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Astron. Astrophys. 322, 846-856 (1997)

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2. Background

In what follows we will assume that RFM exists. In other words, the radiation received at a particular frequency, [FORMULA], can be clearly mapped to a certain altitude above the neutron star surface, [FORMULA], where the emission takes place. We are interested in the absolute value of the emission altitude as well as its dependence on frequency, generally assumed to be a power law,

[EQUATION]

Under this assumption, we might be able to observe a difference in the times-of-arrival (TOAs) of a pulse profile observed at different frequencies. Several contributions to this time difference are possible. Following Phillips (1991b; 1992, hereafter P92), we take four major effects into account.

Retardation: The most obvious contribution is a simple retardation delay. Let us consider a pulse emitted at two frequencies [FORMULA] and [FORMULA] with [FORMULA]. If the corresponding radiation is created at radii [FORMULA] and [FORMULA], respectively, then we expect a delay, [FORMULA] due to the different path lengths as simply given by

[EQUATION]

Here we have chosen the sign in the sense that [FORMULA] is positive if

[EQUATION]

is fulfilled.

Dispersion: Pulses propagating through the interstellar medium suffer a frequency dependent dispersion delay, so that the difference in the arrival times of pulses observed at two different frequencies, [FORMULA] and [FORMULA], is given by

[EQUATION]

with [FORMULA] as the electron column density along the path to the pulsar, which is called dispersion measure and usually quoted in units of cm-3 pc. The dispersion constant, [FORMULA], is defined as [FORMULA], where [FORMULA] and e are the mass and charge of an electron, respectively. Dispersion measures are usually quoted for a value of [FORMULA] cm-3 pc s, rather than for the physical value of [FORMULA] cm-3 pc s (Manchester & Taylor 1977).

Aberration: The rotation of the neutron star causes a bending of the radiation beam towards the rotational direction (cf. BCW). The corresponding deflection angle [FORMULA] depends on the emission altitude, [FORMULA], and is given (after P92) by

[EQUATION]

In this expression, [FORMULA] is the inclination angle between spin and magnetic axis and P the pulse period. An outside observer will, therefore, receive the pulse earlier by a fraction of [FORMULA] of the pulse period if [FORMULA] is measured in radians. Translated into time, the difference between the TOAs of signals received at two different frequencies is given by

[EQUATION]

where the sign is chosen in the sense of Eq. (3). In the case that the emission region is located well inside the light cylinder, i.e. [FORMULA], the relation simplifies to

[EQUATION]

Magnetic field sweep-back: Particles outflowing the open magnetic field lines will cause a toroidal component of the magnetic field near the light cylinder. As a result, the field lines will be swept back, i.e. bend towards a direction opposite to the sense of rotation. Following Shitov (1983) and P92, the sweep-back angle is given by

[EQUATION]

which gives rise to a time delay of approximately

[EQUATION]

Adding these major contributions we obtain for the total time difference

[EQUATION]

Within the canonical model that high frequency emission originates from closer to the neutron star than low frequency emission, i.e. validity of relation Eq.(3), the first and third term are positive while the second and last contribution are negative. In other words, while retardation and aberration effects are delaying the high frequency emission (with respect to low frequency emission), dispersion and magnetic field sweep back act in just the opposite way.

Since previous results indicate that even the low frequency emission takes place well inside the light cylinder (e.g. Cordes 1978, Matese & Whitmire 1980, BCW, P92 or Kijak & Gil 1996), we make use of the simpler expression for the aberration delay and obtain

[EQUATION]

where we have introduced

[EQUATION]

and used [FORMULA] 1. Knowing the exact dispersion delay, [FORMULA], and the magnetic inclination, [FORMULA], on the right hand side, we can measure the total time delay to determine or constrain [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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