nenupy.astro.astro_tools.dispersion_delay

nenupy.astro.astro_tools.dispersion_delay(frequency, dispersion_measure)

Dispersion delay induced to a radio wave of frequency (\(\nu\)) propagating through an electron plasma of uniform density \(n_e\).

The pulse travel time \(\Delta t_p\) emitted at a distance \(d\) is:

\[\Delta t_p = \frac{d}{c} + \frac{e^2}{2\pi m_e c} \frac{\int_0^d n_e\, dl}{\nu^2}\]

where \(\mathcal{D}\mathcal{M} = \int_0^d n_e\, dl\) is the Dispersion Measure (dispersion_measure). Therefore, the time delay \(\Delta t_d\) due to the dispersion is:

\[\Delta t_d = \frac{e^2}{2 \pi m_e c} \frac{\mathcal{D}\mathcal{M}}{\nu^2}\]

and computed as:

\[\Delta t_d = 4140 \left( \frac{\mathcal{D}\mathcal{M}}{\rm{pc}\,\rm{cm}^{-3}} \right) \left( \frac{\nu}{1\, \rm{MHz}} \right)^{-2}\, \rm{sec}\]

Parameters:

• frequency (Quantity) – Observation frequency.

• dispersion_measure (Quantity) – Dispersion Measure (in units equivalent to \({\rm pc}/{\rm cm}^3\)

Returns:

Dispersion delay in seconds.

Return type:

Quantity

Example:

from nenupy.astro import dispersion_delay
import astropy.units as u

dispersion_delay(
    frequency=50*u.MHz,
    dispersion_measure=12.4*u.pc/(u.cm**3)
)