# Alan's baseball aerodynamics model

I look at it and dig through it for cross-references all the time but never write it down. So here's how Alan Nathan does the physics of a baseball, for my reference, and possibly for yours as well!

First, the drag model is given by:

\[

C_d= C_{d, 0} + C_{d, \omega} |\boldsymbol\omega(t)|

\]

where, in Nathan's spreadsheet, $\omega$ is the spin rate in $\textrm{rev min}^{-1}$ per-thou (i.e. thousandths of an RPM).

The spin rate decays as:

\[

\boldsymbol\omega(t)= \boldsymbol\omega_0 \exp(-\frac{t}{\tau} \frac{|\mathbf{V}|}{V_\mathrm{set}})

\]

where $\mathbf{V}$ is the velocity in feet per second, and the constants are given by $\tau= 30 \textrm{ s}$ and $V_\mathrm{set}= 100 \textrm{ mph}$. The exponential term is spin decay term with a decay parameter and scaled with velocity.

The lift model is given by:

\[

C_L= \frac{C_{L, 2} S}{C_{L, 0} + C_{L, 1} S}

\]

with parameters $C_{L, 2}= 1.120$, $C_{L, 1}= 2.333$, and $C_{L, 0}= 0.583$.

Last but not least, there's an approximator for the density that he uses, which corrects for altitude, barometric pressure, altitude, and relative humidity:

$$

\rho= \rho_0 \left( \frac{T_0}{T + T_0} \frac{p}{p_0} \exp(-\beta h_0) - 0.3783 x_{RH} \frac{p_{SVP}}{p_0} \right)

$$

where $\rho_0= 1.2929 \mathrm{~kg/m^3}$ is a reference density, $T_0= 273 \mathrm{~K}$ and $p_0= 101,325 \mathrm{~Pa}$ are the standard sea level temperature and pressures, $h_0$ is the elevation of the stadium above sea level in meters, $\beta= 121.7 \times 10^{-6} \mathrm{~m^{-1}}$ is a pressure decay rate. Last but not least, $x_{RH}$ is the relative humidity fraction, and $p_{SVP}$ is the saturation vapor pressure of air in Pascals.

The saturation vapor pressure is a function of the temperature by:

$$

p_{SVP}= 4.5841 \exp(\frac{18.687 \mathrm{~K} - T}{234.5 \mathrm{~K}} \frac{T}{257.14 \mathrm{~K} + T})

$$