal with NetCDF files. Do not roll your own.

The only thing you need to do is to transform these eastward and northward components of the wind from the local north-east-down frame (or the local east-north-up frame) to ECEF to yield $$\vec v_\text{wind}$$, thereby making $$\vec v_\text{rel} = \vec v_\text{spacecraft} - \vec v_\text{wind}$$.

After that, it's just a matter of calculating the drag force (or drag acceleration) and then transforming to inertial. Whether or not you are modeling winds, the transport theorem is not needed because as I mentioned at the top, drag is a real force. Simply transform to inertial.

• Okay, I will not be modelling wind for now so I will simply transform from ecef to eci once I get the values, thank you so much! Commented Feb 9 at 5:45
• Surely even at VLEO, the wind force is negligible anyway? In the thermosphere, particle movement is dominated by atmospheric tides rather than the physical interaction between molecules. Commented Feb 9 at 15:51
• @DarthPseudonym Those atmospheric tides (mostly nongravitaational) result in significant updrafts and downdrafts. It has been thought (and debated) since the 1970s whether the thermosphere is super-rotating. Both of these qualify as a "wind". Whether one looks at a gas as a "gas" or a bunch of particles depends very much on frequency. Even the extremely thin interstellar gas acts as a "gas" at extremely low frequencies. The thermosphere acts as a gas at diurnal frequencies, but acts as a bunch of particles at frequencies corresponding to an orbiting spacecraft passing through the thermosphere. Commented Feb 9 at 17:38