The potential temperature $\theta$ of a parcel is defined to be the temperature the parcel would have if it were moved, adiabatically, from some pressure level $P$ (where the parcel's temperature is $T$) to a reference pressure level $P_0$. In most cases, the reference pressure level is the surface pressure ($P_0$=1000 mb).

The derivation of the formula is based on the First Law of Thermodynamics in the form shown below

$dq = c_pdT - \alpha dP$

An adiabatic process implies $dq = 0$, so the equation becomes

$c_pdT = \alpha dP$

The following form of the Ideal Gas Law is used.

$PV = mRT$

If $m=1$, then the volume $V = \alpha$, the specific volume of the parcel.

Consequently, $\alpha = \frac{RT}{P}$, and the First Law takes the form

$c_pdT = RT\frac{dP}{P} \Rightarrow \frac{dT}{T} = \frac{R}{c_p}\frac{dP}{P}$

Because the parcel is moved from the level of pressure $P$ and temperature $T$ to the reference level with pressure $P_0$ and potential temperature $\theta$, we integrate the previous differential expression in the following way

$\displaystyle{\int_T^{\theta}\frac{dT}{T}} = \frac{R}{c_p}\displaystyle{\int_P^{P_0}\frac{dP}{P}}$

$ \Rightarrow \left. \ln T \right|_T^{\theta} = \frac{R}{c_p}\left. \ln P \right|_P^{P_0}$

$ \Rightarrow \ln \theta - \ln T = \frac{R}{c_p}\left(\ln P_0 - \ln P \right)$

$ \Rightarrow \ln \left(\frac{\theta}{T}\right) = \frac{R}{c_p}\ln \left( \frac{P_0}{P} \right)$

$ \Rightarrow \frac{\theta}{T} = \left(\frac{P_0}{P}\right)^{R/c_p}$

$ \Rightarrow \theta =T \left(\frac{P_0}{P}\right)^{R/c_p}$

The value of $\frac{R}{c_p}$ is $\frac{287}{1004}$ = 0.2859