2.2 Estructura económica
3.1.1 Breve semblanza del debate
-0.2 0.0 0.2 0.4 0.6
Distance from WD (a)
2.0 1.0 0.5 +++ synthetic data model fit residuals 0.0 0.90 0.95 1.00 1.05 1.10 1.15 Orbital phase White dwarf radii
0 13 26 39 52 64
1.5
Ballistic stream Top magnetic stream Bottom magnetic stream
1.0
Thick curves: input Thin curves: results
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0.0•U ■ . ■ » * * ■ > ■ i . » ■ ... i ... » ■ ■ * ■ ■ . . . - ^ J . . . i . ... ... i ■ .
0.00 0.10 0.20 0.30 0.40 0.50 0.60
Radial distance (a)
0.1 Result from algorithm
L l -
0.0
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Distance from WD (a)
Figure 4.2: As for Fig. 4.1 b u t for a stream accreting onto b o th footpoints of a dipole m agnetic field line. The residuals in th e second panel are p lotted on a scale of —15 to 15 and are shifted vertically for clarity.
C H A P T E R 4. IN D IR E C T IM A G IN G OF T H E A C C R E T IO N S T R E A M 115 0.3 0.2 0.1 0 <§ o o 'LI illliiiiiiiu iiiiiiiiiiiiii; -0.1 S -0.2 # = 0.0005 -0.3 0.6 0.4 0.2 0.0 a/c db 2.0 3 3 .a
•s
£ i.o o a 0.5 # = 0.0005 : o.o 1.15 1.10 1.00 1.05 0.95 0.90 0.3 0.2 0.11
-0.1 CO Q -0.2 -0.3 0.0 0.2 0.4 0.6 a/c db 2.0 •a<L> a 1.0 o ° 0.5 d<|)=0.001 o.o 1.15 1.00 1.05 1.10 0.90 0.95Phase Distance from WD (a)
Figure 4.3: T he effect of lowering th e phase resolution of th e light curve is shown for phase resolutions d</> of 0.0005 (top) and 0.001 (bottom ). As th e phase resolution is decreased, th e num ber and am plitude of th e artefacts in th e stream brightness d istribution increases. T he residuals are p lo tted on a scale of -30 to 30 (displaced vertically by + 15 for clarity) for th e d<j> = 0.0005 case, and -300 to 300 (displaced by +150) for th e d(j> = 0.001 case. The regions of th e stream labelled ‘a ’, {b ’, ‘c ’, ‘d ’, ‘x ’ and ‘y ’ are discussed in the text; th e lines above th e light curves indicate the approxim ate phases a t which th e stream points around ‘a ’, cb ’, Cc’, ‘d ’, ‘x ’ and ‘y 5 are eclipsed or come into view.
R educed phase resolution
For a fixed num ber of emission points along the stream , the artefact number and am plitude is increased when the phase resolution is decreased. Fig. 4.3 contrasts the results for the two-pole stream example in section 4.2.1 a t reduced phase resolutions
d<j>
of 0.0005 and 0.001 (d^ = 0.0005 corresponds to ~ 4 s and d<j>
= 0.001 to ~ 8 s in a binary w ith P Crb = 2 h). These results should be com pared to the results in Fig. 4.2 where the light curve hasd<j)
= 0.0001. No noise is added to the light curves, and the d a ta have a signal-to-noise of 1000. The fit to th e d a ta w ith d ^ = 0.0005 has x 2/ N = 6, while the fit in thed(j>
= 0.001 case is much worse, w ith %2/ N = 85. The model w ithd(j>
= 0.0005 has located the bright regions of th e input stream and has retrieved their am plitudes and shapes, b u t there is an excess of brightness around 0.33a (labelled ‘c’ in the top right-hand panel of Fig. 4.3) and a deficit on the lower half of the m agnetic trajecto ry ju st below th e junction between the ballistic and the m agnetic trajectories (labelled ‘b ’ on th e top right-hand panel of Fig 4.3). W ithd<j)
— 0.001 the artefact problem is worse: th e model has located th e two bright regions, b u t their shapes are distorted and th eir am plitudes are reduced. In addition, there is an excess of brightness on th e ballistic stream near 0.33a (labelled ‘c’ on th e b o tto m right-hand panel of Fig. 4.3), and th e region near th e white dwarf is badly misrepresented.The artefacts in the brightness distributions, th e distortion of th e shapes of the bright regions on the stream , and th e larger of the residuals in the fits can be explained by examining how well the model is able to resolve certain critical parts of the stream . These critical p arts lie ju st below th e orbital plane near the lower pole (labelled ‘a ’ on th e right-hand panels of Fig. 4.3), and ju st below th e junction between the ballistic and the magnetic trajectories (labelled ‘b ’). The group of
f
points at £a ’ lies parallel to th e limb of th e secondary during eclipse ingress a t phase 0 ~ 0.965, and the group of points a t ‘b ’ is parallel to th e limb of the secondary during eclipse egress at <t> ~ 1.075. At these phases, several adjacent points a t ‘a ’ or
C H A P T E R 4. IN D IR E C T IM A G IN G O F T H E A C C R E T IO N S T R E A M 117
£b ’ are eclipsed or come into view in a single phase bin. As a result, th e m odel is not able to assign brightnesses unam biguously to these points. E rrors are propagated into th e points on th e ballistic stream th a t are eclipsed or th a t come into view in th e sam e phase bin (in order to preserve th e x 2 of th e m odel a t th a t phase). For exam ple, a t <f> ~ 0.965 as th e points a t ‘a ’ are eclipsed, th e po in ts along th e ballistic tra je cto ry a t ~ 0.32a (labelled £c ’ on th e rig h t-h an d panels of Fig. 4.3) are also being eclipsed. T he points on th e ballistic stream th a t come into view when th e points a t £b ’ egress, are a t ~ 0.23a (labelled £d ’ in Fig. 4.3). T he brightnesses of emission points a t £c ’ an d £d ’ are thus n o t w ell-determ ined, an d th is accounts for th e artefacts on th e m odel stream s a t these positions (an excess of brightness a t £c’ and dim inished flux a t £d ’). T he largest residuals in th e fit occur where th e points £a ’,£b ’,£c’ and £d ’ ingress and come into view: these phases are indicated by th e relevant letters above th e light curves in Fig. 4.3.
T he artefacts on th e low resolution d 0 = 0.001 m odel are p articu larly severe around th e w hite dwarf, and th e largest residuals occur in th e light curve between 0.961 ~ (j) ~ 0.964. T his phase interval com prises ju s t four phase bins during which 16 adjacent points ju s t below th e lower pole are eclipsed, th e u p p er pole is eclipsed, and five points ju s t above th e up p er pole are eclipsed. T his large num ber of poorly- resolved points around th e w hite dw arf leads to th e spurious stru c tu re in th e stream brightness d istrib u tio n in th is region, and hence to th e po o r fit to th e light curve in th e interval 0.961 ~ <j> ~ 0.964.
T he artefacts in th e m odel stream are created as th e algorithm a tte m p ts to preserve th e x 2 a t a p artic u la r phase, b u t these artefacts resu lt in a decrease of th e
entropy of th e m odel light curve. T his is because th e knots of excess or dim inished
flux in th e artefacts reduce th e local sm oothness of th e stream brightness. Because th e algorithm m axim izes th e entropy of th e brightness d istrib u tio n (by m inim izing —S, see equation 4.1) as well as m inim izing th e x 2 of th e fit, th e decrease in entropy caused by th e artefacts m ust be com pensated for by an increase in entropy elsewhere on th e stream . T his occurs a t th e brig h t region on th e field line below th e orbital
plane (labelled ‘x ’ on the model stream s). In the input stream (shown inxthe top panel of Fig. 4.2) this feature extends over 19 emission points and has a sharply peaked shape. In the low-resolution models, th e feature is spread over a larger num ber of emission points, and is increasingly distorted (flattened) as the phase resolution is decreased. Large residuals in th e model fits occur where this bright region is eclipsed and comes into view: these phases are labelled ‘x ’ in Fig. 4.3 (the ingress of ‘x ’ is not labelled due to lack of space: it occurs between the ingress of features ‘d ’ and ‘b ’). The algorithm also sm oothes and spreads out the feature near the L I point in order to increase the entropy of the model stream : th e ingress and egress of this feature are labelled ‘y ’ in Fig. 4.3.
To check w hether this explanation for the origin of th e artefacts is correct, a fit was perform ed to a low phase resolution (d<^> = 0.001) eclipse profile generated using a stream where the emission points have a uniform brightness. The artefacts should disappear in this case, despite the low phase resolution. This is because the algorithm should not be able to com pensate for th e decrease in entropy (caused by artefacts) by sm oothing other regions of th e stream , since the stream brightness dis tribu tio n is already the sm oothest possible. Indeed, th e algorithm does not produce artefacts on th e model stream (see Fig. 4.4).
Lowering th e phase resolution of the light curve thus has two m ain consequences: the m is-representation of the stream brightness a t ambiguous points along the stream , and the sm oothing of local extrem a in th e stream brightness distribution to com pensate for the resulting decrease in entropy. Com paring th e results of the
d(j) = 0.0005 and d 0 = 0.001 models, it can be seen th a t th a t the quality and relia bility of th e model decreases dram atically if d 0 ~ 0.001 for this num ber of emission points along the stream (see also section 4.2.2).
W hen applying the model to real d ata, it is possible to distinguish between artefacts and real brightenings along th e stream . Once a model stream trajectory is found th a t produces acceptable fits to the original d ata, th e ambiguous points in th a t stream can be identified. An analysis similar to th e one above can then
C H A P T E R 4. IN D IR E C T IM A G IN G O F T H E A C C R E T IO N S T R E A M 119 2.5 2.0 S .3 1 1.0 a 0.5 0.0 0.90 0.95 1.00 1.05 1.10 1.15 Phase 0.1 a 00 § I Q -0.1 -0.2 0.0 0.2 0.4
Distance from WD (a)
0.6
Figure 4.4: T he lack of artefacts in th e stream image derived from an eclipse profile generated a t low phase resolution (top panel) b u t using a stream w ith a uniform brightness d istribution. A low phase resolution tends to cause artefacts in th e solu
tion found by th e algorithm . However, since this stream brightness d istribution is already th e sm oothest possible, th e algorithm cannot com pensate for th e decrease in entropy (caused by artefacts) by sm oothing th e stream . T he resulting stream image (b o tto m panel) is thus free of artefacts.
be perform ed to identify th e points on th e stream th a t are likely to have poorly- determ ined brightnesses, and thus th e regions of th e stream th a t are likely to show artefacts.
The precise location of th e ambiguous points on th e stream depends on the system geom etry i.e. th e values of M i, q, i, (3, f and R ^ . For th e stream trajectory assumed in section 4.2.1, the one-pole stream has no points th a t lie parallel to the limb of th e secondary during ingress or egress. This explains why th e artefact level in the one-pole case is much lower th a n in the two-pole exam ple in section 4.2.1, even though th e two light curves have th e sam e phase resolution and neither have added noise.
N o isy d ata at h igh phase resolu tion
A noisy light curve is constructed by m ultiplying th e norm alized intensities of the synthetic light curve by a constant to convert th em to ‘counts’ and then adding a background ‘sky’ value to each point. The sky contribution is assumed to be a constant fraction of th e m ean intensity of the light curve. T he level of noise added to each point depends on its location in th e eclipse profile. One of th e dom inant sources of noise in a polar light curve is th e flickering from th e accretion region; the phase intervals m ost affected by flickering are thus before th e accretion region ingress and after its egress. T he accretion stream ingress and th e points during to ta lity are not affected (since th e accretion region is obscured during these phases) and therefore have a much lower noise level. For this reason, lower noise levels are assigned to the phases where th e accretion region is obscured th a n when it is in view. The noise level for phases where th e accretion region is in view is calculated using
noisej = G R N D x model; + sky (4.4) where G R N D is a random num ber taken from a G aussian distrib u tio n w ith zero m ean and u n it variance; the noise level for th e phases where th e accretion region
UPLA^lPjJti 4. U VU lK tiU l 1MAUUVU UP PPIPj AUUPLPj l l U l y blU PJAM 121
Input stream brightness
a L l - 0.0 £ o <D I -0-1 -0.2 0.0 0.2 0.4 0.6
Distance from WD (a) 2.0 c 0.5 + + + sy n th etic data m o d el fit resid u als o a 0.0 0.90 0.95 1.00 1.05 1.10 1.15 Orbital phase White dwarf radii
0 13 26 39 52 64
B a llistic stream T op m agn etic stream B ottom m agn etic stream
T h ick curves: input T h in curves: resu lts
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Radial distance (a)
Result from algorithm
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Distance from WD (a)