# CV stuff

### Accuracy of orbital period measurement

Let’s say you have measured the $HJD_0$ at times $t_1$ and $t_2$, separated by N orbital cycles. Then the orbital period, $P$, becomes:

$P=\dfrac{t_{2} - t_{1}}{N}$…(1)

From error propagation theory we know that if:

$P=f(t_1,t_2,N)$

then:

$\sigma^2_P=\left(\dfrac{\partial P}{\partial t_{1}} \sigma_{t_{1}}\right)^2+\left(\dfrac{\partial P}{\partial t_{2}} \sigma_{t_{2}}\right)^2$

in which we are ignoring the partial derivative of $P$ with respect to $N$ because we consider $\sigma_N$ to be zero. The equation becomes:

$\sigma^2_P=\left(\dfrac{-\sigma_{t_1}}{N}\right)^2+ \left(\dfrac{\sigma_{t_2}}{N}\right)^2=\dfrac{\sigma_{t_1}^2+ \sigma_{t_2}^2}{N^2}$

and thus:

$\sigma_P=\dfrac{1}{N}\sqrt{\sigma_{t_1}^2+ \sigma_{t_2}^2}$…(2)

Then using (1) in (2):

$\sigma_P=\dfrac{P}{t_2 - t_1}\sqrt{\sigma_{t_1}^2+ \sigma_{t_2}^2}$…(3)

In real life  $t_1$ and $t_2$ are typically reported with uncertainties of the order $1 \times 10^{-4}$ d. Therefore, for a system with period $\sim6$ h and $t_1$, $t_2$ separated by one year we expect an uncertainty of $\sim 1 \times 10^{-7}$ d for the orbital period.