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: http://www.naic.edu/~pfreire/orbits/
Дата изменения: Sat Dec 20 23:36:29 2008 Дата индексирования: Tue Oct 2 00:07:55 2012 Кодировка: Поисковые слова: microwave background |
The discovery of a pulsar whose period changes significantly during the observation usually indicates that it is a member of a binary system. It is then important to determine the Keplerian orbital parameters of the system in order to investigate the astrophysics of the two stars and to obtain a coherent timing solution for the rotation of the pulsar.
The usual procedure for obtaining orbital parameters involves fitting a Keplerian model to a series of period measurements specified in time. Such a procedure works well, provided that it is possible to determine the rotational period on several occasions during a single orbit. However, there are often circumstances where this is not the case, such as where interstellar scintillation permits only sparse positive detections of a pulsar. This is the case of most of the millisecond pulsars in 47 Tucanae.
In what follows, we present a simple procedure for estimating the Keplerian orbital parameters of a binary that is completely independent of the distribution of the epochs of the individual observations. The only thing that is required is that the period derivatives (or accelerations) are known in each observation. Estimates of accelerations or period derivatives are normally provided by acceleration surveys, and can be easily refined using the technique of pulsar timing
We first present the equations for the period and acceleration of a pulsar in an eccentric binary system as a function of its position in the orbit. We then provide an analysis of the circular case. In Freire, Kramer & Lyne (2001) (from where most of this text and following equations are taken), we describe in more detail how circular binary orbits can be determined from the observed periods and accelerations of the pulsar.
Binary orbits in the acceleration/period plane
Figure 1 represents the orbit of a binary pulsar (point P)
around the centre of mass of the system (O) projected onto a plane that
contains the direction towards the Earth (i.e., perpendicular
to the plane of the sky, ), and the line of nodes where the
orbital plane intersects
. IN THIS PLANE, the orbit is an ellipse of
eccentricity
and projected semi-major axis
(
is the unknown inclination of the orbit relative to the plane of the
sky). Still in the same projected plane, r is the vector connecting O to
the pulsar's projected position, P, and
is the distance of the pulsar to O,
is the
projection of this to the line-of-sight.
is the longitude of
periastron and
is the angle of the pulsar to the periastron measured
at O, also called "true anomaly". According to Roy (1988), the
equation for
as a function of
is given by
ERRATUM: Note that the last term in this equation includes a
The time derivative of is the line-of-sight (or radial) velocity,
. Using the results in Roy (1988):
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(2) |
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(3) |
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(4) |
we obtain:
where is the orbital period of the binary.
The apparent rotational period of the pulsar
as a function of
and the intrinsic rotational period
is given by
if the total velocity of the pulsar, is small compared to
.
Differentiating equation 6, we obtain the acceleration of the pulsar along the line of sight:
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(7) |
This is how we convert the period derivatives to accelerations.
Differentiating equation 5 in time, we obtain as a function of
Plotting as a function of
, we obtain a parametric curve
that does not depend on time, and therefore does not require the solution
of Kepler's equation. This curve is illustrated in Figure
2 for a binary pulsar of spin period 10 ms and orbital
parameters
= 0.9,
,
= 1 day and
(this figure, taken from Freire, Kramer & Lyne (2001),
under-estimates the accelerations by the
term that was missed in eq. 4 of that paper).
and
The track followed by such pulsars in the period/acceleration space is
thus an ellipse centred on the point (, 0) and having as
horizontal and vertical semi-axes the values
and
respectively, with the pulsar moving in a clockwise direction.
Using these newly determined orbital parameters, we can calculate the
angular orbital phase for each th data point, i.e. for each pair of
acceleration and period measured (
,
):
Since the time of each observation is also known, we can determine the
time of the nearest ascending node, or simply
for each
observation:
We can use these values to achieve the correct orbit count between any two observations. One metod of doing this is descibed in Freire, Kramer & Lyne (2001), another more intuitive method is described in Chapter 4 of my Ph.D. Thesis (.ps.gz,.pdf.gz).
Using these methods, we can find the orbital parameters of any binary pulsar, no matter how badly under-sampled, as long as the accelerations are measurable. But this method is useful in a more general situation. It can provide a scientifically sound starting point in a "Time-Period" fit, even when the orbit is reasonably well sampled. Until now, this starting solution always involved some degree of guessing.