To find the periods of the X-ray emission pulses for GK Per and Cen X-3 using data sets from the Chandra X-ray Observatory and ds9 analysis tools and then to use this information to determine if these sources could be white dwarfs or neutron stars.

Background:

ds9 can produce a light curve for data gathered by the Chandra X-ray Observatory. A light curve is a graph of the brightness of an object versus time. For stellar objects which change brightness over time, such as supernovae, novae and variable stars, a light curve can help astronomers classify the object.

GK Per and Cen X-3 are rotating variable stars whose brightness in X-rays changes as a “hot spot” rotates in and out of our view. In the case of a white dwarf, such a “hot spot” could occur if a white dwarf with a magnetic field accretes matter from a companion star. The accretion disk is disrupted at small radii by the white dwarf’s magnetosphere. Material leaves the disk and travels along magnetic field lines. At some distance to the surface, a strong shock occurs where freefall kinetic energy is converted to thermal energy. Below this, material settles onto the white dwarf near the magnetic poles. As this material cools, it releases x-rays. If the magnetic axis is offset from the spin axis, the X-ray emission will pulse with the spin period of the white dwarf. A sun-like star eventually becomes a white dwarf when the core, left behind after a red giant puffs off its outer layers, collapses. An object the size of an olive made of white dwarf material would have the same mass as an automobile!

Accreting White Dwarf Credit: CXC/M.Weiss

illustration of an accretion powered pulsar

The central part of a more massive star will collapse even further to form a neutron star. Electrons are pushed into protons to form neutrons and the result is a tiny star with very little empty space. A neutron star would have the same density as 10 million full-sized African elephants in the space of a thimble. A neutron star in a binary system can become an accretion powered pulsar, producing a pulsing X-ray emission in much the same way as described above for the white dwarf. A neutron star, however, can spin faster (have a shorter period) because its higher mass and smaller size generates a stronger gravitational field that can prevent a fast spinning pulsar from breaking apart.

Procedure:

1. Install ds9.

2. Open ds9. Go to Analysis>Virtual Observatory>Rutgers X-ray Analysis Server and click Load the new GK Per Image on the screen that comes up. (Do not close this window – you will be loading another image later.)

3. Regions have been pre-drawn for you. If you wish, play with different Scales and Colors to see the details of GK Per. Go to Analysis>Chandra Ed Analysis Tools>FTOOLS Light Curve and click OK. On the light curve generated, zoom in on an area on the graph by either left clicking and dragging a box around a region of the graph or by changing the x-axis range under Graph>Axis Range until you can determine if there appears to be a periodic x-ray pulse.

4. Go to Analysis>Chandra Ed Analysis Tools>FTOOLS Power Spectrum and click OK. This command does a fast Fourier Transform on the data to search for periodicities. If the data isn’t periodic, you will not see one large peak as you will for GK Per. Zoom in on the peak until you can determine its frequency. Convert the frequency to period (period = 1/frequency).

5. Go to Analysis>Chandra Ed Analysis Tools>FTOOLS Period Fold. This will help you to check the accuracy of the period you found in step 4. Enter the value for this period in the window that comes up and click OK. To understand what this command is doing, picture a drawing of a sine wave on a long piece of paper. Cut this paper into sections, each one period long, and put each cycle on top of the first one, adding all the sine waves together. If your cuts are not exactly every period, when you add the sine waves, parts of the waves would cancel out and your composite wave would have a smaller amplitude. How does your Period Fold graph look? Try it again with a period several seconds different than the one you used before. Which graph is more sine-like and has the highest amplitude? If you need to, try Period Fold again with different periods until you produce the best graph. Record this period. This is equal to GK Per’s spin period as discussed in the background information.

6. Go to the basho.rutgers.edu window and hit the back arrow. Click Load the Cen X-3 Image. Make a light curve, power spectrum, and period fold for this data to find the period of Cen X-3.

Calculations and Interpretations:

The acceleration due to gravity (g) on the surface of a star (according to Newton’s Universal Law of Gravitation) is given by

g = (GM)/R² where G = 6.67 X 10^-11 Nm²/kg², M=star’s mass and R = star’s radius

Centripetal acceleration (a_c) of an object on the surface of a star at its equator is given by

a_c = V²/R and since V = 2pR/T for an object moving in a circle

a_c = 4p²R/T², where R = star’s radius and T = star’s spin period

If the centripetal acceleration on material on the star’s surface for a given period is less than the acceleration due to gravity, the gravitational force would be enough to hold the material on the surface and the star could sustain such a period.

Let the mass of a white dwarf be approximately one solar mass or 2.0 X 10³⁰ kg and its radius, approximately that of Earth or 6.4 X 10⁶ m. Let the mass of a neutron star be two solar masses or 4 X 10³⁰ kg and its radius be 10 km.

7. Assume GK Per is a white dwarf. Calculate both g and a_c for GK Per (using the period you found using ds9). According to your calculations, can GK Per be a white dwarf? Why or why not?

8. If your answer above is “no”, repeat #7 assuming GK Per is a neutron star.

9. Repeat #7 and #8 for Cen X-3.

Resources:

FTOOLs Graphs of GK Per and Cen X-3 to be used if computers are not available
http://www.oocities.org/lewiston_stargazer/ds9/gkperintro.html

Chandra Education Data Analysis Software And Activities
http://chandra-ed.harvard.edu/

Chandra Field Guide
http://chandra.harvard.edu/field_guide.html

Variable Star Of The Month Nov. '00: GK Persei (Nova Persei 1901)
http://www.aavso.org/vstar/vsots/1100.shtml

The Encyclopedia of Astrobiology, Astronomy, and Spaceflight - Cen X-3
http://www.daviddarling.info/encyclopedia/C/Centaurus_X-3.html

ASTRONOMY AND ASTROPHYSICS YY Draconis and V709 Cassiopeiae
http://physics.open.ac.uk/~ajnorton/papers/yydra_v709cas.pdf