Calculate the frequency of peak synchrotron emission, ν_pk.

    This is Dulk (1985) equation 43, which is a fitting function assuming a
    power-law electron population. Arguments are:

    b
      Magnetic field strength in Gauss
    ne
      The density of electrons per cubic centimeter with energies greater than 10 keV.
    delta
      The power-law index defining the energy distribution of the electron population,
      with ``n(E) ~ E^(-delta)``. The equation is valid for ``2 <~ delta <~ 5``.
    sinth
      The sine of the angle between the line of sight and the magnetic field direction.
      It's not specified for what range of values the expressions work well.
    depth
      The path length through the emitting medium, in cm.
    E0
      The minimum energy of electrons to consider, in MeV. Defaults to 1 so that
      these functions can be called identically to the gyrosynchrotron functions.

    The return value is peak frequency in Hz.

    No complaints are raised if you attempt to use the equation outside of its
    range of validity.

def calc_synch_nu_pk(b, ne, delta, sinth, depth, E0=1.):
    """Calculate the frequency of peak synchrotron emission, ν_pk.

    This is Dulk (1985) equation 43, which is a fitting function assuming a
    power-law electron population. Arguments are:

    b
      Magnetic field strength in Gauss
    ne
      The density of electrons per cubic centimeter with energies greater than 10 keV.
    delta
      The power-law index defining the energy distribution of the electron population,
      with ``n(E) ~ E^(-delta)``. The equation is valid for ``2 <~ delta <~ 5``.
    sinth
      The sine of the angle between the line of sight and the magnetic field direction.
      It's not specified for what range of values the expressions work well.
    depth
      The path length through the emitting medium, in cm.
    E0
      The minimum energy of electrons to consider, in MeV. Defaults to 1 so that
      these functions can be called identically to the gyrosynchrotron functions.

    The return value is peak frequency in Hz.

    No complaints are raised if you attempt to use the equation outside of its
    range of validity.

    """
    coldens = ne * depth
    return (3.2e7 * sinth *
            E0**((2 * delta - 2) / (delta + 4)) *
            (8.7e-12 * (delta - 1) * coldens / sinth)**(2./(delta + 4)) *
            b**((delta + 2) / (delta + 4)))