Physical parameter estimation with MCMC from observations of Vela X-1 3
one or two parameters, while the tuning of parameters scales
as N (N + 1)/2 in an N -dimensional space for traditional
algorithms. It thus reduces computational costs during the
fitting procedure. A complete discussion of the MCMC
methods and the algorithms can be found in Ref. [14] and
references therein.
3. Application to the spectra of Vela X-1
In this section, our method is applied to the spectral study
of Vela X-1. Vela X-1 is a well-studied X-ray binary, with
extensive astronomical simulations of observations
[4, 6, 7]
,
several linked laboratory experiments
[18]
and related theo-
retical modeling
[19, 20]
, which provides a chance to test the
performance of this method through a thorough comparison.
Vela X-1 is a pulsing, eclipsing high mass X-ray binary
(HMXB) system with highly ionized gases, which was
first identified by Ulmer et al.
[21]
. Previous observations
and theoretical studies have given a reasonably complete
description of the system properties. The global structure of
the stellar wind in the system was first modeled by S99 using
the spectra in the Eclipse phase observed by ASC A Solid-
State Image Spectrometer. Other physical properties of Vela
X-1 were provided by the high-resolution X-ray spectra
during different phases observed with the Chandra High-
Energy Transmission Grating Spectrometer (HETGS)
[22, 23]
.
G04 first presented the variation of spectra in three orbital
phases: Eclipse, φ = 0.25, and φ = 0.5. In addition, W06
probed the stellar wind dynamics and ionization structures
by a quantitative analysis of Doppler shift and line intensities
with the same X-ray spectra. The previous literature studies
are considered to be sufficiently convincing to be used as
benchmarks. With the results of these studies, it is possible
for us to verify our analysis method.
3.1. Observations
The datasets of Vela X-1 used in our analysis were observed
with the Chandra HETG/ACIS-S – the High-Energy Trans-
mission Grating/Advanced Imaging Spectrometer, during
three different orbital phases centered on Eclipse, φ = 0.25
and φ = 0.5 in 2001, with Chandra ObsIDs 1926, 1928 and
1927, respectively. The observation and instrument details
can be found in Ref. [24], Ref. [22] and G04. Table 1
summarizes the observation dates and exposure times. The
full datasets including counts spectra, ancillary response
files (ARFs), and redistribution matrix file (RMFs) are all
obtained from public archive Transmission Grating Catalog
and Archive (TGCat)
1
. Only the first-order events of the
high-energy grating (HEG) data are used in our analysis. The
data of positive and negative spectral orders are combined
1
http://tgcat.mit.edu/.
Table 1. A summary of observation information taken from
Chandra data archive
a
.
ObsID Orbital phase Start date Exposure [s]
1926 Eclipse 2001-02-11 21:19:13 83.15
1927 0.5 2001-02-07 09:56:13 29.43
1928 0.25 2001-02-05 05:28:51 29.57
a
http://cda.harvard.edu/chaser/dispatchOcat.do.
to decrease the error bars of observation. Flux correction
was done with the Interactive Spectral Interpretation System
(ISIS, version 1.6.2-30)
2
for the observed data to have
the same dimensions as the models and to improve the
agreement with physical predictions. The background for the
observations is not measured independently, which is diffi-
cult to be resolved from the source of interest. Practically,
the background could be subtracted from the observation
signals, as was done in W06. But it is treated as a correction
of observation error in the present work because subtracting
the background could cause loss of useful signals.
3.2. Model
The spectra of Vela X-1 are considered to be composed
of continuum and line emission (W06). Therefore, in the
present work, for the likelihood function Equation (2), a
linear sum of parameterized continuum (F
cont
) and line
(F
line
), that is, F
mod
= F
cont
+ F
line
, is used to model the
spectra of Vela X-1. For F
line
, we mainly focus on He-
like triplets which have relatively high S/N -ratio and are
commonly used in photoionization plasma diagnostics.
3.2.1. Continuum
As to the continuum emissions of Vela X-1, we adopt a two-
component model (S99), having the form
F
cont
(E) = A
scat
exp
−σ (E)N
scat
H
E
−Γ
scat
keV
+ A
dir
exp
h
−σ (E)N
dir
H
i
E
−Γ
dir
keV
, (5)
where F
cont
is the energy-resolved photon flux
(photon · cm
−2
· s
−1
· keV
−1
). Equation (5) indicates that the
intrinsic continuum radiation of Vela X-1 consisted of two
components: (1) the first term (labeled with ‘scat’) represents
the intrinsic continuum radiation, which is strongly absorbed
when the neutron star passes through the surrounding stellar
wind; (2) the second term (labeled with ‘dir’) is the direct
continuum which is blocked by the companion. A
scat
and
A
dir
, (units: photon · cm
−2
· s
−1
· keV
−1
), are normaliza-
tions corresponding to the two components. σ (E) is the
2
see http://space.mit.edu/asc/isis/.
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