Low frequency QPOs and possible change in the accretion geometry during the outbursts of Aquila X$-$1
Wenda Zhang and Wenfei Yu
C. D. Biemesderfer\altaffilmark{4
5
R. J. Hanisch\altaffilmark{5
We have studied the evolution of the Low Frequency Quasi-Periodic Oscillations (LFQPOs) during the rising phase of seven outbursts of the neutron star Soft X-ray Transient (SXT) Aql X$-$1 observed with the Rossi X-ray Timing Explorer (RXTE). The frequency correlation between the low frequency break and the LFQPO sampled on the time scale of $\sim$2 days was seen. Except for the peculiar 2001 outburst, the frequency of the LFQPOs increased with time before the hard-to-soft state transition up to a maximum $\nu_{max$ at $\sim$31 Hz, a factor of $\sim$5 higher than those seen in black hole transients such as GX 339$-$4, making the maximum QPO frequency a likely indicator of the mass of the central compact object. The characteristic frequencies increased by around ten percent per day in the early rising phase and accelerated to nearly one hundred percent per day since $\sim$2 days before the hard-to-soft state transition. We examined the dependence of the frequency $\nu_{LF$ on the source flux $f$ and found an anti-correlation between the maximum frequency of the LFQPOs and the corresponding X-ray luminosity of the hard-to-soft transition (or outburst peak luminosity) among the outbursts. We suggest that X-ray evaporation process can not be the only mechanism that drives the variation of the inner disk radius if either of the twin kHz QPO corresponds to the Keplerian frequency at the truncation radius. %We suggest that if the LFQPOs correspond to %Lense-Thirring precession, the rate-of-change of the frequency of the %LFQPOs then actually measure the evolution of the Bardeen-Petterson %radius during outbursts.
aqlx1_150505.tex
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%\shortauthors{Djorgovski et al.}

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\title{Low frequency QPOs and possible change in the accretion
geometry during the outbursts of Aquila X$-$1}

%% Use \author, \affil, and the \and command to format
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\author{Wenda Zhang and Wenfei Yu} 
\affil{Shanghai Astronomical Observatory and Key Laboratory for
Research in Galaxies and Cosmology, Chinese Academy of Sciences, 80
Nandan Road, Shanghai, 200030, China; wenfei@shao.ac.cn}

%\author{C. D. Biemesderfer\altaffilmark{4,5}} \affil{National
%Optical Astronomy Observatories, Tucson, AZ 85719}
%\email{aastex-help@aas.org}

%\and

%\author{R. J. Hanisch\altaffilmark{5}} \affil{Space Telescope
%Science Institute, Baltimore, MD 21218}

%% Notice that each of these authors has alternate affiliations, which
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%\altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American
%Observatory. CTIO is operated by AURA, Inc.\ under contract to the
%National Science Foundation.} \altaffiltext{2}{Society of Fellows,
%Harvard University.} \altaffiltext{3}{present address: Center for
%Astrophysics, 60 Garden Street, Cambridge, MA 02138}
%\altaffiltext{4}{Visiting Programmer, Space Telescope Science
%Institute} \altaffiltext{5}{Patron, Alonso's Bar and Grill}

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\begin{abstract}
We have studied the evolution of the Low Frequency Quasi-Periodic
Oscillations (LFQPOs) during the rising phase of seven outbursts of
the neutron star Soft X-ray Transient (SXT) Aql X$-$1 observed with
the \textit{Rossi X-ray Timing Explorer (RXTE)}. The frequency
correlation between the low frequency break and the LFQPO sampled on
the time scale of $\sim$2 days was seen. Except for the peculiar 2001
outburst, the frequency of the LFQPOs increased with time before the
hard-to-soft state transition up to a maximum $\nu_{max}$ at $\sim$31
Hz, a factor of $\sim$5 higher than those seen in black hole
transients such as GX 339$-$4, making the maximum QPO frequency a
likely indicator of the mass of the central compact object. The
characteristic frequencies increased by around ten percent per day in
the early rising phase and accelerated to nearly one hundred percent
per day since $\sim$2 days before the hard-to-soft state transition.
We examined the dependence of the frequency $\nu_{LF}$ on the source
flux $f$ and found an anti-correlation between the maximum frequency
of the LFQPOs and the corresponding X-ray luminosity of the
hard-to-soft transition (or outburst peak luminosity) among the
outbursts. We suggest that X-ray evaporation process can not be the
only mechanism that drives the variation of the inner disk radius if
either of the twin kHz QPO corresponds to the Keplerian frequency at
the truncation radius.

%We suggest that if the LFQPOs correspond to
%Lense-Thirring precession, the rate-of-change of the frequency of the
%LFQPOs then actually measure the evolution of the Bardeen-Petterson
%radius during outbursts. 
\end{abstract} 

\keywords{X-rays: binaries}

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%\keywords{globular clusters: general --- globular clusters:
%individual(NGC 6397, NGC 6624, NGC 7078, Terzan 8}

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%% Otherwise, LaTeX will issue an error. 
\section{Introduction}
Low Mass X-ray Binaries (LMXBs) are binary systems that contain a
black hole (BH) or neutron star (NS) primary and low mass companion.
The primary accretes from the companion through an accretion disk
around the compact object which emits predominately in the X-ray band
at high mass accretion rates. Soft X-ray transients (SXTs) are
transient LMXBs. They spend most of the time in the quiescent state
at low luminosity, and occasionally turn into outbursts during which
their luminosity increases by several orders of magnitude. During
outbursts, SXTs show distinctive states with coupled spectral and
timing properties: in the Low-Hard State (LHS) the soft X-ray (e.g.,
0.3--10 keV) energy spectrum is dominated by a power-law spectral
component and the Fourier power spectrum at low frequency is
dominated by broad band noise component and quasi-periodic
oscillations (QPOs), and the variability amplitude is quite large at
a few tens percent level, while in the High-Soft State (HSS) the soft
X-ray energy spectrum is dominated by a thermal disk component and
the Fourier power spectrum is dominated by a power-law noise
component with little variability \citep[see,
e.g.,][]{belloni_evolution_2005,remillard_x-ray_2006}.

Systematic studies of the state transitions in SXTs during the rising
phase of transient outbursts suggest that the accretion process is
non-stationary: the luminosity of the hard-to-soft state transitions
can occur at quite different luminosity, spanning two orders of
magnitude which correlates with the rate-of-increase of the
luminosity as well as the peak luminosity of the corresponding
outbursts
\citep{yu_correlation_2004,yu2007,yu_state_2009,tang_rxte/asm_2011}.
This indicates that the mass accretion rate cannot be the only
parameter that determines the transition between the accretion states
and the rate-of-change of the mass accretion rate is one important
parameter which determines the regimes of the accretion states in SXT
outbursts.

The most popular picture for the X-ray energy spectra in the distinct
states of black hole and neutron star SXTs is the truncated disk
picture, such as the model proposed in
\citet{esin_advection-dominated_1997}. In this picture the change of
the energy spectrum is associated with the change of the accretion
geometry. A geometrically-thin but optically-thick disk is truncated
at the truncation radius. The region within the radius is filled with
a hot flow such as the radiatively-inefficient accretion flow. In the
quiescent and the hard state, the truncation radius is quite large.
During the rising phase of transient outbursts the truncation radius
moves inward with increasing mass accretion rate, and reaches the
innermost stable orbit during the soft state. The detailed process
that truncates the disk is not clear, and proposals for the physical
mechanism include magnetosphere-disk interaction \citep[][for neutron
star systems only]{ghosh_accretion_1977} and disk evaporation
\citep[e.g.,][]{meyer_accretion_1994,liu_evaporation_1999,meyer_re-condensation_2007}.

It was suggested that timing features, especially the quasi-periodic
oscillations (QPOs) shown in the power spectra, might be the probe of
the accretion geometry and its change in X-ray binaries several
decades ago
\citep[e.g.,][]{klis85a,klis85b,miller_sonic-point_1998,stella_lense-thirring_1998,osherovich_kilohertz_1999,ingram2009}.
The variation in the characteristic frequencies of the QPOs or noise
components may reflect the change in the inner most edge of the
accretion disk or the size of the corona. Low Frequency
Quasi-Periodic Oscillations (LFQPOs) are the most obvious feature in
the Fourier power density spectra (PDS) of black hole SXTs in the
hard and the intermediate state. For black hole transients, the
LFQPOs can be classified into A, B and C-type QPOs
\citep{wijnands_complex_1999,remillard_characterizing_2002}. Among
them, the C-type QPO appears in the hard and hard-intermediate states
and its characteristic frequency appears to correlate with the
luminosity. There are similar QPOs and broad noise components in
neutron star LMXBs. For example, \citet{yu_hard_2003} showed that the
1--20 Hz LFQPOs and broad noise components in the neutron star
transient Aql X$-$1 are associated with the LFQPOs and broad noise
components commonly seen in black hole transients during outbursts.

Aql X$-$1 is a well-known neutron star transient with very frequent
outbursts
\citep[e.g.,][]{priedhorsky_long-term_1984,kitamoto_unstable_1993,simon_recurrence_2002}.
Its neutron star nature was revealed by the detection of type-I X-ray
bursts from the source
\citep[e.g.,][]{koyama_discovery_1981,czerny_persistent_1987}. Its
neutron star spin was detected as well
\citep{zhang98,casella_discovery_2008}. Based on its spectral and
temporal properties \citep{reig_correlated_2000}, Aql X$-$1 has been
classified as an ``Atoll'' source, of which great similarities of
both timing and spectral properties to those black hole X-ray
binaries have been seen. This makes Aql X$-$1 an ideal source for the
study of the evolution of the characteristic timescales in the
accretion flow during the rising phase of their outbursts, during
which drastic changes in the accretion geometry are expected. In this
$Paper$ we study the evolution of the LFQPOs during the rising phase
of seven outbursts of the neutron star SXT Aql X$-$1 with the
observations performed with the \textit{RXTE}. 

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\section{Observations and Data Reduction}

We made use of the \textit{RXTE}/ASM and the MAXI long-term X-ray
monitoring observations of Aquila X$-$1 to identify X-ray outbursts
in this study. \textit{RXTE}/PCA pointed observations were used to
study X-ray timing and spectral properties. We have analysed a total
of 611 \textit{RXTE} observations of Aquila X$-$1 performed from 1996
to 2011, starting from the observation 10072-06-01-00 to the
observation 96440-01-10-05.

\subsection{Long-term Light Curves and Evolution of the Hardness Ratio}

Figure 1 shows the long-term X-ray light curve of Aquila X-1. We
converted the \textit{RXTE}/ASM 2--12 keV flux in unit of Crab by
dividing the ASM 2-12 keV count rate by the mean Crab count rate of
75.6 cts/s. Similarly, for the later period when \textit{RXTE}
stopped operation, we obtained the source flux in 2--10 keV by
combining the 2--4 keV and the 4--10 keV MAXI data which are publicly
available.

Figure 2 shows the X-ray light curve and the X-ray hardness ratio
seen with the pointed observations of the \textit{RXTE}/PCA. We
extracted source light curves from the PCA standard
products\footnote{http://heasarc.nasa.gov/docs/xte/recipes/stdprod\_guide.html}
which provide data with a time resolution of 16 seconds from the STD
2 PCA data. The light curves in the PCA standard products have five
energy bands, namely 2--9, 2--4, 4--9, 9--20 and 20--60 keV. We
identified the time intervals that were associated with Type-I bursts
following the screening method mentioned in
\citet{klein-wolt_identification_2008} and excluded these time
intervals. Then we calculated the mean count rates per PCU in the
five energy bands, and the hardness ratio (HR), which was defined as
the count rate ratio between the 9--20 keV and the 2--9 keV in each
observation.

\subsection{Timing Analysis}

We made use of the PCA data with a time resolution better than 0.25
ms and combined all available data in different energy bands whenever
possible. We extracted the data and rebinned the light curves into
1/4096 s resolution and then generated the Fourier power spectrum for
every 128 s segment. In the analysis of each power spectrum, we took
the average power above 1800 Hz to estimate the white noise level,
which was then subtracted. We fit each power spectrum with a model
composed of several Lorentzians in which zero-centered Lorentzian
represents the band-limited noise components and narrow Lorentzians
represents the QPOs following the previous approach
\citep{belloni_unified_2002}. A power-law component was included to
fit the Very Low Frequency Noises (VLFNs) when necessary. These
model fits were performed using a custom package making use of the
MPFIT
package\footnote{\url{http://cow.physics.wisc.edu/~craigm/idl/}}.
The best-fit model was achieved by minimizing the $\chi^2$, and the
uncertainty was obtained from the distribution of $\chi^2$.
Throughout this paper the error quoted for the power spectral fit
represents the 1$\sigma$ uncertainty.

\subsection{Spectral Analysis}

In order to estimate the X-ray flux of the source, we fit the 3--20
keV Standard 2 PCA energy spectra of the observations when the source
was in the hard state during the rising phase of each outburst. The
X-ray spectral model we used consisted of four components, namely a
power-law component representing the emission from the hot accretion
flow, a blackbody component for the emission from the neutron star
surface, a Gaussian line component with its central energy fixed at
6.5 keV representing a broad Iron emission line, and the absorption
by Galactic neutral hydrogen. We fixed the column density of the
neutral hydrogen to $3.4\times10^{21}~\rm atoms~cm^{-2}$, according
to \citet{dickey_h_1990}. The 2--20 keV flux was then obtained from
the spectral fits in XSPEC 12.8.2. 

\section{Results} 
\subsection{Outburst Sample}

As shown in Figure~\ref{fig:pcalc}, the combined \textit{RXTE}/ASM
and MAXI 2--10 keV light curve of Aql X$-$1 indicates 16 outbursts in
total in a period of 16 years, of which the \textit{RXTE}
observations covered the hard X-ray state during the rising phase of
seven outbursts, which occurred in 1999, 2000, 2001, 2004, 2009, 2010
and 2011, respectively, as indicated by the triangles in the plot.

\begin{figure}
\plotone{asmlc.ps}
\caption{\textit{RXTE}/ASM and MAXI 2--10 keV light curve of Aql
X$-$1. Light curve after MJD 55500 is MAXI/2--10 keV light curve.
Triangles indicate the seven outbursts we analyzed.
\label{fig:pcalc}}
\end{figure}

%20141019

The PCA 2--20 keV count rate and the HR during the seven outbursts
are shown in Figure~\ref{fig:lchr}. The peak count rate ranged from
$\sim$200 (the 2001 outburst) to $\sim$1800 (the 2011 outburst) $\rm
counts~s^{-1}~PCU^{-1}$, spanning nearly one order of magnitude in
the \textit{RXTE} observations. For the 1999, 2000, 2004, 2010 and
2011 outbursts, the hard-to-soft state transitions are clearly shown
by the abrupt decrease of the HR from $\sim$1.0 to $\sim$0.2. For the
2001 outburst, the hard-to-soft state transition was identified in
\citet{yu_hard--soft_2007}, which was made based on HEXTE(15--250
keV)/PCA(2--9 keV) hardness ratio. So we used the previous
identification. The time of the hard-to-soft state transition
corresponds to the observation 60054-02-04-00 on MJD 52095. It is
also worth noting that the 2001 outburst was peculiar since it was
not a Fast Rise with Exponential Decay (FRED) outburst as others. In
addition, the hard-to-soft state transition in the 2001 outburst was
also peculiar since it occurred when the luminosity declined
\citet{yu_hard--soft_2007}. For the 2009 outburst, we identified the
time corresponding to the hard-to-soft state transition based on
source timing properties. This is based on the fact that for other
atoll sources, such as 4U 1608$-$52, the state transition is
accompanied by the transition of properties of the power spectra (see
for example, Figure 7 of \citealt{van_straaten_atoll_2003}). In Aql
X$-$1 we saw such a transition in the power spectra from the
observation 94076-01-04-04 taken on MJD 55152.2 to the observation
94076-01-04-05 taken on MJD 55153.1, hence we can conclude that the
source left the hard state between these two observations.

% based on timing properties. 

\begin{figure*}
\plotone{allhr.ps}
\caption{Evolution of the PCA count rates and the X-ray hardness
  ratio of the seven outbursts for which hard states during rising
  part of the outburst were covered by the \textit{RXTE}. The PCA
  Std2 2--20 keV light curves are plotted in the upper panels while
  the corresponding 9--20 keV/2--9 keV hardness ratios are plotted in
  the lower panels. The year when the outbursts began are indicated
  in the upper right corner of each panel.\label{fig:lchr}}
\end{figure*}

%Along with the transition in energy spectra, the power spectra also
%transitioned from multi-Lorentzian to VLFN dominated spectra. 

\subsection{The Evolution of the Power Spectra}

Figure~\ref{fig:pdsevo} shows the evolution of the power spectra in
the rising phase of the seven outbursts during which the LFQPOs and
the band-limited noise components were seen. The most obvious
component below 100 Hz is the LFQPO component of which the frequency
evolved gradually from several Hz to tens of Hz. The LFQPO component
also corresponds to the maxima in the $\nu P_{\nu}$ vs. Frequency
plot below 100 Hz. We have found that the LFQPOs had the following
properties:

\begin{enumerate}
  \item In all except the 2001 outburst, the frequency increased with
	time before the state transition. This indicates that the
	characteristic timescale corresponding to the LFQPO decreased
	with time during the rising phase of the outbursts.

 \item In all except the 2001 outburst, the frequency appeared to
increase more rapidly just before the hard-to-soft state transition
than during the earlier stages. This is best-illustrated in the 1999
outburst. Notice that the power spectra are plotted in the
logarithmic scale and the power spectra are shifted by multiplying a
factor proportional to the time separation between the observations.
\end{enumerate}

\begin{figure*}
%\epsscale{0.9} 
\plotone{evo_all.ps} 
\caption{The evolution of the Fourier power spectra in the seven
  X-ray outbursts of Aql X$-$1. The power has been shifted vertically
  by a factor proportional to the time separation between
  observations. The years when these outbursts started are indicated
  in the upper-left corner of each panel. For the 2001, 2004, 2009,
  and 2010 outbursts, the power spectra seen in several observations
  obtained at the beginning of the outbursts are not shown due to
  limited space in the figures. \label{fig:pdsevo}}
\end{figure*}

To quantitatively analyze the frequency evolution of the LFQPO and
other noise components, we fit the power spectra with
multi-Lorentzians following the previous approach
\citep{belloni_unified_2002}. By doing so the power spectra can be
decomposed into several Lorentzian components empirically
\citep[e.g.,][]{belloni_unified_2002,van_straaten_multi-lorentzian_2002,van_straaten_atoll_2003,
klein-wolt_identification_2008}, allowing us to study frequency
correlations among the characteristic frequencies. Some examples of
our model fits can be seen in Figure~\ref{fig:pdsfit1}, in which we
show the power spectra with the best-fit models corresponding to
three observations performed during the 1999 outburst. The components
with the lowest frequency can be identified as the noise component,
$L_b$, as defined in previous studies. Two distinct components, a
narrow component at a lower frequency and a prominent broader
component with a higher frequency, were found in the frequency range
from several Hz to tens of Hz. The broader and stronger components
are the LFQPOs. In more than half of the observations, the narrow
feature is not required in the power spectral fit. The components
with higher frequencies are identified as $L_l$ and $L_u$, following
the conventions used in \citet{belloni_unified_2002}.

\begin{figure}
%\epsscale{0.7}
\plotone{threepds.ps}
\caption{The power spectra corresponding to three observations during
  the 1999 outburst of Aql X$-$1. The data are plotted with solid
  lines and the best-fit models are over-plotted. We identified those
  components and plotted the $L_b$, $L_{lf}$, the harmonics of
  $L_{lf}$, $L_l$ and $L_u$ in dotted, dash-dot, dashed,
  dash-dot-dot-dot, and long dashed lines, respectively.
  \label{fig:pdsfit1}}
\end{figure}

It has been found that the characteristic frequencies of broad noise
and narrow QPO components are correlated
\citep[e.g.][]{wijnands_broadband_1999,psaltis_correlations_1999,belloni_unified_2002}.
To compare with the results obtained from other sources and to study
how these characteristic frequencies are correlated during single
outbursts in the \textit{RXTE} era, we plotted the LFQPO
characteristic frequency $\nu_{LF}$ vs. break frequency $\nu_b$,
along with the data shown in the WK99 relation
\citep{wijnands_broadband_1999} and the BPK02 relation
\citep{belloni_unified_2002} in Figure~\ref{fig:corr}. The
correlation of $\nu_{LF}$ and $\nu_b$ in Aql X$-$1 is consistent with
the correlation in other sources as expected. 

\begin{figure}
\epsscale{1.0}
\plotone{corr.ps}
\caption{The correlation between $\nu_{LF}$ and $\nu_b$ in Aql X$-$1.
The open diamonds represent the correlation in other black
hole/neutron star SXTs reported in \citet{wijnands_broadband_1999}
and \citet{belloni_unified_2002}, and the solid circles represent our
results on Aql X$-$1. The correlation in Aql X$-$1 agrees well with
the correlations established in other SXTs. \label{fig:corr}}
\end{figure}

\subsection{Maximum LFQPO Frequency and the Evolution Pattern}
\label{sec:evo}

The evolution of the characteristic frequency of the LFQPOs in seven
outbursts is shown in Figure~\ref{fig:qpoevo}. First we examined the
maximum LFQPO frequency the source can reach in the hard states; this
frequency may correspond to a certain radius or timescale in the
accretion flow when the state transition occurs. As seen from
Figures~\ref{fig:qpoevo}, the maximum LFQPO frequency varies among
outbursts, from $6\pm0~\rm Hz$ in the 2000 outburst to $38\pm12~\rm
Hz$ in the 2001 outburst, although the uncertainty of the latter is
quite large. In the 2009 outburst, the maximum frequency was
$30\pm5~\rm Hz$. We took the weighted average of the maximum LFQPO
frequency of the 1999 and the 2001 outburst, i.e., $\sim31~\rm Hz$,
as the estimate of the maximum LFQPO frequency seen in Aql X$-$1 in
the following discussion.

\begin{figure}
\plotone{qpoevo.ps}
\caption{The evolution of the frequency of the LFQPO $\nu_{LF}$
during the outbursts of Aql X$-$1. Zero day corresponds to the time
of the last observation before the corresponding soft-to-hard state
transition. The evolution of $\nu_{LF}$ in the 1999 (blue) and the
2009 (black) outburst was fit with an exponential quadratic
polynomial and a broken exponential model, and the best-fit models
are plotted in dotted and dashed lines,
respectively.\label{fig:qpoevo}}
\end{figure}

The evolution pattern of $\nu_{LF}$ was quite similar among the
outbursts. In the following analysis, for the 2004 outburst, we
concentrated on the last four observations in the hard state before
the hard-to-soft transition, since earlier observations corresponded
to a special flare in the light curve, as can be seen from
Figure~\ref{fig:lchr}. For the peculiar 2001 outburst, the frequency
was around 10 Hz at the beginning, then increased rapidly to around
30 Hz at about 6 days before the state transition, and then decreased
to around 24 Hz at about just one day before the transition, and
finally turned to around 40 Hz at the time of the state transition.
For all the other outbursts, the frequency of the LFQPO increased
with time consistently during the rising phase of the outbursts.

In our model-independent approach, we estimated the rate-of-change of
the LFQPO frequency, following $\dot{\nu}=(\nu_2-\nu_1)/(t_2-t_1)$,
where $\nu$ is the LFQPO frequency, $t$ is the time, and the
subscripts 1 and 2 denote two consecutive observations. Then we
calculated the fractional rate-of-change of the frequency
$2\dot{\nu}/(\nu_1+\nu_2)$, corresponding to the time at
$(t_2+t_1)/2$. Figure~\ref{fig:qpoder} shows the time evolution of
the fractional rate-of-change. The fractional rate-of-change
increased from 10\% day$^{-1}$ at the beginning of the outburst, to
$\sim$100\% day$^{-1}$ just before the state transition.

\begin{figure}
\plotone{qpoder.ps}
\caption{The fractional rate-of-change of
  $\nu_{LF}$ vs. time during the seven outbursts of Aql X$-$1, as
  shown in different colors. Day zero corresponds to the time of the
  last observation just before the hard-to-soft state transition. The
  fractional rate-of-change is the rate-of-change of the frequencies
  divided by the frequencies themselves.\label{fig:qpoder}}
\end{figure}

The pattern of the increase of the LFQPO frequency deviates from a
single exponential pattern, which would follow a straight line if
plotted in logarithmic scales, and the fractional rate-of-change
would be constant with time. We tried two phenomenological models to
fit the pattern. The first one is an exponential quadratic
polynomial, i.e. $\nu=e^{at^2+bt+c}$. This function is continuous and
the fractional rate-of-change increases with time when the parameter
{\it a} is positive; the second one is of a broken exponential
function, i.e., the frequency first increases with a constant
e-folding timescale $t_1$, then afterwards the frequency increases
with another constant e-folding timescale $t_2$. We fit the models to
the data of the 1999 outburst and the 2009 outburst which have the
most number of data points available (which is also more than the
number of model parameters). The dashed and the dotted lines show
the best-fit broken exponential and exponential quadratic polynomial
models, respectively. Notice that the time at zero corresponds to
the time when the last observation of the source in the hard state
was taken.

For the 2009 outburst, the $\chi^2$ were 8.76 with 6 Degrees of
Freedom (DOF) and 34.25 with 7 DOF for the broken exponential
function and exponential quadratic polynomial function, respectively,
indicating that the former model describes the pattern of the
frequency evolution better. The result from the best-fit broken
exponential model showed that the frequency increased with an e-fold
timescale of $5.13\pm0.17$ and $1.31\pm0.17$ days before and after
the moment at $2.80\pm0.22$ days before the state transition,
respectively. For the 1999 outburst, the model fit with an
exponential quadratic polynomial gave a good fit with $\chi^2=1.89$
with 2 DOF, while the fit with the broken exponential model yielded a
$\chi^2=0.11$ with 1 DOF, indicating we were lacking statistics to
constrain the later model. The trend in both the 2009 outbursts
suggests that the frequency increased with time at different speed in
the early and in the late stage of the hard state during the rising
phase of an outburst, further supporting the idea that the transition
is associated with non-stationary accretion during these outbursts.

\subsection{LFQPO Frequency and Its Dependence on the X-ray Flux}
\label{sec:frevsflux}

In Figure~\ref{fig:frevsflux} we show how the LFQPO frequency evolved
with the 2--20 keV flux. During the rising phase of an outburst, the
frequency in general increased with the X-ray flux, except for the
peculiar 2001 outburst. The frequency-flux relation for each outburst
deviates from a straight line on a log-log plot, showing that it does
not follow a simple power-law relation. The power-law model was only
acceptable for the 2000 outburst, with $\chi^2=0.59$ and 1 DOF, while
for other outbursts the model was rejected. The model parameters in
the fits with the power-law model are tabulated in
Table~\ref{tab:fit}. We fit the data with a broken power-law model
instead. Such a model consists of four free parameters: the
normalization, the break frequency, and the power-law indices before
and after the break. We fit the model to the data corresponding to
the 1999 and the 2009 outburst which have more than four data points. 

\begin{figure}
\plotone{mulblnrate.ps}
\caption{The $\nu_{LF}$ vs. 2--20 keV source flux relation of Aql
X$-$1. Notice that the same QPO frequency corresponds to different
X-ray fluxes, and vice versa. We fit the data corresponding to the
1999 (blue) and the 2009 (black) outburst with a power-law model and
a broken power-law model, and the best-fit models are plotted as
dotted and dashed lines, respectively.\label{fig:frevsflux}}
 \end{figure}

\begin{deluxetable}{rrrrr}
\tabletypesize{\scriptsize}
%%\rotate
\tablecaption{Power-law model fit to the frequency vs. flux
relation\label{tab:fit}}
\tablewidth{0pt}
\tablehead{
\colhead{Year} & \colhead{$\alpha$} & \colhead{N} &
\colhead{$\chi^2$} & \colhead{DOF}}
\startdata
1999 & $2.45\pm0.08$ &$0.47\pm0.05$ & 36.80 & 3	\\
2000 & $0.88\pm0.12$ & $0.27\pm0.07$ & 3.48 & 2	\\
2004 & $2.50\pm0.22$ & $1.04\pm0.23$ & 98.50 & 2	\\ 
2009 & $2.20\pm0.09$ & $2.23\pm0.13$ & 39.36 & 7	\\
2010 & $1.81\pm0.12$ & $0.79\pm0.11$ & 0.59 & 1	\\
2011 & $1.26\pm0.12$ & $0.45\pm0.05$ & 4.89 & 1	
\enddata
%
\tablecomments{The parameters of best-fit power-law model of
 frequency vs. 2--20 keV flux: $\nu_{10}=Nf_{3}^{\alpha}$, where
 $\nu_{10}=\nu_{LF}/10~\rm Hz$ and $f_3$ is flux in
 $3\times10^{-9}~\rm erg~s^{-1}cm^{-2}$.}
%% Text for table notes should follow after the \enddata but before
%% the \end{deluxetable}. Make sure there is at least one \tablenotemark
%% in the table for each \tablenotetext.
%%\tablecomments{Table \ref{tbl-1} is published in its entirety in the 
%%electronic edition of the {\it Astrophysical Journal}. A portion is 
%%shown here for guidance regarding its form and content.}
%%\tablenotetext{a}{Sample footnote for table~\ref{tbl-1} that was generated
%%with the deluxetable environment}
%%%\tablenotetext{b}{Another sample footnote for table~\ref{tbl-1}}
\end{deluxetable}


%20141024

For the 1999 outburst, the broken power-law model yielded a
$\chi^2=1.36$ with 1 DOF while the power-law model gave a
$\chi^2=36.80$ with 3 DOF, showing a preference for the broken
power-law model. In that model, the break occurred at $\sim 2~\rm
Hz$. The index before the break was not well constrained, giving
$0.64\pm0.43$, while the index after the break was well-constrained
as $2.89\pm0.12$. Fitting with the broken power-law model yielded a
$\chi^2=12.35$ with 5 DOF, while the power-law model fit gave a
$\chi^2=39.36$ with 7 DOF. The break frequency occurred at $\sim7~\rm
Hz$. The power-law indices were $1.34\pm0.22$ and $3.58\pm0.34$
before and after the break, respectively. The best-fit power-law and
broken power-law models are plotted as dashed and dotted lines,
respectively, in Figure~\ref{fig:frevsflux}. 

Among different outbursts, the power spectra show that the source can
reach significantly different characteristic QPO frequencies at
similar X-ray fluxes. For instance, as shown in
Figure~\ref{fig:frevsflux}, at a source flux of $2\times10^{-9}~\rm
erg~cm^{-2}~s^{-1}$, the frequency of the LFQPO was at around 2 Hz
for the 1999 and the 2011 outburst, but at around 30 Hz for the 2001
outburst. It is interesting to notice that for a certain
characteristic frequency, the X-ray flux is usually higher in the
outbursts in the years of 1999, 2000 and 2011, which were associated
with higher state transition fluxes, while the X-ray flux
corresponding to the same QPO frequency was lower in the outbursts in
the years of 2001, 2004, 2009 and 2010, which were associated with
lower state transition fluxes. There is another anti-correlation
between the maximum LFQPO frequency and the state transition flux (or
the outburst peak X-ray luminosity), with the Spearman correlation
coefficient of -0.96 at a significance level of $4.5\times10^{-4}$.
This is shown in the upper-right corner of Fig.~\ref{fig:frevsflux}.
Actually the maximum frequency of the LFQPO was reached during the
2001 and the 2009 outburst, which was associated with the lowest
state transition fluxes.

%As would be seen in the proceeding sections, the normalization of
%the broken powerlaw is also of physical importance. Apparently the
%normalization is not same for different outbursts, as shown in the
%upper panel of Figure~\ref{frevsflux}. To evaluate the
%normalization, we fixed the two indices to be the weighted mean of
%the best-fit indices of the 1999, 2000 and 2009 outbursts:
%$1.49\pm0.16$ before break and $3.69\pm0.13$ after the break. 

%We assume a simple powerlaw dependence of the frequencies on source
%flux $\nu=bf^a$ and calculate the index $a$ and the normalization
%$b$ for the 7 outbursts, from the lowest and highest LFQPO
%frequencies and fluxes. The results are listed in
%Table~\ref{tab:frevsflux}. Both $a$ and $b$ change among outbursts.
%The indices have a mean value of $1.72\pm0.07$. The normalization
%$b$ is more than one order of magnitude larger in the 2001 outburst,
%which has the lowest hard-to-soft state transition luminosity and
%lowest peak luminosity. This indicate a trend of higher
%normalization of frequency-flux dependence in outbursts with lower
%hard-to-soft state transition luminosity.

 %Note that the maximum frequency of an outburst doesn't depend on
 %the amplitude of the outburst: the 2000 outburst has a peak 2-20
 %keV count rate of around 1500 counts/s/PCU, while that of the 2009
 %outburst was around 600. The maximum frequency of the dim 2001
 %outburst also reached $\sim$30 Hz.

%The frequency seemed to change faster near before state transition.
%To evaluate the ``speed'' of increase of LFQPO frequency, we
%measured the time derivative of the frequency and show them in
%Figure~\ref{fig:qpoder}.

%To show how the frequency is correlated with the accretion rate, in
%Figure~\ref{fig:frevsflux} we show how the frequency evolve with
%2--20 keV flux in the seven outbursts. 

%\begin{figure} \plotone{frevsflux.ps} \caption{The frequency v.s.
%2--20 keV flux} \end{figure}



\section{Conclusion and Discussion}

We have analyzed the observations obtained during the rising phase of
seven outbursts of the neutron star soft X-ray transient Aql X$-$1
with the \textit{RXTE}/PCA when the source was in the hard state. As
expected, in the power spectra, the characteristic break frequencies
of the band-limited noise component and the frequency of the LFQPOs
were correlated, consistent with the overall correlation established
in a large sample of black hole and neutron star X-ray binaries
previously found.

The frequency evolution of LFQPOs is shown in
Figure~\ref{fig:pdsevo}. Both the characteristic frequency of the
band-limited noise and the frequency of the LFQPOs increased with the
PCA/2--20 keV count rate before the hard-to-soft state transitions,
except for the peculiar 2001 outburst. We found that the maximum
frequency of the LFQPOs in Aql X$-$1 can reach 31~{\rm Hz}. We
measured the rate-of-change of the LFQPO frequency during the rising
phase of the outbursts and found that the frequencies increased by
around one tenth per day in the early stage of the outbursts to
nearly one hundred percent per day nearly two days before the
hard-to-soft state transition. In the 2009 outburst, for which we
have the best data available, we found that the frequency of the
LFQPOs actually increased with time with an e-folding timescale of
$5.13\pm0.17$ and $1.31\pm0.17$ days in the early rising phase and in
the later rising phase, respectively.

In Figure~\ref{fig:corr} we also show that the frequency correlation
between band-limited noise and the LFQPOs holds down to the timescale
of one to two days, since the \textit{RXTE} observations were taken
roughly every other day. It would be interesting to investigate the
correlation on shorter time scales with future X-ray missions to test
if such a correlation holds on shorter time scales. In addition, we
found that the characteristic frequency vs. flux relation is
significantly different among the seven outbursts. Higher
characteristic frequency is reached by the source at similar X-ray
flux levels in the outbursts with lower state transition and outburst
peak fluxes. In individual outbursts, the relationship between the
frequency and the source flux deviated from a simple power-law form.
The frequency depended on the source flux. For the 2009 outburst, If
we model it with a broken power-law function, we found that the
power-law indices were $1.34\pm0.22$ and $3.58\pm0.34$ before and
after the break, respectively. The observed multiple relation between
the LFQPO frequency and X-ray flux and the fast variation of the
frequency would potentially put strong constraints on the
Lense-Thirring precession model of the LFQPO proposed by
\citep{ingram2009}, if quantitative prediction of the relation
between the transitional radius, the mass accretion rate and the
Lense-Thirring precession frequency can be delivered. 

\subsection{Maximum LFQPO Frequency: Indicator of the Mass of
the Compact Object ?}

In this work, we found that the maximum frequency of the LFQPOs in
Aql X$-$1 observed with the \textit{RXTE} was $\sim$ 31 Hz. In a
similar study of the black hole transient GX 339$-$4 we found that
the maximum frequency of the LFQPOs was around 5 Hz, which is about
six times lower than that of Aql X$-$1. \cite{ozel_mass_2012} showed
that the mass of the neutron stars in the recycled pulsars has a mean
value of $1.48~\rm M_{\odot}$ with a dispersion of $0.2~\rm
M_{\odot}$. We know that the black hole binary GX 339$-$4 has a mass
function of $5.8~\rm M_{\odot}$ \citep{hynes_dynamical_2003}, so we
do not know the exact mass ratio. If we take its mass as a typical
value of 10 solar masses in black hole X-ray binaries, the mass ratio
between the compact stars in the two systems seems consistent with
the ratio between their maximum frequencies of the LFQPOs. This
suggests that the maximum frequency of the LFQPOs may be an indicator
of the mass of the compact star. By observing the maximum frequency
of the LFQPOs during outbursts, we may be able to determine the
nature of the compact star in a new transient LMXBs empirically.

\subsection{Tracing the Inner Disk Radius with LFQPO ?}

Using the observations we analysed, we can investigate whether it is
possible to trace the evolution of the truncation radius using the
LFQPO based on current disk truncation models and to constrain these
models with observations. It has been believed that the frequency of
one of the twin kHz QPOs might be associated with the Keplerian
frequency at the inner disk edge
\citep[e.g.,][]{miller_sonic-point_1998,stella_lense-thirring_1998,osherovich_kilohertz_1999}.
Therefore we can probably associate the frequency of the LFQPOs with
the inner disk edge (or the truncation radius in some popular models)
based on the empirical frequency correlation between the kHz QPOs and
the LFQPOs. The empirical correlations found in the \textit{RXTE}
data suggest that for the lower kHz QPO at frequency $\nu_1$, the
frequency of the LFQPO is $ \nu_{lf} \approx (42\pm3~\rm Hz)(\nu_1
/500 ~\rm Hz)^{0.95 \pm 0.16}$ \citep{psaltis_correlations_1999}. The
empirical relation between the lower and the upper kHz QPO is $\nu_1
= (724\pm3)\left( \nu_2/1000~\rm Hz \right )^{1.9\pm0.1}~\rm Hz$
\citep{psaltis_beat-frequency_1998}. This yields the relation between
the LFQPO and the upper kHz QPO as $\nu_{LF} \propto
\nu_2^{1.85\pm0.21}$.

In the magnetospheric model
\citep{ghosh_accretion_1977,ghosh_accretion_1979}, the inner edge of
the disk is where the ram pressure is balanced by the magnetic
pressure. \citet{ghosh_accretion_1979} studied the structure of a
geometrically thin disk around a magnetized neutron star and found
that the disk is truncated at $r_0\approx C r_A$, where $r_A$ is the
Alfv\'{e}n radius, i.e., $r_A =\mu^{4/7} (2GM)^{-1/7}
\dot{M}^{-2/7}$, here $\mu=BR^3$ is the magnetic dipole moment and
$R$ is the neutron star radius, and $C$ is a parameter which does not
affect configuration a lot. The magnetospheric model therefore
predicts that the truncation radius $r_{tr} \propto \mu^{4/7}
\dot{M}^{-2/7}$.

In the disk evaporation model, the heat generated in the corona
formed above the disk is conducted down to the lower layer of the
corona and radiated away. If the density of the lower layer is too
low to efficiently radiate away the heat, the thin disk would be
heated and then evaporated, leading to a steady mass flow from the
disk to the corona. In a stationary state, the outer disk would be
truncated at a radius $r_{tr}$ where the evaporation is balanced by
the mass inflow from the outer disk. This truncation radius lies at
$r_{tr} \propto \dot{M}^{-1/1.17}$. Further development of the model
included the effect of the viscosity and the accretion disk magnetic
field \citep{taam_disk_2012,liu_constraints_2013}, which gives the
truncation radius $r_{tr} \propto \dot{M}^{-0.886} \alpha^{0.07}
\beta^{4.61}$, where $\alpha$ is the viscous parameter and $\beta$ is
the ratio of gas pressure to total pressure. The truncation radius
depends only weakly on the viscosity.

It is worth noting that the parallel-track phenomenon of the LFQPO
shown in Fig.~8, which is similar to that seen for the kHz QPOs,
actually suggests that mass accretion rate is not the only parameter
that determines the QPO frequency. However, let's consider the
simplest case where mass accretion rate is the only parameter in the
accretion system. As we know, both disk truncation models predict
that the truncation radius is a function of the mass accretion rate,
i.e., $r_{tr} =c \dot{M}^\delta$, where $c$ is the normalization
coefficient. Then we get the relation between the LFQPO frequency and
the mass accretion rate as $\nu_{LF} \propto
\dot{M}^{1.43\delta}=\dot{M}^{\epsilon_1}$ if taking the lower kHz
QPO as the orbital frequency at the inner disk edge and as $\nu_{LF}
\propto \dot{M} ^{2.71\delta} =\dot{M}^{\epsilon_2}$ if taking the
upper kHz QPO frequency as the Keplerian frequency at the inner disk
edge. For the former case, the index $\epsilon$ is $1.22\pm0.20$ in
the evaporation model and $0.57\pm0.10$ in the magnetospheric model;
while in the later case which corresponds to the upper kHz QPO
assumption, the index $\epsilon$ is $2.31\pm0.26$ in the evaporation
model and $0.77\pm0.10$ in the magnetospheric model. In the above
estimates we have assumed that the bolometric correction factor was a
constant during those observations and that the bolometric luminosity
was proportional to the mass accretion rate (i.e., the radiation
efficiency did not change a lot).

In Section~\ref{sec:frevsflux} we showed that for the well-sampled
2009 outburst, the frequency vs. the X-ray flux relation was best-fit
by a broken power-law model with indices of $1.34\pm0.22$ and
$3.58\pm0.34$ before and after the trend break. Both of the values
are larger than that predicted by the magnetospheric model. As for
the disk evaporation model, we found that for the case corresponding
to the lower kHz QPO assumption, during the early rising phase of the
outburst, the observed index before the trend break is consistent
with the model, while after the break the observed index is much
larger than that predicted by the disk evaporation model. This might
indicate that the disk evaporation model is consistent with the
observed evolution of the inner disk edge at the early rising phase
of those outbursts but not later on. For the case corresponding to
the upper kHz QPO assumption, the disk evaporation model
overestimates the index in the early stage of the outburst but
underestimates the index in the later stage. This suggests that at
least additional physics should be considered in the current disk
evaporation model. We also realised that additional constraints can
be put on the evaporation model as well from the so-called
`parallel-track' phenomenon of the LFQPO. In the improved evaporation
model investigated in \citet{taam_disk_2012}, the truncation radius,
$r_{tr}$, depends on the magnetic field in the accretion disk as
$r_{tr} \propto \dot{M}^{-0.886} \alpha^{0.07} \beta^{4.61}$. In
order to explain the observations showing that lower characteristic
frequency can be observed under similar mass accretion rates in the
outbursts with a higher state transition luminosity, the disk
evaporation theory with the consideration of the magnetic field has
to be consistent with the fact the brighter an outburst is, the lower
the magnetic pressure in the accretion disk should be, due to the
fact that there is empirical correlation between the luminosity of
the hard-to-soft transition and the peak luminosity of the outburst
or the following soft state
\citep{yu_correlation_2004,yu2007,yu_state_2009}.


%% If you wish to include an acknowledgments section in your paper,
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\acknowledgments
We would like to thank the anonymous referee for useful comments
which have improved the work a lot.
The authors would like to thank Michiel van der Klis, Diego
Altamirano and Ronald Taam for useful discussions and comments. This
work was supported in part by the National Natural Science Foundation
of China under grant No. 11333005, 11073043, and 11350110498, by
Strategic Priority Research Programme ``The Emergence of Cosmological
Structures'' under Grant No. XDB09000000 and the XTP project under
Grant No. XDA04060604, by the Shanghai Astronomical Observatory Key
Project and by the Chinese Academy of Sciences Fellowship for Young
International Scientists Grant.
This research has made use of data and software provided by the
High Energy Astrophysics Science Archive Research Center (HEASARC),
which is a service of the Astrophysics Science Division at NASA/GSFC
and the High Energy Astrophysics Division of the Smithsonian
Astrophysical Observatory.

%% To help institutions obtain information on the effectiveness of their
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%\bibliography{/home/wenda/research/astroph_zo}
%
\begin{thebibliography}{}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
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\bibitem[{Hynes {et~al.}(2003)Hynes, Steeghs, Casares, Charles, \&
 O'Brien}]{hynes_dynamical_2003}
Hynes, R.~I., Steeghs, D., Casares, J., Charles, P.~A., \& O'Brien, K. 2003,
 \apj, 583, L95
 
 \bibitem[Ingram et al.(2009)]{ingram2009} Ingram, A., Done, C., 
\& Fragile, P.~C.\ 2009, \mnras, 397, L101

\bibitem[{Kitamoto {et~al.}(1993)Kitamoto, Tsunemi, Miyamoto, \&
 Roussel-Dupre}]{kitamoto_unstable_1993}
Kitamoto, S., Tsunemi, H., Miyamoto, S., \& Roussel-Dupre, D. 1993, \apj, 403, 315

\bibitem[{Klein-Wolt \& van~der Klis(2008)}]{klein-wolt_identification_2008}
Klein-Wolt, M., \& van~der Klis, M. 2008, \apj, 675, 1407

\bibitem[{Koyama {et~al.}(1981)Koyama, Inoue, Makishima, Matsuoka, Murakami,
 Oda, Osgawara, Ohashi, Shibazaki, Tanaka, Marshall, Kondo, Hayakawa, Kunieda,
 Makino, Masai, Nagase, Tawara, Miyamoto, Tsunemi, \&
 Yamashita}]{koyama_discovery_1981}
Koyama, K., Inoue, H., Makishima, K., {et~al.} 1981, \apj, 247, L27

\bibitem[{Liu \& Taam(2013)}]{liu_constraints_2013}
Liu, B.~F., \& Taam, R.~E. 2013, \apjs,
 207, 17

\bibitem[{Liu {et~al.}(1999)Liu, Yuan, Meyer, Meyer-Hofmeister, \&
 Xie}]{liu_evaporation_1999}
Liu, B.~F., Yuan, W., Meyer, F., Meyer-Hofmeister, E., \& Xie, G.~Z. 1999, \apjl, 527, L17

\bibitem[{Meyer {et~al.}(2007)Meyer, Liu, \&
 Meyer-Hofmeister}]{meyer_re-condensation_2007}
Meyer, F., Liu, B.~F., \& Meyer-Hofmeister, E. 2007, \aap, 463, 1

\bibitem[{Meyer \& Meyer-Hofmeister(1994)}]{meyer_accretion_1994}
Meyer, F., \& Meyer-Hofmeister, E. 1994, \aap, 288, 175

\bibitem[{Miller {et~al.}(1998)Miller, Lamb, \&
  Psaltis}]{miller_sonic-point_1998}
Miller, M.~C., Lamb, F.~K., \& Psaltis, D. 1998, \apj,
508, 791

\bibitem[{Osherovich \& Titarchuk(1999)}]{osherovich_kilohertz_1999}
  Osherovich, V., \& Titarchuk, L. 1999, \apj, 522,
  L113

\bibitem[{Priedhorsky \& Terrell(1984)}]{priedhorsky_long-term_1984}
Priedhorsky, W.~C., \& Terrell, J. 1984, \apj, 280, 661

\bibitem[{Psaltis {et~al.}(1999)Psaltis, Belloni, \& van~der
 Klis}]{psaltis_correlations_1999}
Psaltis, D., Belloni, T., \& van~der Klis, M. 1999, \apj,
 520, 262

\bibitem[{Psaltis {et~al.}(1998)Psaltis, Méndez, Wijnands, Homan, Jonker,
  van~der Klis, Lamb, Kuulkers, van Paradijs, \&
  Lewin}]{psaltis_beat-frequency_1998}
Psaltis, D., Méndez, M., Wijnands, R., {et~al.} 1998, \apj,
501, L95

\bibitem[{Reig {et~al.}(2000)Reig, M\'{e}ndez, van~der Klis, \&
 Ford}]{reig_correlated_2000}
Reig, P., Méndez, M., van~der Klis, M., \& Ford, E.~C. 2000, \apj, 530, 916

\bibitem[{Remillard \& McClintock(2006)}]{remillard_x-ray_2006}
Remillard, R.~A., \& McClintock, J.~E. 2006, \araa, 44, 49

\bibitem[{Remillard {et~al.}(2002)Remillard, Sobczak, Muno, \&
  McClintock}]{remillard_characterizing_2002}
Remillard, R.~A., Sobczak, G.~J., Muno, M.~P., \& McClintock, J.~E. 2002,
\apj, 564, 962

\bibitem[{Sanna {et~al.}(2012)Sanna,
	M\'{e}ndez, Belloni, \&
Altamirano}]{sanna_time_2012}
Sanna, A., M\'{e}ndez, M., Belloni, T., \& Altamirano, D.
  2012, \mnras
424, 2936

\bibitem[{Simon(2002)}]{simon_recurrence_2002}
Simon, V. 2002, \aap, 381, 151

\bibitem[{Stella \& Vietri(1998)}]{stella_lense-thirring_1998}
  Stella, L., \& Vietri, M. 1998, \apj, 492, L59

\bibitem[{Taam {et~al.}(2012)Taam, Liu, Yuan, \& Qiao}]{taam_disk_2012}
Taam, R.~E., Liu, B.~F., Yuan, W., \& Qiao, E. 2012, \apj,
 759, 65

\bibitem[{Tang {et~al.}(2011)Tang, Yu, \& Yan}]{tang_rxte/asm_2011}
Tang, J., Yu, W.-F., \& Yan, Z. 2011, Research in Astronomy and
Astrophysics, 11, 434

\bibitem[{van Straaten {et~al.}(2002)van Straaten, van~der Klis, di~Salvo, \&
 Belloni}]{van_straaten_multi-lorentzian_2002}
van Straaten, S., van~der Klis, M., di~Salvo, T., \& Belloni, T. 2002, \apj, 568, 912

\bibitem[{van Straaten {et~al.}(2003)van Straaten, van~der Klis, \&
 Méndez}]{van_straaten_atoll_2003}
van Straaten, S., van~der Klis, M., \& Méndez, M. 2003, \apj, 596, 1155

\bibitem[van der Klis et al.(1985)]{klis85a} van der Klis, M., 
Jansen, F., van Paradijs, J., et al.\ 1985, \nat, 316, 225 

\bibitem[van der Klis 
\& Jansen(1985)]{klis85b} van der Klis, M., \& Jansen, F.~A.\ 1985, \nat, 313, 768 


\bibitem[{Wijnands \& van~der Klis(1999)}]{wijnands_broadband_1999}
Wijnands, R., \& van~der Klis, M. 1999, \apj, 514, 939

\bibitem[{Wijnands {et~al.}(1999)Wijnands, Homan, \& van~der
  Klis}]{wijnands_complex_1999}
  Wijnands, R., Homan, J., \& van~der Klis, M. 1999, \apj,
526, L33

\bibitem[{Yu \& Dolence(2007)}]{yu_hard--soft_2007}
Yu, W., \& Dolence, J. 2007, \apj, 667, 1043

\bibitem[{Yu {et~al.}(2003)Yu, Klein-Wolt, Fender, \& van~der
 Klis}]{yu_hard_2003}
Yu, W., Klein-Wolt, M., Fender, R., \& van~der Klis, M. 2003, \apjl, 589, L33

\bibitem[{Yu {et~al.}(2004)Yu, van~der Klis, \& Fender}]{yu_correlation_2004}
Yu, W., van~der Klis, M., \& Fender, R. 2004, \apj, 611, L121

\bibitem[Yu \& Dolence(2007)]{yu2007} Yu, W., \& Dolence, J.\ 2007, \apj, 667, 1043 

\bibitem[{Yu \& Yan(2009)}]{yu_state_2009}
Yu, W., \& Yan, Z. 2009, \apj, 701, 1940

\bibitem[{\"{O}zel {et~al.}(2012)\"{O}zel, Psaltis, Narayan, \&
 Santos~Villarreal}]{ozel_mass_2012}
 \"{O}zel, F., Psaltis, D., Narayan, R., \& Santos~Villarreal, A. 2012, \apj, 757, 55
 
 \bibitem[Zhang et al.(1998)]{zhang98} Zhang, W., Jahoda, K., 
Kelley, R.~L., et al.\ 1998, \apjl, 495, L9
 
 
\end{thebibliography}

%% Use the figure environment and \plotone or \plottwo to include
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%% table environment. Use of deluxetable is preferred.
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%% Three table samples follow, two marked up in the deluxetable environment,
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1412.0960.extracted_bib
in the reference list below. We have %% chosen the first three characters of the first author's name plus %% the last two numeral of the year of publication as our KEY for %% each reference. %% Authors who wish to have the most important objects in their paper %% linked in the electronic edition to a data center may do so by tagging %% their objects with \objectname{ or \object{. Each macro takes the %% object name as its required argument. The optional, square-bracket %% argument should be used in cases where the data center identification %% differs from what is to be printed in the paper. The text appearing %% in curly braces is what will appear in print in the published paper. %% If the object name is recognized by the data centers, it will be linked %% in the electronic edition to the object data available at the data centers %% %% Note that for sources with brackets in their names, e.g. [WEG2004] 14h-090, %% the brackets must be escaped with backslashes when used in the first %% square-bracket argument, for instance, \object[\[WEG2004\] 14h-090]{90). %% Otherwise, LaTeX will issue an error. \section{Introduction Low Mass X-ray Binaries (LMXBs) are binary systems that contain a black hole (BH) or neutron star (NS) primary and low mass companion. The primary accretes from the companion through an accretion disk around the compact object which emits predominately in the X-ray band at high mass accretion rates. Soft X-ray transients (SXTs) are transient LMXBs. They spend most of the time in the quiescent state at low luminosity, and occasionally turn into outbursts during which their luminosity increases by several orders of magnitude. During outbursts, SXTs show distinctive states with coupled spectral and timing properties: in the Low-Hard State (LHS) the soft X-ray (e.g., 0.3--10 keV) energy spectrum is dominated by a power-law spectral component and the Fourier power spectrum at low frequency is dominated by broad band noise component and quasi-periodic oscillations (QPOs), and the variability amplitude is quite large at a few tens percent level, while in the High-Soft State (HSS) the soft X-ray energy spectrum is dominated by a thermal disk component and the Fourier power spectrum is dominated by a power-law noise component with little variability \citep[see, e.g.,][]{belloni_evolution_2005,remillard_x-ray_2006. Systematic studies of the state transitions in SXTs during the rising phase of transient outbursts suggest that the accretion process is non-stationary: the luminosity of the hard-to-soft state transitions can occur at quite different luminosity, spanning two orders of magnitude which correlates with the rate-of-increase of the luminosity as well as the peak luminosity of the corresponding outbursts \citep{yu_correlation_2004,yu2007,yu_state_2009,tang_rxte/asm_2011. This indicates that the mass accretion rate cannot be the only parameter that determines the transition between the accretion states and the rate-of-change of the mass accretion rate is one important parameter which determines the regimes of the accretion states in SXT outbursts. The most popular picture for the X-ray energy spectra in the distinct states of black hole and neutron star SXTs is the truncated disk picture, such as the model proposed in \citet{esin_advection-dominated_1997. In this picture the change of the energy spectrum is associated with the change of the accretion geometry. A geometrically-thin but optically-thick disk is truncated at the truncation radius. The region within the radius is filled with a hot flow such as the radiatively-inefficient accretion flow. In the quiescent and the hard state, the truncation radius is quite large. During the rising phase of transient outbursts the truncation radius moves inward with increasing mass accretion rate, and reaches the innermost stable orbit during the soft state. The detailed process that truncates the disk is not clear, and proposals for the physical mechanism include magnetosphere-disk interaction \citep[][for neutron star systems only]{ghosh_accretion_1977 and disk evaporation \citep[e.g.,][]{meyer_accretion_1994,liu_evaporation_1999,meyer_re-condensation_2007. It was suggested that timing features, especially the quasi-periodic oscillations (QPOs) shown in the power spectra, might be the probe of the accretion geometry and its change in X-ray binaries several decades ago \citep[e.g.,][]{klis85a,klis85b,miller_sonic-point_1998,stella_lense-thirring_1998,osherovich_kilohertz_1999,ingram2009. The variation in the characteristic frequencies of the QPOs or noise components may reflect the change in the inner most edge of the accretion disk or the size of the corona. Low Frequency Quasi-Periodic Oscillations (LFQPOs) are the most obvious feature in the Fourier power density spectra (PDS) of black hole SXTs in the hard and the intermediate state. For black hole transients, the LFQPOs can be classified into A, B and C-type QPOs \citep{wijnands_complex_1999,remillard_characterizing_2002. Among them, the C-type QPO appears in the hard and hard-intermediate states and its characteristic frequency appears to correlate with the luminosity. There are similar QPOs and broad noise components in neutron star LMXBs. For example, \citet{yu_hard_2003 showed that the 1--20 Hz LFQPOs and broad noise components in the neutron star transient Aql X$-$1 are associated with the LFQPOs and broad noise components commonly seen in black hole transients during outbursts. Aql X$-$1 is a well-known neutron star transient with very frequent outbursts \citep[e.g.,][]{priedhorsky_long-term_1984,kitamoto_unstable_1993,simon_recurrence_2002. Its neutron star nature was revealed by the detection of type-I X-ray bursts from the source \citep[e.g.,][]{koyama_discovery_1981,czerny_persistent_1987. Its neutron star spin was detected as well \citep{zhang98,casella_discovery_2008. Based on its spectral and temporal properties \citep{reig_correlated_2000, Aql X$-$1 has been classified as an ``Atoll'' source, of which great similarities of both timing and spectral properties to those black hole X-ray binaries have been seen. This makes Aql X$-$1 an ideal source for the study of the evolution of the characteristic timescales in the accretion flow during the rising phase of their outbursts, during which drastic changes in the accretion geometry are expected. In this $Paper$ we study the evolution of the LFQPOs during the rising phase of seven outbursts of the neutron star SXT Aql X$-$1 with the observations performed with the RXTE. %% In a manner similar to \objectname authors can provide links to dataset %% hosted at participating data centers via the \dataset{ command. The %% second curly bracket argument is printed in the text while the first %% parentheses argument serves as the valid data set identifier. Large %% lists of data set are best provided in a table (see Table 3 for an example). %% Valid data set identifiers should be obtained from the data center that %% is currently hosting the data. %% %% Note that AASTeX interprets everything between the curly braces in the %% macro as regular text, so any special characters, e.g. "#" or "_," must be %% preceded by a backslash. Otherwise, you will get a LaTeX error when you %% compile your manuscript. Special characters do not %% need to be escaped in the optional, square-bracket argument. %% In this section, we use the \subsection command to set off %% a subsection. \footnote is used to insert a footnote to the text. %% Observe the use of the LaTeX \label %% command after the \subsection to give a symbolic KEY to the %% subsection for cross-referencing in a \ref command. %% You can use LaTeX's \ref and \label commands to keep track of %% cross-references to sections, equations, tables, and figures. %% That way, if you change the order of any elements, LaTeX will %% automatically renumber them. %% This section also includes several of the displayed math environments %% mentioned in the Author Guide. \section{Observations and Data Reduction We made use of the RXTE/ASM and the MAXI long-term X-ray monitoring observations of Aquila X$-$1 to identify X-ray outbursts in this study. RXTE/PCA pointed observations were used to study X-ray timing and spectral properties. We have analysed a total of 611 RXTE observations of Aquila X$-$1 performed from 1996 to 2011, starting from the observation 10072-06-01-00 to the observation 96440-01-10-05. \subsection{Long-term Light Curves and Evolution of the Hardness Ratio Figure 1 shows the long-term X-ray light curve of Aquila X-1. We converted the RXTE/ASM 2--12 keV flux in unit of Crab by dividing the ASM 2-12 keV count rate by the mean Crab count rate of 75.6 cts/s. Similarly, for the later period when RXTE stopped operation, we obtained the source flux in 2--10 keV by combining the 2--4 keV and the 4--10 keV MAXI data which are publicly available. Figure 2 shows the X-ray light curve and the X-ray hardness ratio seen with the pointed observations of the RXTE/PCA. We extracted source light curves from the PCA standard products\footnote{http://heasarc.nasa.gov/docs/xte/recipes/stdprod\_guide.html which provide data with a time resolution of 16 seconds from the STD 2 PCA data. The light curves in the PCA standard products have five energy bands, namely 2--9, 2--4, 4--9, 9--20 and 20--60 keV. We identified the time intervals that were associated with Type-I bursts following the screening method mentioned in \citet{klein-wolt_identification_2008 and excluded these time intervals. Then we calculated the mean count rates per PCU in the five energy bands, and the hardness ratio (HR), which was defined as the count rate ratio between the 9--20 keV and the 2--9 keV in each observation. \subsection{Timing Analysis We made use of the PCA data with a time resolution better than 0.25 ms and combined all available data in different energy bands whenever possible. We extracted the data and rebinned the light curves into 1/4096 s resolution and then generated the Fourier power spectrum for every 128 s segment. In the analysis of each power spectrum, we took the average power above 1800 Hz to estimate the white noise level, which was then subtracted. We fit each power spectrum with a model composed of several Lorentzians in which zero-centered Lorentzian represents the band-limited noise components and narrow Lorentzians represents the QPOs following the previous approach \citep{belloni_unified_2002. A power-law component was included to fit the Very Low Frequency Noises (VLFNs) when necessary. These model fits were performed using a custom package making use of the MPFIT package\footnote{\url{http://cow.physics.wisc.edu/~craigm/idl/. The best-fit model was achieved by minimizing the $\chi^2$, and the uncertainty was obtained from the distribution of $\chi^2$. Throughout this paper the error quoted for the power spectral fit represents the 1$\sigma$ uncertainty. \subsection{Spectral Analysis In order to estimate the X-ray flux of the source, we fit the 3--20 keV Standard 2 PCA energy spectra of the observations when the source was in the hard state during the rising phase of each outburst. The X-ray spectral model we used consisted of four components, namely a power-law component representing the emission from the hot accretion flow, a blackbody component for the emission from the neutron star surface, a Gaussian line component with its central energy fixed at 6.5 keV representing a broad Iron emission line, and the absorption by Galactic neutral hydrogen. We fixed the column density of the neutral hydrogen to $3.4\times10^{21~\rm atoms~cm^{-2$, according to \citet{dickey_h_1990. The 2--20 keV flux was then obtained from the spectral fits in XSPEC 12.8.2. \section{Results \subsection{Outburst Sample As shown in Figure~\ref{fig:pcalc, the combined RXTE/ASM and MAXI 2--10 keV light curve of Aql X$-$1 indicates 16 outbursts in total in a period of 16 years, of which the RXTE observations covered the hard X-ray state during the rising phase of seven outbursts, which occurred in 1999, 2000, 2001, 2004, 2009, 2010 and 2011, respectively, as indicated by the triangles in the plot. \begin{figure \plotone{asmlc.ps \caption{RXTE/ASM and MAXI 2--10 keV light curve of Aql X$-$1. Light curve after MJD 55500 is MAXI/2--10 keV light curve. Triangles indicate the seven outbursts we analyzed. \label{fig:pcalc
1412.0960.extracted_bib
-\cite %% cross-referencing. Each reference is preceded by a %%
1412.0960.extracted_bib
command that defines in curly braces the KEY that corresponds %% to the KEY in the \cite commands (see the first section above). %% Make sure that you provide a unique KEY for every
1412.0960.extracted_bib
or else the %% paper will not LaTeX. The square brackets should contain %% the citation text that LaTeX will insert in %% place of the \cite commands. %% We have used macros to produce journal name abbreviations. %% AASTeX provides a number of these for the more frequently-cited journals. %% See the Author Guide for a list of them. %% Note that the style of the
1412.0960.extracted_bib
labels (in []) is slightly %% different from previous examples. The natbib system solves a host %% of citation expression problems, but it is necessary to clearly %% delimit the year from the author name used in the citation. %% See the natbib documentation for more details and options. %\bibliography{/home/wenda/research/astroph_zo % \begin{thebibliography{ \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1\fi
1412.0960.extracted_bib
[{Belloni {et~al.(2005)Belloni, Homan, Casella, van~der Klis, Nespoli, Lewin, Miller, \& Méndez]{belloni_evolution_2005 Belloni, T., Homan, J., Casella, P., {et~al. 2005, \aap, 440, 207
1412.0960.extracted_bib
[{Belloni {et~al.(2002)Belloni, Psaltis, \& van~der Klis]{belloni_unified_2002 Belloni, T., Psaltis, D., \& van~der Klis, M. 2002, \apj, 572, 392
1412.0960.extracted_bib
[{Casella {et~al.(2008)Casella, Altamirano, Patruno, Wijnands, \& van~der Klis]{casella_discovery_2008 Casella, P., Altamirano, D., Patruno, A., Wijnands, R., \& van~der Klis, M. 2008, \apj, 674, L41 %
1412.0960.extracted_bib
[{Casella {et~al.(2005)Casella, Belloni, \& Stella]{casella_abc_2005 %Casella, P., Belloni, T., \& Stella, L. 2005, \apj, 629, % 403
1412.0960.extracted_bib
[{Czerny {et~al.(1987)Czerny, Czerny, \& Grindlay]{czerny_persistent_1987 Czerny, M., Czerny, B., \& Grindlay, J.~E. 1987, \apj, 312, 122
1412.0960.extracted_bib
[{Dickey \& Lockman(1990)]{dickey_h_1990 Dickey, J.~M., \& Lockman, F.~J. 1990, \araa, 28, 215
1412.0960.extracted_bib
[{Esin {et~al.(1997)Esin, McClintock, \& Narayan]{esin_advection-dominated_1997 Esin, A.~A., McClintock, J.~E., \& Narayan, R. 1997, \apj, 489, 865
1412.0960.extracted_bib
[{Ghosh \& Lamb(1979)]{ghosh_accretion_1979 Ghosh, P., \& Lamb, F.~K. 1979, \apj, 232, 259
1412.0960.extracted_bib
[{Ghosh {et~al.(1977)Ghosh, Pethick, \& Lamb]{ghosh_accretion_1977 Ghosh, P., Pethick, C.~J., \& Lamb, F.~K. 1977, \apj, 217, 578
1412.0960.extracted_bib
[{Hynes {et~al.(2003)Hynes, Steeghs, Casares, Charles, \& O'Brien]{hynes_dynamical_2003 Hynes, R.~I., Steeghs, D., Casares, J., Charles, P.~A., \& O'Brien, K. 2003, \apj, 583, L95
1412.0960.extracted_bib
[Ingram et al.(2009)]{ingram2009 Ingram, A., Done, C., \& Fragile, P.~C.\ 2009, \mnras, 397, L101
1412.0960.extracted_bib
[{Kitamoto {et~al.(1993)Kitamoto, Tsunemi, Miyamoto, \& Roussel-Dupre]{kitamoto_unstable_1993 Kitamoto, S., Tsunemi, H., Miyamoto, S., \& Roussel-Dupre, D. 1993, \apj, 403, 315
1412.0960.extracted_bib
[{Klein-Wolt \& van~der Klis(2008)]{klein-wolt_identification_2008 Klein-Wolt, M., \& van~der Klis, M. 2008, \apj, 675, 1407
1412.0960.extracted_bib
[{Koyama {et~al.(1981)Koyama, Inoue, Makishima, Matsuoka, Murakami, Oda, Osgawara, Ohashi, Shibazaki, Tanaka, Marshall, Kondo, Hayakawa, Kunieda, Makino, Masai, Nagase, Tawara, Miyamoto, Tsunemi, \& Yamashita]{koyama_discovery_1981 Koyama, K., Inoue, H., Makishima, K., {et~al. 1981, \apj, 247, L27
1412.0960.extracted_bib
[{Liu \& Taam(2013)]{liu_constraints_2013 Liu, B.~F., \& Taam, R.~E. 2013, \apjs, 207, 17
1412.0960.extracted_bib
[{Liu {et~al.(1999)Liu, Yuan, Meyer, Meyer-Hofmeister, \& Xie]{liu_evaporation_1999 Liu, B.~F., Yuan, W., Meyer, F., Meyer-Hofmeister, E., \& Xie, G.~Z. 1999, \apjl, 527, L17
1412.0960.extracted_bib
[{Meyer {et~al.(2007)Meyer, Liu, \& Meyer-Hofmeister]{meyer_re-condensation_2007 Meyer, F., Liu, B.~F., \& Meyer-Hofmeister, E. 2007, \aap, 463, 1
1412.0960.extracted_bib
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