**Abstract**

The aim of this project is to find out more about the variability
of the X-ray flux of the Black Hole Candidate Cygnus X-1. Because of
the large distance to earth [2.5 kpc,
m] we cannot actually *see* what happens around the black
hole. We can measure though the x-ray flux and its fluctuations.
Fluctuations of the X-ray flux occur on timescales from years down to
milliseconds. New technics of data analysis have shown that the
usually applied shot-noise-model does not explain all caracteristics
of mesuared data. The objective of this project is to analyse the
x-ray flux with modern technics of data analysis and to determine
physical processes of the accretion disc.

**The black hole candidate Cygnus X-1**

Cygnus X-1 an X-ray source in the constellation of Cygnus (Swan) in the nothern sky was discovered in 1972/73.

Greek mythology tells about Hercules, son of Zeus, who had to hunt the so called stymphalic birds. Three of them an eagle, a vulture and a swan could escape. Up to today we can see Hercules hunt these three birds in the northern sky.

**What are Black Holes ?**

In 1783 Reverend John Michell, an British geologist and astronomer
published the hypothesis that a star with the same density as the sun
but 500 times its volume would have such a high gravitation that all
light emitted by such a body would drop back down on it. 12 years
later Pierre-Simon Laplace drew the same conclusion:*It is
therefore possible that the greatest luminous bodies in the universe
are on this very account invisible.* The Theory of General
Relativity made it possible to learn more about Black Holes. For
example that a black hole of the mass of the sun has a radius of only
3km. [The sun's radius is about 700000 km!] Not only in theory but
also experimentally Black Holes are a difficult matter as it is
difficult to detect them due to their invisibility. The following
cartoon visualizes the problem.

*It's black and it looks like a hole. I guess it's a Black
Hole.*

We learn about them by the so called accretion disk
that forms in close binary systems.

**Accretion disks**

In order to be able to detect Black Holes one has to look at binary star systems, where a Black Hole has a companion star. Both stars turn around each other. The Black Hole attracts gas from the companion star. The gas spirals in towards the Black Hole's event horizon forming a huge accretion disc. Falling down the gas heats up an begins to emit radiation.

**Observations**

As mentioned above the X-ray flux of Cygnus X-1 variates on all timescales from years to milliseconds. That is why at first Cygnus X-1 attracted attention of scientists. There are huge variations, like explosions of the flux.

These sets of data were measured by the ASM [All Sky Monitor] from
1996-2000. There are different states. Usually one distinguishes low-
, intermediate- and high-states.

All these states are up to now
not well understood. Though there are speculations that the spinning
sense of the accretion disc may change in relation to the spin of the
Black Hole. Then it could approach to the event horizon and so more
potential energy could be converted to radiation. Anyway we are more
interested in short time variability on timescales of milliseconds.
These sets of data are provided by the RXTE satellite.

**Short-Time-Scale-Observations**

Data used for this analysis were taken by the RXTE satellite. The three data files contain the number of counts of the photons in the energy band from 2-14 keV detected in time units of 4 ms. Each file consists of 390000-870000 data points.

We find huge fluctuations on timescales of some milliseconds.
This is one of the reasons why Cygnus X-1 is supposed to be a Black
Hole. Timescales of milliseconds correspond to Kepler-orbits of only
a couple of Schwarzschild radii.

**Data analysis**

Measured data of the X-ray flux at a first glance look like a
noise signal. In order to find out if there is dynamics in it we
performed an analysis called Q-statistics. Therefore we calculate the
asymmetry of the sets of data using

Where
is the n-th data point of the set. Now we calculate the significance
of the results. Therefore we use surrogate data. That means that we
change some qualities (the phases) of the sets of data at random and
calculate the significance

This
is the significance of the quality asymmetry *Q*(*m*) with
respect to noise. For the intermediate- and the low-state we find

**Shot Noise Model**

In order to describe observations one usually
uses a shot noise model. This is a formal mathematical model that
does not make use of physical ideas. A shot noise process is a
special point process. Given a Poisson distributed set of time points
with corresponding random weights
and a non-negative integrable function *f* with
in the interval
.
Then the density

defines
a shot noise process.

This means that shots
at times
decay according to *f*. In this special problem one usually uses
an exponential decay. This formal model mirrors well the short time
variability of the measured data. Anyway, it neither explains where
the shots come from nor reproduces the peak in the Q-statistics.

**Shot Noise Model with memory**

So as to model this peak we translate (3)
to a differential equation and introduce a *memory-function*.
This leads to

Here
is a memory- or weight-function,
is the exponential parameter and
is the input corresponding to the
.
In this equation we can consider a peak-peak interaction. A physical
process may be that after an explosion in the accretion disk it takes
some time until another instability can form and explode.

This
memory function induces - if carefully chosen - peaks in the
Q-statistics similar to measured ones. Anyway it cannot reproduce the
temporal behavior of these peaks. Therefore it will be necessary to
consider processes in the accretion disc.

**Physical Processes**

As Cygnus X-1 is at a large distance we cannot look at it with any spacial resolution. That's why little is known about accretion discs. In modern simulations it is not possible to attain a time resolution of some milliseconds. But some of the processes in accretion discs are at least qualitatively known. The first is viscosity. It produces turbulence that can lead to eruptions in the X-ray flux.

The second is a process based on magnetic loops. If they recombine, they possibly accelerate particles so that they emit a huge amount of radiation.

**Modeling**

We also develop what we call conceptual models. This means that we
try to build basic models that only consider one or few fundamental
processes. One of these models is a *traffic jam model*. It is
known that in accretion discs in-falling matter is retarded by an
outward force from radiation. Radiation that is produced by
in-falling matter. So we get a kind of feedback system similar to a
traffic jam. So we construct a set of differential equations that is
easy to handle.

**Outlook**

Further data analysis based on nonlinear methods of the 'Nonlinear Dynamics Group' at Potsdam University.

Analysis of other X-ray sources in cooperation with the AIP.

Construction of conceptual models.

**Reference**

J. Timmer, U. Schwarz, H.Voss, I. Wardinski, T. Belloni, G.
Hasinger, M. van der Klis, J. Kurths; *Linear and nonlinear time
series analisis of the black hole candidate Cygnus
X-1*,Phys.Rev.E61(2000), February

Mon Feb 26 12:14:35 MET 2001