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Variable Stars
http://www.wolfram.com/applications/astronomer/index.html
setup
<<Astronomer`HomeSite`
setup algolAnimate
rA=2.74;rB=3.60;dAB=14.00;
mA=3.70;mB=0.81;
dA = +mB*dAB/(mA + mB);
dB = -mA*dAB/(mA + mB);
e=0.2;
algolAnimate[] :=
Do[c = Cos[t];
s = Sin[t];
xA = dA*c; yA = dA*s*e;
xB = dB*c; yB = dB*s*e;
algolA = {RGBColor[1., 1., 0.8],
Disk[{xA, yA}, rA]};
algolB = {RGBColor[0.4, 0.4, 0.],
Disk[{xB, yB}, rB]};
Show[Graphics[If[t < 3.14,
{algolA, algolB},
{algolB, algolA}]],
Background -> GrayLevel[0],
PlotRange ->
{{-1, 1}*dAB + {-1, 1}*rB,
{-e, e}*dAB + {-1, 1}*rB}*1.2,
AspectRatio -> Automatic],
{t, 0., N[2*Pi], N[(2*Pi)/24]}]
setup NextEclipse, NextPulsate functions
NextEclipse::usage="NextEclipse[variableStar, neardate] computes the date of the next "<>"eclipse of the eclipsing variable star (eg Beta.Perseus). "<>"Eclipsing variable stars quickly fade to minimum brightness during "<>"the eclipse, and then quickly rise to maximum brightness and remain "<>"there for most of the cycle.";
NextPulsate::usage="NextPulsate[variableStar, neardate] computes the date of the next "<>"pulsate of the pulsating variable star (eg Omicron.Cetus). "<>"Pulsating variable stars quickly rise to maximum brightness during "<>"the pulsate, and then slowly fade to minimum brightness during the "<>"rest of the cycle.";
NextEclipse[variableStar_]:=NextEclipse[variableStar,Date[]]
NextPulsate[variableStar_]:=NextPulsate[variableStar,Date[]]
NextEclipse[variableStar_,date_]:=varNext[varEclipseData[variableStar],date]
NextPulsate[variableStar_,date_]:=varNext[varPulsateData[variableStar],date]
varNext[False, d_] := False
varNext[{p_, t_}, d_] :=
LocalDate[t + Ceiling[(ModifiedJulianDay[d] - t)/p]*p]
varEclipseData[Beta.Perseus]={2.8673043,2.0709};
varEclipseData[Epsilon.Auriga]={9892.0,825.5};
varEclipseData[Lambda.Taurus]={3.9529478,0.5633};
varEclipseData[Zeta.Phoenix]={1.6697671,1.4224};
varEclipseData[Beta.Lyra]={12.9138340,8.213};
varEclipseData[_]:=False
varPulsateData[ZZ.Carina]={35.53584,24.99};
varPulsateData[Beta.Dorado]={9.842600,9.60};
varPulsateData[Delta.Cepheus]={5.366341,3.789};
varPulsateData[Eta.Aquila]={7.176641,1.715};
varPulsateData[Zeta.Gemini]={10.150730,9.107};
varPulsateData[Omicron.Cetus]={331.96,275.1};
varPulsateData[Chi.Cygnus]={408.05,189.2};
varPulsateData[_]:=False
Variable Stars
Most stars remain at a fixed magnitude or brightness throughout most of their lifes. However some stars have a varying magnitude. These stars are naturally known as Variable Stars.
Variable stars fall into two main types - Eclipsing and Pulsating.
Eclipsing Variables
Eclipsing variables occur when the star is part of a binary system with the second component regularly passing in front of the main component.
Two types of eclipsing variable can occur depending on the relative sizes, orientation and magnitude of the components.
An Algol type variable is a binary system with one component much brighter than the other, and the diameters somewhat less than the separation of the two components. In such a case the system will appear to stay at maximum brightness for most of the time, and only fade for a short time during the eclipse. The prototypical example is Algol (or Beta.Perseus) with a period of 2.87 days.
A Beta Lyrae type variable is a close binary system with the components having large diameters almost equal to the separation. In such a case the system will vary continuously in brightness throughout the cycle. The prototypical example is Beta.Lyra with a period of 12.91 days.
Eclipsing - Algol Type
Mag Period Mid-Eclipse
Asc Dec Max-Min (days) (JD)
Beta.Perseus 03h08.2m +40?57' 2.12-3.39 2.8673043 2445641.5135
Epsilon.Auriga 05h02.0m +43?49' 2.92-3.83 9892.0 2435629.0
Lambda.Taurus 04h00.7m +12?29' 3.37-3.91 3.9529478 2421506.8506
Zeta.Phoenix 01h08.4m -55?15' 3.91-4.42 1.6697671 2441643.6890
Eclipsing - Beta.Lyra Type
Mag Period Mid-Eclipse
Asc Dec Max-Min (days) (JD)
Beta.Lyra 18h50.1m +33?22' 3.25-4.36 12.913834 2408247.950
Prediction
In the setup section at the start of this notebook various data about the period and phase of some eclipsing variables is given. A function called NextEclipse is also defined to return the date of the next mid-eclipse of any of those eclipsing variables.
?NextEclipse
For example the first eclipse of Algol in 1996 occurs at
NextEclipse[Algol,{1996,1,1}]
That is 1996 January 3 at 7pm (Melbourne time). During the eclipse, which lasts about 4 hours either side of the mid-eclipse time above, the magnitude of Algol will fade from about 2.1 down to 3.4.
Algol, like most eclipsing variables, has a short period of just a few days. However there are long period eclipsing variables, with the best example being Epsilon.Auriga with a period of 27 years.
NextEclipse[Epsilon.Auriga,{1996,1,1}]
The next eclipse of Epsilon.Auriga therefore occurs in late 2010. During the eclipse, which lasts about 6 months either side of the mid-eclipse time above, the magnitude of Epsilon.Auriga will fade from about 2.9 down to 3.8.
Algol Animation
The best known eclipsing binary is Algol.
Algol is the star about 20 degrees north of Pleiades, and about 25 degrees west of Capella.
Good comparison stars for viewing the variability of Algol are Kappa.Perseus (mag=3.80) to the north, Epsilon.Perseus (mag=2.89) to the east and Gamma.Andromeda (mag=2.14) to the west.
Do not use the nearby Rho.Perseus to the south, as it is a pulsating variable.
During the eclipse about 72% of the smaller brighter component is obscured by the larger fainter component.
algolAnimate[]
Pulsating Variables
Pulsating variables occur when the star is physically changing its diameter, with the result that its magnitude changes.
Several types of pulsating variables can occur depending on the reason for the change in diameter.
A Cepheid type variable is typically an old yellow giant star which has gone unstable due to most of its hydrogen and helium being exhausted. The magnitude variation is usually only about 1. The prototypical example is Delta.Cepheus with a period of 5.37 days.
A Mira type variable is typically a red giant star which pulsates similar to a Cepheid. However much larger magnitude changes of around 10 are typical. The prototypical example is Mira (or Omicron.Cetus) with a period of 331.96 days.
There are also semi-regular variable stars, whose period is mostly regular but not fixed. The prototypical example is Betelgeuse (or Alpha.Orion) with a period averaging roughly 6.4 years.
There are also slow irregular variable stars, whose variability is completely unpredictable, but changes happen only over timescales of years. The prototypical example is Epsilon.Pegasus.
Pulsating - Cepheid Type
Mag Period Brightest
Asc Dec Max-Min (days) (JD)
ZZ.Carina 09h45.2m -62?30' 3.28-4.18 35.53584 2440736.9
Beta.Dorado 05h33.6m -62?29' 3.46-4.08 9.84260 2440905.3
Delta.Cepheus 22h29.2m +58?25' 3.48-4.37 5.366341 2436075.445
Eta.Aquila 19h52.5m +01?00' 3.48-4.39 7.176641 2436084.656
Zeta.Gemini 07h04.1m +20?34' 3.62-4.18 10.150730 2443805.927
Pulsating - Mira Type
Mag Period Brightest
Asc Dec Max-Min (days) (JD)
Omicron.Cetus 02h19.4m -02?59' 2.00-10.10 331.96 2444839.0
Chi.Cygnus 19h50.6m +32?55' 3.30-14.20 408.05 2442140.0
Pulsating - Semi-Regular
Mag Period
Asc Dec Max-Min (days)
Alpha.Orion 05h55.2m +07?24' 0.00-1.30 2335.0
L.Puppis 07h13.5m -44?38' 2.60-6.20 140.6
Alpha1.Hercules 17h14.6m +14?23' 2.74-4.00 100.0
Eta.Gemini 06h14.9m +22?30' 3.15-3.90 232.9
Rho.Perseus 03h05.2m +38?50' 3.30-4.00 50.0
Mu.Cepheus 21h43.5m +58?47' 3.43-5.10 730.0
Pulsating - Slow Irregular
Mag
Asc Dec Max-Min
Epsilon.Pegasus 21h44.2m +09?53' 0.70-3.50
Beta.Pegasus 23h03.8m +28?05' 2.31-2.74
Prediction
In the setup section at the start of this notebook various data about the period and phase of some pulsating variables is given. A function called NextPulsate is also defined to return the date of the start of the next pulsation of any of those pulsating variables.
?NextPulsate
For example the first pulsation of Mira (Omicron.Cetus) in 1996 occurs at
NextPulsate[Omicron.Cetus,{1996,1,1}]
That is on 1996 March 8. During the pulsation, the magnitude of Omicron.Cetus will quickly rise to its maximum magnitude of 2.0. Over the next 332 days it will slowly fade down its minimum magnitude of 10.0, after which another pulsation will occur and it will brighten rapidly again.
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