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.