Presented by Arkajit Dey, Matthew Low, Efrem Rensi, Eric Prawira Tan, Jason Thorsen, Michael Vartanian, Weitao Wu.

Transcription

1 Presented by Arkajit Dey, Matthew Low, Efrem Rensi, Eric Prawira Tan, Jason Thorsen, Michael Vartanian, Weitao Wu.

2 I ntroduction Transient Chaos Sim ulation and Anim ation Return Map I Return Map I I Modified DHR Model Fixed Points Recap Acknowledgem ent

3 Artist s View of Neutron Star (L) Accreting Matter From Com panion Star (R)

4 Under such extrem e conditions, standard m odels break down, so...

5 Constant accretion into cells Diffusion from neighbors Cell drips when full Result: chaos

6 Original Model wit h Recent Observat ions

7 Miller & Lam b Effect of Radiation Forces on Accret ion Outward radiation force causes tim e-varying accret ion Radiation drag force causes asym m etric diffusion

8 Ext ended Model wit h Recent Observat ions

9 Original m odel accounts for chaos and low-frequency oscillations in recent observat ions Our extended m odel m ay help explain high-frequency oscillations as well

10 Scargle & Young: original model displays chaos only for limited ( transient ) times How does the power spectrum of our extended model evolve over long periods?

11 Chaotic initial spectrum at t = 1 Non-chaotic Periodic spectrum at t = 50

12 Chaotic spectrum with high-frequency oscillations at t = 1 Unchanged spectrum at t = 50

13 Transient Chaos in the original m odel : Significant change in the power spectrum over a period of tim e Perm anent Chaos in the extended m odel: The power spectrum stays the sam e indefinitely - advantage

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15 I nner edge of disk represented as cells, Each cell having a state. Density

16 Cells accrete m ass (state values increse) Diffusion occurs bet ween cells Cell density resets at a threshold value

17 Return m ap is a m isnom er. Com pare m ass at a particular tim e x n to the m ass at a future tim e x n+k x n vs. x n+k Return m ap I : Random init ial condit ions n and k both fixed Return m ap I I : Sam e initial condition n varies, k fixed.

18 Mass at a certain tim e vs. one tim e step later We don t expect m uch change

19 Variability increases as tim e m oves forward

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21 Bands form in the lower-right-hand corner Mass appears to discretize

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23 Higher accretion rate Pattern repeats itself once

24 Where the dots are m ore concentrated, the cell s m ass is m ore likely to be located in that area. After enough tim e, the m ass in a cell becom es discretized, i.e., can only take on one of finitely m any values I t would be interesting to exam ine raw astronom ical data to confirm t hese observat ions.

25 Single cell s m ass at tim e n vs. at tim e n+ 5 Going through cycles with sm all shifts

26 Total m ass of the cells at tim e n vs. at tim e n+ 1. Showing fract als

27 Adding onto Young & Scargle s DHR m odel, we have the following discrete dynam ical system. The tim e variable is discret e. X n 1 f ( X n) f : H N H N f ( X ) AX b I n the extended m odel we added a constant > 0 to m odel dynam ic accretion. Then the m odified m atrix, A, is as shown above.

28 Each vector X has n coordinates all with values between 0 and 1 (i.e. X H N ) that is the density of the corresponding cell. One of the first ways to investigate a dynam ical system is by finding eigenvalues. X H N Adding the constant m akes the m odified eigenvalues i i. This guarantees that at least one eigenvalue is greater than 1 contributing to perm anent chaos. The m odified m atrix has the sam e eigenvectors as the original m atrix does.

29 A fixed point will satisfy: The solution is: I f m is an integer and every com ponent has value between 0 and 1. I f there is no tim e- varying accretion, fixed points do not exist.

30 Our extended model shows promise of explaining recent observat ions Our visualizat ion and return map studies give valuable new ways of extracting info Our abstract study has given a deeper understanding of the underlying dynam ics

31 Dr. J. Scargle Dr. S. Sim ic (NASA) (SJSU Math) The Woodward Fund

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