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Estimating the age of

Estimating the age of. Cassiopeia A: Chandra's 1st light - August 19, 1999.

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Estimating the age of

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  1. Estimating the age of

  2. Cassiopeia A: Chandra's 1st light - August 19, 1999 Extraordinary first images from NASA's Chandra X-ray Observatory traced the aftermath of a gigantic stellar explosion in such stunning detail that scientists could see evidence of what might be a neutron star or black hole near the center.

  3. How old is Cas A? In 1680 the British astronomer John Flamsteed observed a star that was near the position of Cas A. This star was never seen again, so it might have been the explosion that produced Cas A. From an x-ray image of Cas A, you will used the observed size of the remnant and an estimated rate of expansion to determine the approximate age of the remnant.

  4. Ds9 software (available from http://chandra-ed.harvard.edu) allows Chandra data to be loaded and analyzed. Left click and drag to draw a circular region around the Cas A SNR to determine its radius in pixels. Note that the jet in the upper left has been excluded from the region – the dynamics of this jet formation are different that those of the overall expansion of the SNR. Record the radius of Cas A in pixels given in the “Circle” information box. How big is Cas A?

  5. Using the small angle approximation To find the radius of Cas A in meters, we must use the small angle approximation. Imagine the lines of sight from Cas A to Earth. These lines form an angle, q. On a Chandra image, 1 pixel corresponds to 0.5 arc seconds of angle. Find the angular size of the radius of Cas A in arc seconds. Convert this to radians using the information in the red box below. lines of sight from Cas A to Earth q Conversion factors: 60 arc sec = 1 arc min 60 arc min = 1 deg 360 deg = 2p rad Note: Image is not to scale!

  6. Check your answer! Finding arc seconds per radian: (60 arc sec / 1 arc min)(60 arc min / 1 deg)(360 deg / 2p rad) = 206,265 arc sec/rad Finding the radius of Cas A in radians: 337 pixels (0.5 arc sec/pixel) (1 rad / 206,265 arc sec) = 0.000817 rad

  7. Arc length is ~ radius of Cas A The lines of sight are the radii of an imaginary circle with Earth at the center and Cas A on the circumference. The radius of this circle is the distance to Cas A. Note that for very small angles, the radius of Cas A is approximately equal to the arc length transcribed by these lines of sight. Therefore, the small angle formula is as follows, where q is in radians: q = (radius of Cas A) / (distance to Cas A) Using the small angle formula and the information in the red box, find the radius of Cas A in meters. Radius of large circle = distance to Cas A q Information and Conversion Factors: distance to Cas A = ~11,100 light years 1 light year = 9.46 × 1015 meters Note: Image is not to scale!

  8. Check your answer! Converting the distance to Cas A to meters: (11,100 light years)(9.46 × 1015 meters/ 1 light year) = 1.05 X 1020 m Finding the radius of Cas A in meters: q = (radius of Cas A) / (distance to Cas A) radius of Cas A = (q)(distance to Cas A) radius of Cas A = (0.000817 rad)(1.05 X 1020 m) = 8.58 X 1016 m

  9. Determining the rate of expansion It is known that the average amount of energy released in a supernova explosion is 1044 Joules, and that only about one quarter of this goes towards pushing the gas outwards. The kinetic energy of the expanding gas is given by KE = ½ MV2 = ¼ X 1044 j M is the mass of the expanding gas and V is its velocity. How can we determine the mass of the expanding gas?

  10. Although the initial explosion throws out the outer layers of the star, most of the gas we see in the remnant is not from the star. As the explosion expands outwards it collects up gas from the interstellar medium, and pushes it forward, building up the shell we see. The amount of gas in the shell is therefore determined by the volume through which the remnant has expanded, and the density of the interstellar medium. On average this density is about 10-21 kg/m3. Think of the Cas A SNR as a sphere. Using the radius calculated previously, determine the mass of the gas within the remnant and then, using the formula for kinetic energy, calculate the velocity of the gas (expansion velocity of Cas A).

  11. Check your answer! Finding the mass of gas contained in Cas A: Density = mass/volume Volume = 4/3 pr3 Mass = (density)(volume) = (10-21 kg/m3 )(4/3 p)(8.58 X 1016 m) 3 = 2.65 X 1030 kg Finding the expansion velocity of Cas A: KE = ½ MV2 = ¼ X 1044 j V = SQRT (2 KE/M) = SQRT [2(¼ X 1044 j) / (2.65 X 1030 kg)] = 4.35 X 106 m/s

  12. Estimating the age of Cas A - finally! How long did it take Cas A to expand to its current size? Calculate this using the expansion velocity and the radius of Cas A. Convert from seconds to years. radius of Cas A V = D/T Could Cas A be the supernova observed by John Flamsteed in 1680? Take into account that we have used many approximations in this exercise and that the expansion of a supernova is a more complex mechanism than suggested by a model of a spherical shell expanding at constant rate.

  13. Check your answer! Estimating the age of Cas A in seconds: V = D/T T = D/V = (8.58 X 1016 m) / (4.35 X 106 m/s) = 2.65 X 1010 s Converting to years: (2.65 X 1010 s)(1 min / 60 s)(1 h / 60 min)(1 d / 24 h) (1 y / 365.25 d) = 625 y

  14. Finding the postion of the core remnant Move the pointer over the core remnant (this will appear as a dark gray dot towards the center of the remnant with Invert Colormap). The box in the upper right corner gives you a close up view of where your pointer is. Record its physical x- and y- coordinates.

  15. What is the kinetic energy of the core remnant? In geometry, the distance between two points is given by d2 = (x2-x1)2 + (y2-y1)2 Find the distance between the core remnant and the center of the SNR using the physical x- and y-coordinates of both. Convert this to arc seconds, then radians and then meters. Do you remember how? Using your estimated age of Cas A in seconds, what is the average velocity of the core? Using this average velocity, find its kinetic energy assuming that the core is a neutron star with a typical mass of about 1.4 solar masses.

  16. Check your answers! Displacement of the core remnant: d = sqrt [ (4292-4286)2 + (4234-4252)2 ] = 19 pixels 19 pixels (0.5 arc sec/pixel) (1 rad / 206,265 arc sec) = 0.0000460 rad (0.0000460 rad)(1.05 X 1020 m) = 4.83 X 1015 m Average velocity and kinetic energy of the core remnant: v = d/t = (4.83 X 1015 m)/( 2.65 X 1010 s) = 1.82 x 105 m/s KE = ½ mv2 = ½ (1.4)(2.0 X 1030 kg)( 1.82 x 105 m/s)2 = 4.6 X 1040 j

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