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, tex2html_wrap_inline194 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.




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.


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 tex2html_wrap_inline206 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 tex2html_wrap_inline210 with corresponding random weights tex2html_wrap_inline212 and a non-negative integrable function f with tex2html_wrap_inline216 in the interval tex2html_wrap_inline218. Then the density
defines a shot noise process.
This means that shots tex2html_wrap_inline212 at times tex2html_wrap_inline210 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 tex2html_wrap_inline226 is a memory- or weight-function, tex2html_wrap_inline228 is the exponential parameter and tex2html_wrap_inline230 is the input corresponding to the tex2html_wrap_inline212. 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.



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.




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

Marco Thiel
Mon Feb 26 12:14:35 MET 2001