Astrophysics Write Up

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Analysis of Eclipsing Binary data - Mini-Project

Diego Granziol

25/01/2014

Abstract

I describe my analysis of the eclipsing binary data, taken by Radhakr-ishnan & Sarma and by Tomkin on the eclipsing binary system R CanisMajoris. I determine the orbital period as 1.139480 0.0000122days anduse this to fit the phase radial velocity data for both stars. The orbitsare found to be nearly circular which allows a deduction of the radii ofboth stars as 1.12 R (Solar Radii) and 0.8943 R and their masses as1.002092105 M (Solar Masses) and 0.156081863 M. I also estimate theeffective black body surface temperatures of the two stars as 7393.076Kand 5189.508K. The measurements on the second star suggest it is nolonger on the main sequence and must have had a far greater mass inthe past. A possible scenario for the evolution of the binary system isdiscussed in the main text.

1 Introduction

The future evolution of a star is largely determined by its mass. Although thesituation is changing now, historically stellar masses could only be determineddirectly for objects in binary systems and inferred for single stars by comparingtheir spectra with those of binaries. The study of binaries, and eclipsing binariesin particular, remains an important way of determining stellar masses and ofproviding insights into the later stages of stellar evolution [ref.Hilditch 2001] Thepurpose of the work descibed here is to verify physical and geometric character-istics of the eclipsing binary system R Canis Majoris using previously reportedradial veloctity and light curves [Ref. Radhakrishnan & Sarma, Tomkin]. Thepurpose of this experiment is to infer fundamental properties about the eclips-ing binary R Canis Majoris from the light and radial velocity curves. We willthen determine paramaters such as the orbital period, the radii, the mass, theeffective temperature and the extent of distortion from a spherical shape for

both stars. I list the experimental objectives hereDetermine the orbital period, that is the time taken by both stars to orbit

their centre-of-mass and produce phased light curves for both the Johnson-Band Johnson-V filters.

Phase radial velocity data for both stars using the derived period.

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Fit sinusoids to both phased radial velocity curves and assess whether ornot the assumption of circular orbits is reasonable.

For circular orbits whose planes lie in the line-of-sight, fitting sinusoidsto the phased radial velocity curves gives orbital speeds; use these to estimatemasses of both stars and their orbital radii.

Deduce the radii of both stars from eclipse timings. Estimate effective temperatures of both stars from (B-V) colours.Compare masses, radii and effective temperatures of both stars with those

of the Sun and deduce whether or not these are Main Sequence stars. Thereality of any conclusion reached will depend on the precision with which themass-ratio is known.

One of the two stars will be identified as having evolved beyond the MainSequence; provide a scenario by which this may have happened.

A detailed fit to the light and radial velocity curve data is needed toestablish the extent to which each star is distorted from its spherical shape; thisshows the extent to which mass-loss and mass- exchange had a role in the earlierevolution of both stars.

Compare residuals in fits to light curves with those expected, given theprecision of the data, and deduce whether or not there is further informationwhich could (in principle) be extracted.

1.1 Period Determination

1.1.1 Introduction to the Experiment

The first estimate of the period is achieved by the method of inspection. We areprovided with values of magnitudes of stellar flux from the Johnson B-band light

curves in the file B.dat. Armed with the formulax1

x2 =

2.5log

F1F2 (Where

x1& x2 refer to magnitudes and F1& F2 refer to the fluxes) we immediately

realize that the smaller the value ofF1 the greater the value of2.5log

F1F2

.

Thus in order to identify the minima, i.e smallest values of F1 we need toidentify the greatest magnitudes ofx1x2. By inspection we find the Times ofthe Minima (heliocentric Julian Date of observation) and in the adjacent columnwe include the difference in Time between supposed minimas.

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1.1.2 Table

Times of Minima (s) Difference in Time between last Minima2444606.2928 N/A2444607.4235 1.13072444647.1883 39.76482444648.3210 1.13272444664.2299 14.77482444672.1742 7.94932444998.1999 326.02072444999.3302 1.13032445015.2392 15.909

1.1.3 Calculation

We begin by assuming that our smallest difference in time corresponds to theperiod. We then refine our value by finding the closest integer value of n towhich our approximate value divides into a larger Time difference. Then dividethis Time difference by that value of n. We do this step twice and the calculationis outlined below. Using 1.1303 days as our first estimate 14.77481.1303 = 13.072 13. We now can gain a superior estimate of 14.774813 = 1.136523077 repeatingthis procedure for the largest time difference of 39.7648 yields 39.76481.136523077 =34.988 35 thus an even more improved estimate of 39.764835 = 1.136137143.This is carried out one final time with our largest time difference of326.0207326.0207

1.136137143 = 286.9664103 287giving us a final estimate of 326.0207

287 = 1.13596to 5dp. The unwary analyst may of course be tempted to skip the first two stepsand proceed immediately to the final step. We note however that 326.02071.1303 =288.4373175which gives a refined value of1.132016319notably different to our

calculate value.

1.1.4 Error analysis

This conclusion can be verified by a simple error calculation. Let us assumethat there exists an exact interval time which corresponds to a complete orbit.Let us call this time Tp and let us say that the smallest time between which weregister two minima can be written as Tp +e1, wheree1is the error which we canmodel as a symmetric distribution centered around 0. For a larger time betweenminima we can write this asnTp + e2, wheren corresponds to the exact numberof revolutions of the secondary around the primary planet. We approximatenas

nTp+e2Tp+e1

=n n, where the delta represents the deviation from the actualvalue ofn. Now, to remove any ambiguity in the integer value ofn, we need

the value of| |to be less than a 1/2. Thus we have= e2ne1

Tp+e1 2 = e22+n2e212ne1e2(Tp+e1)2 < 2 >=

+n22nTp++2Tp

+n2Tp+

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Where we have assumed that e1 and e2 are independent.

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1.2.4 Using the V light Curve to improve the estimate further

We repeat the same procedure with the V data. Finding that we have a min-imum at 1.13594443 days which gives a string length of 8.427 .204 and byrepeating the same method of taking the mid point between the values twostandard deviations away we get a value of

TV lightcurve= 1.135940+1.135957

2 = 1.1359485 0.0000085days

1.2.5 Final Period Result

Averaging the two values and summing the errors in quadrature we getTfinal = 1.139480 0.0000122days

1.2.6 B and V phase light plots

We display the plots of the B and V light curves using Dipso

1.2.7 B light curve (1)

Where the magnitude of the B light curve data is plotted on y against the phase(measured in multiples of Pi radians) is plotted on x. This plot has been zoomedin around radians to make for more accurate measurements for calculationsused in Section 2.

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1.2.8 B light curve (2)

More general overview of the B Light curve, with phase measured in radiansplotted on the x axis and the magnitude plotted on the y axis.

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1.2.9 V light Curve (1)

Identical Plot as in 1.2.7 except this time using Johnson V-band data.

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1.2.10 V light curve (2)

Identical Plot as in 1.2.8 except using Johnson V-band data.

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1.3 Radial Velocity Curve Fitting

1.3.1 Introduction

We fit functions of the form v = s+ a sin(2 + p) (i.e assuming perfectlycircular orbits) to the observed radial velocity curves for both stars using dipso.For circular orbits we expect the phase difference p to be an integer multiple of (including 0). Completing the procedure we get two sets of results, one forthe primary star (the larger one) and one for the secondary.

1.3.2 Radial Velocity data

Phase in units of radians

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1.3.3 Approximate values of Radial Velocities and Phase offsets

From the graph above we see that for the secondary the difference betweenmaximum and minimum is around 375 so we get an approximate amplitudeof 187.5 which we use as our first estimate for the secondary radial velocity.Repeating the the procedure for the Primary we get a radial velocity of about30. We find that it seems to be centered around roughly -32 so we take that asour systemic velocity first guess. Whilst the program is not so sensitive to phaseas it is to velocity estimates, we note that if we trace a sinusoidal curve ontothe secondary, that it roughly reaches the 0 point (i.e -32 in our approximation)because of a design feature in the program not dealing well with an input of 0 wegive it a small non zero value (i.e 0.1). Repeating the procedure for the primary,we note that tracing a sinusoidal curve through the points and following throughwe get an inverted sine curve. Thus we take as a starting point 3.14 as our phaseoffset for the primary, the justification for which is provided below.

sin(x + ) = sin(x) cos() + cos(x)sin() = sin(x)

1.3.4 Results from non linear least squares fit

We use the program bnryrvfit to carry out a non-linear least squares fit. We loadthe input file binary.dat with the already preset template. We use our previouslyobtained estimates for the Systemic Velocity, Radial velocity (for both primaryand secondary), the Phase offset (for both primary and secondary) and thePeriod in days (Tp as calculated earlier). Included in the results are also the

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graphical plots obtained from the fit program. For completeness, and in linewith the method above, we also quote the range which encompasses a 95%

confidence interval around the result. Initially we plot the radial phase velocityobservations for both stars. The secondary is shown in white and the primaryin red. (Fig 1.3.2)

1.3.5 Secondary star results

Systemic Velocity =32.2270938 kms1Radial Velocity = 185.390930 0.659569 kms195% confidence interval = 183.978858 186.003002 kms1Phase Offset = 0.0864732908 0.020116radians95% confidence interval = 0.0387876481 0.134158933radians

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1.3.6 Primary star results

Systemic Velocity =32.2270938 kms1Radial Velocity = 28.8757506 0.54489 kms1

95% confidence interval = 27.7390724 30.0124289 kms1

Phase Offset = 3.24332685 0.012718radians95% confidence interval = 3.21679608 3.26985762radians

1.3.7 Comments on Results

It is interesting to note that for the primary star that, even with the 95% con-fidence interval the phase offset does not encompass the value (3.14..) Weare forced to conclude that the orbit is clearly not circular, but obviously theellipse does not have too large an eccentricity (or the value would be much moredistorted from ) so the approximation will still be of use in further calcula-tions. The systemic velocity is not quoted with a standard deviation as it is notparticularly fundamental to our calculations. It will be allowed to vary withinthe fitting programs to achieve the best fit and then lowest chi-squared value. Itis also interesting to note that a moments thought would have revealed that fora circular orbit, the two radial velocities would have always been out of phasewith each other. Assuming we can treat the binary stars as a closed system,quite clearly by Newtons 3rd law the Centre of mass experiences no net forcesin this closed system and hence cannot move in space with anything other than

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a uniform velocity from our reference frame. In that frame of reference, giventhat there are no net forces, the centre of mass cannot move. Thus by whatever

angle the first star has moved through, the other will have had to move throughthat same angle, but in the opposite direction (i.e out of phase).

2 Light Curve Synthesis

2.1 Estimating component masses and orbital radii

We find that the mass ratio r = m1m2

= v2v1

= a2a1

= 185.3909328.8875706 = 6.420298214andwe find the variance given by

2(r)r2

= 2(a1)

a21

+ 2(a2)

a22

= 2(r)

(6.420298214)2 = (0.659569)2

(185.390930)2 + (.54489)2

(28.8757506)2

(r) = 0.1232862899

r= 6.42

0.123

2.1.1 Derivation of mass ratio

Assuming circular motion for both stars

F= m1v

2

1

r1=

m2v2

2

r2= Gm1m2(r1+r2)2

wherer1 andr2 specify the distance from stars 1 and 2 to the C.O.M1 = 2 = (or centre of mass would move)v= rm1r12 =m2r22m1m2

= r2r1

= v2v1

2.1.2 Calculation of the inter-stellar radii

Given that we know the radial velocities (calculated in section 1.3.6) and wehave an estimate for the period of the orbit we can estimate radii with thesimple relationship r1 =

v1

= v1Tp2 and similarly forr2 yielding the results

r1= (1.135948243600)(28.8757806

2 ) = 451050956m= 0.648527614Rr2= 2895881646m= 4.163740685Rr1+ r2 = 4.812268299RWhere r1&r2 signify the distances between stars 1 and 2 to the centre of

mass of the binary system.

2.1.3 Error analysis of the inter-stellar radii

Applying basic calculus of 2 variables and summing in quadrature (i.e assumingindependence between the two variables) gives us a total error e of

e= v2(T)2+T2(v)22 (where v is either v1 or v2 depending and the sigmasrefer to the errors in v and T respectively)

we thus get on inserting the correct values of of v, T, (T)and (v)e(r1) = 1.8870162%e(r2) = 0.3557736%

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2.1.4 Calculation of the stellar masses

Manipulating the basic circular motion algebra and remembering the definitionof r as the mass ratio/radius ratio and r1/r2 as the distance from star 1/2 fromthe centre of mass, we get expressions for the masses of both stars

M1 = r3

12r(1+r)2

G = 1.993261405 1030kg = 1.002092105M

M2 = r3

22(1+r1)2

Gr = 3.104624332 1029kg = 0.156081863M

M1+ M2= 1.158173968M

2.1.5 Error analysis of the stellar masses

The calculation is analogous to that above except for having 3 variables. Addingin quadrature we get an error for M1 and M2 which is best expressed in per-centages

e(M1) = 5.29842%

e(M2) = 7.65355%

2.1.6 Estimating Effective Radii

From the images 1.2.9 and 1.2.10 we make an estimate of the time in which thesecondary star is blocking the incoming light from the primary and the estimatethe time in which it does this and additionally drops outside of the field of view,this will be clarified by the algebra below. Where we assume that the stars canbe modeled as disks and that the eclipses are short with respect to the orbitaldurations and thus we can model the trajectories as linear.

2(r1+r2)v2+v1

=long=

= 0.2091246306( 21.135948243600

)= 3266.611692

2(r1r2)v2+v1

=short =

= 0.02338661688( 21.135948243600

)= 365.308457

r1= 1.12 Rr2= 0.8943R

2.2 Estimating Temperatures

2.2.1 Primary Star Preamble

We measure the B and V magnitude from figures 1.2.8 and 1.2.9 and we calculateB-V.

V(magnitude) = 0.31B(Magnitude) = 0.54 B V = 0.23

2.2.2 Determination of TemperatureAs given by the regression coefficients of Lynas and Gray in the astronomicalliterature [Ref.2] we calculate the Temperature of the star

T =A + Bp + Cp2

wherep = B V

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T= 8515 + (5270)(0.23) + 1440(0.23)2T1 = 7393.076K

2.2.3 Secondary Star Preamble

Unlike the primary star, assuming a synchronized binary system, the secondarystar will have one hemisphere permanently directed towards the primary. Thusthe observer would see the hot hemisphere immediately before and after thesecondary eclipse (as explained in section 6.2 of the manuscript). Thus thesecondary effective temperature requires a slightly more involved calculationwhich is outlined in Appendix B [Ref.1]. Following the calculation and usingthe values inferred from the figures 1.2.8 and 1.2.9 for the secondary eclipse weget

100.4V2 + 100.4V1 = 100.4V12

100.4V2 + 100.4(0.305) = 100.4(0.54)

V2 = 3.02798623100.4B2 = 100.4B12 100.4B1 B2 = 3.481377662B2 V2 = 0.453391432

2.2.4 Determination of Temperature

Using a slightly different set of coefficients as we know the mass of the secondaryis much smaller than that of the sun but it still has a roughly solar radius, itmust be on the Red Giant branch, using tabulated values from Lynas and Gray[Ref.2].

T =A1+ B1p + C1p2

T= 4842 + 1413(0.453391432)

1426(0.453391432)2

T2 = 5190K

3 Fitting Light and Radial Velocity Curves with

Nightfall

3.0.5 Editing the V.dat file

Following the instructions laid out in section 6.4 and 6.5 respectively we insertthe following information into the beginning of the V.dat file

#B VThis tells the program that this is the Johnson-V band data, we repeat this

procedure for the B band data replacing this with the instruction #B B

#P 1.13948This is the refined period time which we found in section 1.2.5#Z 2444648.3283#N 0.25#S 0.0

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3.0.6 Setting up the configuration file for nightfall

A template file called rcma.cfg will have been copied into the setup. We needto modify all the entries with the following values (as in type in the numericalvalues listed below)

MassRatio r= 6.42(section 2.1)Primary Temperature T2 = 5190K(section 2.2.2)convention of program is to have the opposite convention for Primary and

Secondary as the usual one, thus we mean by this the approximate effectivetemperature of the secondary (i.e cooler star)

Secondary Temperature T1 = 7393.076section (2.2.4)Absolute MassM1+ M2 = 1.158173968section (2.1.4)Absolute Distance r1+ r2= 4.812268299section (2.1.2)Absolute PeriodTfinal = 1.139480(1.2.5)

3.0.7 Fitting Light and Radial Velocity Curves with Nightfall

We then find that the Chi-Squared is extremely sensitive to changes in s and p.Where roughly S increases the width of the fit and P the depth. We get a bestfit of

2 = 12.78828compared to a value of 10 in the manuscript

p= 0.795s= 0.575

3.0.8 Table showing the range of P and S used and the varying Chi-Squared

p s

2

0.5 0.5 507.950.6 0.5 205.40.7 0.5 47.379020.8 0.5 1760.7 0.6 1530.8 0.6 22.33653

3.0.9 Plane of the Orbit Inclined to the line of sight

We try out different values of p and s for different inclination angles. Notingthat the circular motion is not a perfect but only a close fit, and the fact thatif the angle of inclination is much less than 90 degrees we would not observe

any eclipses, we try for 85, 80 and 75 degrees respectively. We also note thatwe only have a component radial velocity of sin(i)v where i is the inclinationangle and v is the speed. Thus clearly if i is smaller than 90 we observe only aportion of the velocity and thus we have underestimated the mass by a factorofsin3(i). Thus we divide through by this factor to get the more appropriatelarger mass. The idea is to find a value of i that gives the smallest value of2

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and then stick with that value of i and try to find the absolute minimum valuepossible.

3.0.10 85 degrees

p s 2

0.83 0.58 12.823850.85 0.59 11.744690.86 0.6 10.831510.87 0.61 10.530.88 0.62 10.69

0.875 0.65 10.5113

3.0.11 80 degrees

p s 2

0.5 0.5 5070.6 0.5 2080.7 0.5 47.390.8 0.5 176.9950.7 0.6 1530.8 0.6 22.33653

3.0.12 75 degrees

p s 2

0.45 0.5 883.195

0.6 0.5 638.970.7 0.5 3950.8 0.5 1890.9 0.5 48.90.9 0.5 43.6021 0.5 16.673511 0.6 30.951971 0.7 13.791

0.99 0.61 12.7910.98 0.61 13.390.98 0.605 12.9

0.985 0.605 12.885

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3.0.13 Retry of 85 degrees

p s 2

0.85 0.6 9.8540.852 0.601 9.70.855 0.605 9.320.857 0.607 9.1850.859 0.609 9.070.861 0.611 9.09

0.8611 .611 8.41994We note out of interest that there is not very much difference between the

chi-squared values obtained at 85 degrees and those at 75 degrees with excessivemanipulation of the values s and p, we also note that the experiment expects amanual adjustment of the parameters which under the time constraints of thelab time and in general seem rather short sighted, we propose as an improvementto the experiment that either the student or the demonstrator create a scriptthat systematically goes through all the values to find an absolute minimumand is allowed to run overnight to get the job done.

3.0.14 Final results

We increase the chi-squared by 11.5 giving a value of 19.919 and attempt tofind that value by varying each of the parameters (temperature of the secondarystar, the value p and the value s) until it reaches such a Chi-squared. This isa 99% confidence interval. We give the results as a range of the variable forthe 99% confidence interval, quoting the Chi-Squared at each limit respectively(which had to extracted manually as well). Gvein that this was a very laboriousprocess and that the chi squared was incredibly sensitive to initial conditions,

occasionally in the interests of time keeping a Chi-Squared slightly different tothe value of 19.919 was recorded.

3758 T 5611 (2 = 19.91448/19.8977)0.887 p 0.8214 (2 = 19.91536/19.96457)0.58624 s 0.65083 (2 = 20.11648/19.95356)

4 Interpretation

4.0.15 Conclusion

We find that the mass of the primary is M6 with error bars small in comparision.However given that it has the temperature and radius of the sun we reason that

it can obviously no longer be a main sequence star. Given that the minimumsolar mass multiple required to get onto the main sequences (fusing of hydrogento helium) is 0.08 (and we are at about .16). We infer that it used to be amain sequence star. At some point when the core was no longer able to burnhydrogen into helium at a point in the core, it expanded into a shell around thecore, pushing the radius of the star outwards until it hit the point of gravitational

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equilibrium between itself and the next star. At this point there would havebeen a transfer of mass (in the form of hydrogen) from one star to another (this

depends on the roche lobe filling factor which should be around 0.99 which sayshow much of this volume has been filled) explaining the observed difference inmass.

5 References

ListAS33p-2 Analysis of eclipsing binary data - Oxford University Practical Man-

agement SchemeBlackwell DE & Lynas-Gray AE, 1998 AAS 129, 505Budding E & Butland R, 2011 MN 418, 1764Dworetsky MM, 1983, MNRAS 203, 917

Hilditch RW, 2001, An Introduction to Close Binary Stars, Cambridge Uni-versity Press.

Radhakrishnan KR & Sarma MBK, 1982, Contr. Nizamiah & Japal-RangapurObs. No. 16.

Ribas I, Arenou F & Guinan EF, 2002, AJ 123, 2033Tomkin J, 1985, ApJ 297, 250Wichmann R, 2002, NIGHTFALL Users Guide

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