## Atmospheric Drag Model

The spacecraft drag tool utilizes the NRLMSISE-00 atmospheric model to calculate the drag force experienced on a spacecraft at various altitudes. Two large factors in the atmospheric density are the solar radio flux, F10.7, and the geomagnetic index, ap, which can vary significantly over the course of a given 11 year solar cycle.

When initially sizing a propulsion system for a given LEO mission its a good idea to look at past atmospheric data and use an average of each parameter over multiple previous 11 year solar cycles. The tool takes data from the last 5 cycles normalized over the duration of the cycle and uses the average spline value at 0% and 50% of the solar cycle to capture average F10.7 and ap values at a low and high point in an average cycle. The normalized variation of F10.7 and ap shown below: The effect of F10.7 and ap values on atmospheric density is shown in the plot below displaying density at a few different points in the average solar cycle. The plot focuses on the 200 - 500 km range as it is most impacted by the differences in F10.7 and ap, and is generally the altitude range of interest for most LEO missions. Described in more detail in the spacecraft drag tool, the drag force depends on altitude (density and orbital velocity terms) and spacecraft design (coefficient of drag and wetted area terms). Given a spacecraft design, the estimated drag force can be calculated using the atmospheric density from average solar cycle above.

## Generalized Impact of Atmospheric Drag

### Normalized Drag Force

It is advantageous to generalize the drag affects to an arbitrary satellite design in order to understand how a propulsion system will need to scale for drag compensation at various mission altitudes. The first general result can be seen when the drag force is normalized by the ballistic coefficient (Cd A), which will allow the result to extend to an arbitrary design(s). The normalized drag force plot is shown below: ### Normalized Drag Power

A physics based analysis of the power that will be required to overcome drag is also useful for sizing. The effect of drag of the spacecraft is to reduce the orbital kinetic energy and the rate at which kinetic energy is removed is known as drag power an expression that describes the power exerted by the atmosphere on the spacecraft. To offset drag and remain in orbit the spacecraft propulsion system must supply at least this much power to compensate for the drag. The power that must be continually supplied to maintain the orbit of a spacecraft having a Cd = 2 and A = 0.5 m2 (ie CdA = 1) m2 is shown below: The dotted lines in the figure above are interesting because they represent the power any EP technology must impart to the spacecraft to maintain orbit. The solid lines represent the power that must be input into an electric propulsion system having a total efficiency of ηt = 30% since low-power EP systems operate at some total efficiency number between 20% and 40% after accounting for all the losses. It is important to note that solid lines represent the steady state power, meaning it is the power that must be supplied continuously in the case that the EP system thrusts continually to exactly offset the drag. If the system is to be operated at some duty cycle less than 100% (which is required for most missions), then the power demand during operation must scale inversely with the duty cycle in order to replace the lost kinetic energy. The total energy consumed is the same whether the thruster fires continuously or intermittently.

### Normalized Drag Impulse

A propulsion system must consume stored propellant to provide impulse at least equal to the drag impulse, and more if the mission requires other uses of propellant such as station-keeping, phasing, COLA, or plane changes. The normalized drag impulse (drag force acting over time) for a year is shown below: ### Normalized Drag Propellant

Now the propellant mass required to compensate for drag can be found by equating the propulsion system impulse to the drag impulse (again, only if no other maneuvering is required). To do this, we make an approximation that the total spacecraft mass does not change appreciably throughout a given drag make-up maneuver. This allows for the calculation of propellant mass without knowledge of spacecraft mass (and therefore without needing the rocket equation). The approximation is assuming the impulse delivered by the propulsion system during an individual burn translates directly into delta-V which offsets the delta-V imposed by drag on the same-mass spacecraft prior to the burn. Compared to reality, this assumption will slightly over-estimate the propellant mass required as a spacecraft's mass will decrease and the same impulse will translate into a larger delta-V, but for our initial sizing we'll take the conservative approach.

The normalized propellant mass to counter drag is shown below: ### Normalized Drag Duty Cycle

Until now we haven't prescribed anything about what propulsion technology is being employed for a given mission, the above modeling and generalization only mentions specific impulse in the propellant mass estimation plot to illustrate its effect of mass required over the range of altitudes.

Now, we will a assume parameter about the propulsion technology being used; a low power electric propulsion system operating a total efficiency of 30%.

For a given electrical power input, the thrust of an EP system is inversely proportional to the specific impulse. This means that high specific impulse systems will have a lower thrust at the same input power and will therefore need to operate over a longer duration to offset drag. At various normalized input powers each altitude of interest can be plotted to assess the required thruster duty cycle in terms of whatever mission time fraction CONOPS allows. The fraction of mission time spent thrusting, and therefore consuming power, at 250 km is shown below for various normalized input power: Here we can see that for a given technology that is 30% efficient operating at 1000 seconds with 100 W of input power (our previous generic spacecraft with a CdA = 1 m2) the duty cycle required to compensate for drag at 250 km is 10% (green line).

### Normalized Drag Over Time

The above generalized results have applied to single points in time relative to the 11.1 year solar cycle and varied the altitude to see the impacts of flying at various mission altitudes. Now we're interested in the change in drag force experienced by an arbitrary spacecraft design throughout an average solar cycle. The figure below shows the variation in drag with time at 250 km: Here, the two lines correspond to starting a mission at either 0% or 50% of an average solar cycle and show the expected normalized drag force for the next 11.1 years of an average cycle at 250 km. As expected, the drag varies appreciably (see variation in F10.7 at the top of this post) and depending on the launch date and desired mission life, a spacecraft can experience very different drag forces which will have an impact on propulsion sizing the could also affect maneuver CONOPS.