Analyzing Binary Star Data                                                         Printable Version

 

Introduction

Procedure

Radial Velocity Curves

Light Curves

Glossary

Stars to Analyze

What to Turn In

Binary Star Data Sheet

 

Introduction Our basic data relating stellar masses and radii are based on analyses of binary star systems. When we discuss the evolution of stars, we use mathematical models that have been verified by comparison with binary stars of known mass and radius.   When stars eclipse one another, we can match what we see to find the stars’ radii, their shapes, and their mutual orbit. When we use the Doppler effect to measure the radial velocity (velocity or approach or recession), we get information about the masses of the stars and the orbit.

 

In this lab you will be using analyzing actual light curves and radial velocity curves of stars. You will use computer simulations to model the light curves and radial velocity curves. You vary the input to the simulations to match the data and  find the  masses of the stars, their distance apart, their radii etc.

 

Astronomers can spend months or years measuring the radial velocities of the stars in a light curve and recording the amount of light a binary star system gives.   Then they model what they see using computer programs and find the orbit, the stars’ masses and radii.

 

You are each assigned two binary star systems to analyze. Links to the star data are below. The numbers of the systems you personally are assigned to do are obtained from within WebCT in the quiz that says “Assignments for Lab 7, HW 5 and Lab 8”. Take that quiz to get the identifiers of the systems to work on. 

 

Then print out or download the two data sheets so that you can look at  the data.


http://webct.dvc.edu/web-ct/en/img/shim.gif                                                                                              Binary Star Systems to Analyze

1

 BH Cas

2

FS Mon

3

GG Ori

4

HD115071

5

HD 135240

6

HV982

7

HV 5936

8

HV2274

9

Kalrath Sample

10

R136-38

11

R136-39

12

R136-42

13

R136-77

14

RT CRB

15

V 380 Cyg

16

V1073 Cyg

17

WW Cam

18

YY Gem

19

EI Cep

Your textbook has a couple of sections about binary stars and what can be found from them.. Read these sections first.   Also read the background information in the Student Guide and Main Content  (the pdf files) from http://astro.unl.edu/naap/ebs/ebs.html. READ it ALL.

 

For each system, you have both a light curve and a radial velocity curve.  These are plots (even though they are called curves). Also you will have the name of the system,  how many days it takes for the stars to orbit one time (called period and abbreviated P), and either the temperatures or the spectral types of the two stars.

 

Phase is plotted on the  y axis. It is labeled from 0 to 1  representing time as a fraction of the orbital cycle of the stars. Sometimes the scale goes past 1 or before 0, showing the same thing over again. .  Regardless of when the observations were actually made, the time is expressed in phase so that all the data will be directly comparable  and the motion of the system can be seen clearly. Normally the hotter star is furthest away at phase zero.

 

Radial velocity curves have km/sec on the y axis. There are two curvy lines on each plot, one for the motion of each of the two stars in the binary system.  Positive velocities mean that the star is  getting further from us; negative velocities mean that it is getting closer.

 

Light curves have magnitude or amount of light as the y-axis and only one line. The amount of light decreases because one star blocks part or all of the light from the other.

 

You will be finding  the physical properties of the stars by using computer models to match the observed radial velocity curves and light curves. You will need to  play around with the  star properties to match the light curves and radial velocity curves. This is basically what astronomers do, except that they use statistics to assess the match and you will  just look at the results.   

 

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Procedure

1. Get the star data for your stars and read the background material you’re your textbook and from  http://astro.unl.edu/naap/ebs/ebs.html.  Print out a data recording sheet for each of the binary systems.

 

2. Model the radial velocity curve using the program karensradvel. Input everything highlighted in yellow color.  You will need try various values for the  eccentricity and the longitude of the line of apsides to get the radial velocity curves to look right. The values highlighted in the pink color are the output. Print out the results, both the material above the plot and the plot itself. The print area is already set up. More instructions can be found below.

 

3. Model the light curve for each of the binary star systems using the eclipsing binary program from  http://astro.unl.edu/naap/ebs/animations/ebs.html

Use the values of the stars’ masses, eccentricity, longitude of line of apsides  and separation from your radial velocity simulation. Use the stars’ temperatures from their spectral types (this program is very sensitive to the difference in temperature and not sensitive to the temperatures themselves).

Play around with the inclination, the difference in temperatures and stars’ radii to match the depth and duration of the eclipses. Print the screen with your final values and the light curve. More instructions can be found below.

 

4 Complete a  binary star data recording sheet for each of the two systems

 

5. Mess around with the simulations to get insight and answer the questions at the end.

6. Write an objective and conclusion.

 

Radial Velocity Curves  Radial velocity is the velocity along our line of sight, toward us or away from us. The special thing about radial velocity is that it produces Doppler shift so we can measure the velocity of the stars.    The radial velocity plots show the velocities of the two stars as a function of the time in the orbit.   The stars’ velocities depend on their masses, the distance between them, and whether the orbit is circular or not.

 

You can tell the radial velocity curve because the Y axis is labeled in km/sec.  Each one has two lines. You are going to use an excel program to match the radial velocity curve. In the process, you will find out the masses of the stars, the distance between them and the shape of their mutual orbit.

This is a sample of the results of the excel program. It  can be found from  karensradvel.

 

The Y axis is the speed that each star is moving. The X axis is time, but rather than showing  days or months or years, it is in  Phase.  The stars go around one another all the time, over and over, just the same every orbit. So rather than looking at all the data in sequence, all the events are shown in terms where they are in one, repetitive orbit. The phase is the fraction of the way through the orbit.

 

At zero phase (the start) one star is in front of the other. The stars’ are moving crosswise as seen from Earth.  Then one star moves away (positive  velocity compared to the center of the system), the other moves forward (more negative  velocity).  At about half way through the orbit, the stars are again one in front of the other and the velocities reverse. The two stars orbit around a common point called the center of mass. Traditionally, phase zero is the time when the cooler star is in front.

 

The radial velocity curve shows the response of the two stars to the force of gravity between them. It depends on their masses and on the distance between them.  Astronomers, and you, simulate the radial velocity curve to find out the stars’ masses and separation.

 

Use the program   karensradvel  to model the radial velocity curve.  It will tell you the stars’ masses, the size, shape and orientation of the orbit.  It is an excel program. You can either download it or use it online.  You need the results of this program for your next step, so be sure to use it first. The data to input and the plot it makes are shown below.

 

Items to enter in radvel The blocks in yellow  are the values that MUST be specified. When you open the program there will be values in these boxes. These are NOT suggested values. They are just left over. You need to enter the proper values for each of the numbers. After you do this (and hit ENTER), the program will calculate the motion that the stars would have and calculate a plot of the speeds of the stars. Look at the plot. Adjust items d  and e until the plot in the radvel program matches the plot given in the data. 

 

Items to enter in the radial velocity program:

a.     The highest V (velocity) and lowest V for each star as read from the plot (4 numbers).

b.    The period, how long the orbit takes in days, is given on the data page. You don’t need to enter every single digit, but the period affects the results strongly.  

c.     The inclination. Set it to 90 degrees and leave it.

d.    Eccentricity could be between 0 and 0.999, but usually it is quite small. The eccentricity is the amount that the orbits of the stars differ from circles. Close binary star systems normally have small or zero eccentricity. But try different values to see what causes the best match of the radial velocity curve.

e.     Longitude of the line of apsides (between 0 and 360 degrees). It tells how the orbit is oriented to the observer. Symmetric radial velocity curves with crossing points at 0 and .5 phase, correspond to 90o longitude and zero eccentricity.   If the eccentricity is not zero, the longitude of the line of apsides changes the shape of the radial velocity curve and changes the phases where the radial velocity curves cross. It can be any value between 0 and 360.  If the eccentricity is zero, this does not matter.

 

Items a and b come directly from the data sheet. Don’t change them.  

Item c, the inclination, can’t be told from the radial velocity curve.  Set it to 90 degrees.

 

Items d and e are ones that you enter and adjust until the radial velocity curve shape matches the observations the best.

 

Be sure to hit return after typing all the numbers to get all the information into the program.   If plots for these programs do not update even though you have input the data and hit enter, click on the curve of the plot you want, go to the formula bar, click after the = sign, then click on the ü.

 

The outputs that you need from this program are the values of inclination, eccentricity, longitude of the line of nodes (these are in yellow), the masses of the stars in solar masses, the mass ratio and the average separation of the stars in units of the sun’s radius. These three values are filled in pink. The program calculates them.

 

When you are happy with the shape of the radial velocity curve, print out the page with the data and the plot of the radial velocity curve (or save it as a screen shot to turn in).

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Light curves are plots of the amount of light we receive versus time. When we look at eclipsing binaries, we cannot see the two stars separately because they are too close together. We just see the total light from the pair. As they orbit, one may eclipse (block the light of) the other, causing the amount of light received to decrease. Because of the symmetry of the system, each star will be blocked in turn.

 

In the following figure, the stars partially eclipse one another. In this light curve, at  phase 0, the hotter star is in back. At phase 0.5, the colder star is in back. We can tell that the hotter one is in back at phase 0.0 because the deeper eclipse (more light lost) occurs at phase 0. A hotter surface gives more energy from each square meter and  the same amount of area is covered at each eclipse. In this example, the total amount of light is constant when the stars are out of eclipse.  The label “V” on the Y axis, means magnitudes in the visual range (typically yellow light).

This light curve shows phase zero  twice. Not every light curve is set up this way, so be prepared to see the phase laid out differently..

 

The amount of light can be expressed as magnitudes (most usual), solar luminosities, or in terms of the total light of the system (like 0.875 when 1 is the most light seen).  The amount of light can change because one star comes in front of the other (an eclipse or a transit, if a small star passes across the front of a larger one). 

 

In cases where the stars are close together the period is short and the stars are often distorted. They can be pulled into shapes like . They look like balloons with the mouths together. When the stars are side by side there is a lot of light. When the stars are end on, they look like smaller circles, and there is less light. The light curve is often curved everywhere, like in the figure to the right.

 

A simulation of a binary star light curve can be found at binary . The .swf file is a simulation of an eclipsing binary star light curve. This is a Java program that runs on the internet. There is a student guide available from this page as well as several other mini-lessons. You may find them helpful. 

 

This is a screen shot of the binary star simulation roughly as you will see it when you enter (the .swf file). Play around with it to get an idea of what happens as you change each value. To change a slider, place your cursor over the “handle”, press the mouse button and keep it down as you slide. The sliders won’t go to places where they conflict with one another (like if you make the stars overlap).

 Mass2 from Radial Velocity Curve

 

 

 

 

 

The red arrows indicate values to enter  using information from the data sheet and radial velocity curve. They should not really change. Chose the temperature of star 1 based on the spectral type given on the data sheet.  

 

The blue arrows indicate things to match up. Do this by changing the following: inclination, radius of each star, and temperature of the second star (indicated by green arrows), until the light curve looks as close to the observed one as possible. Rather than trying to change everything at once randomly, try changing one thing at a time to see what effect it has.   There are some questions about this at the end.

Examples: The “preset button” leads to sample binary stars which demonstrate several issues.

 

Example 1 Two stars the same. These are partial eclipses, since the amount of light changes all the time. See what happens if you change inclination away from 90o. Then set inclination back to 90 o and see what happens with different radii for the stars.

 

Example 2 Two stars, same temperature, very different radii. The same temperature means that the eclipses are the same depth. At phase 0, the smaller star is in back. This is a total eclipse. The smaller is in front at phase 0.5. This is called a transit. Notice the flat bottoms of the eclipses. This is the indication of total eclipses and transits. Now look at  example 14. AD Her. The dots are the real data and the lines are the simulation output. The real data shows eclipses that are not exactly flat (because of limb darkening and non-spherical stars), but the simulation cannot model this.

 

Example 3 Two stars, different temperatures, same radius. Note the effect on the depths of eclipse. See what happens if you change the temperatures even more.

 

Example 4  The effect of different masses. This makes little to the light curves in this program, but start the simulation and watch the motions of the two stars. Adjust the masses to see how the motion changes.

 

Example 5 Shows the effect of eccentricity on the light curve. To see what is happening with the stars’ motion, change the inclination away from 90 degrees and look at the animation (there won’t be   any eclipses in some of these situations).

 

Example 6 Same as 5 but different masses.

 

Systems 15, 16, 17 These are real star systems with stars that are large compared to the separation. The stars’ shapes are actually quite distorted by the other one’s gravity. That make the light curve vary smoothly.  You can see this as you look a the observational points in the plot. The simulation doesn’t match this, because it won’t model the distortion. There are some stars like this among our samples. So be prepared.

 

As you look at the simulated light curve, the vertical scale is in terms of the amount of light. The most light, both stars unblocked, is 1.0. Your observational light curve is probably in terms of magnitude. Normally smaller magnitude values are toward the top of the plot and represent more light.

 

 The magnitude level out of eclipse is toward the top of your data sheet, but it may have any value. To compare with the simulated light curve, look at your data sheet and  figure out how many magnitudes down the eclipse is compared to the level outside of eclipse. If your maximum light is curved, try to get the light level without the “hump”. Now compare to the level  on the simulation using the plot that follows.

 

 

Example: The first example light curve in this write up shows a primary (deeper) eclipse with light changing from 7.5 to 8.15 magnitudes. That is a change of 0.65 magnitudes. Look at the plot and find 0.65 magnitudes on the X axis. Then go up to the line and find that a decrease in light of 0.65 magnitudes corresponds to a remaining light of 0.55. So look for a level of 0.55 on the simulated light curve for the deeper eclipse. The secondary minimum of the same example goes from 7.5 to 8.0 or a change of 0.5 magnitudes. Again looking on the x axis at 0.5 magnitudes and going to the curve, we find a light level of 0.63.

 

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http://webct.dvc.edu/web-ct/en/img/shim.gifBinary Star Systems to Analyze

1

 BH Cas

2

FS Mon

3

GG Ori

4

HD115071

5

HD 135240

6

HV982

7

HV 5936

8

HV2274

9

Kalrath Sample

10

R136-38

11

R136-39

12

R136-42

13

R136-77

14

RT CRB

15

V 380 Cyg

16

V1073 Cyg

17

WW Cam

18

YY Gem

19

EI Cep

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What to do/turn in

Experiment with the programs so that you can answer the questions below.

 

Work with the computer programs until you have the best match to the radial velocity and light curves that you can get.  Print out (or send as a file) the page from the radial velocity analysis and from the light curve analysis for each of the two systems. Label each one with the name of the binary star system it represents. Complete the  Binary System Data Sheet (below) for each of the binary star systems.

 

Attach your objective and conclusion.

 

Questions

 

A)    What happens to the light curve of a system if the inclination changes from 90o to lower values?

B)    What happens to the light curve if you make the radii of the stars larger and smaller?

C)    What happens to the radial velocity curve if you change the eccentricity from zero, but leave the longitude of the line of nodes, w, at 0o or 180o?

D)    What happens to the radial velocity curve if you have high eccentricity (like 0.5) and change the value of longitude to 90o or 270o from 0o?

 

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Binary System Data Sheet   Turn in one for EACH of your two systems.

1) System name                                   

 

2) System number from assignment list

 

3) Period (with units)

 

4) Spectral Types and Temperatures (you decide which is #1, but after that BE CONSISTANT)

(get correspondence between spectral type and temperature from textbook or HR diagram homework)

 

Star 1      Spectral type                                        Surface temperature

 

Star 2      Spectral type                                        Surface temperature

 

5) Which star is IN FRONT at phase zero?

 

6) Are the eclipses  partial or  total + transit?

 

From your simulations (from light curve, mainly)

 

7) Are the maxima flat or rounded?

 

8) Inclination, i (in degrees)

 

9) Radius of star 1 (specify units)

 

10) Radius of star 2 (specify units)

 

From your simulations (from radial velocity curve mainly)

 

11) Star 1  Maximum radial velocity                                Minimum radial velocity

 

12) Star    Maximum radial velocity                                 Minimum radial velocity

 

13) System Center of Mass Velocity

14) Eccentricity, e

 

15) Longitude of the line of nodes,w

(if the eccentricity is zero, then this doesn’t matter and you should say N/A)

 

16) Mass ratio

 

17) Mass of star 1 in solar masses

 

18) Mass of star 2 in solar masses

 

19) Semimajor axis of the relative orbit, a

 

20) Can you detect any major differences between the data and the your simulations  of the light curve and radial velocity curve?  If so, describe the differences.  You should experiment with the parameters  of the system until you get as good a match as possible, But some of the programs don’t model all the effects, so it may not be possible to get a really good match.

 

21) Print or trace the  simulated radial velocity curve and the simulated light curve and attach. Use a separate sheet for EACH system.

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Glossary

 

Center-of Mass Velocity The speed that the entire system is moving compared to the Sun. Earth’s orbital motion is averaged out before the plots are made.  The units are usually km/sec. It is the average speed of the system. Normally it is the value where the radial velocity curves cross.

 

The center of mass velocity is not related to the mass, radius etc of the stars. The center of mass is between the two stars. As the stars orbit one another (as shown in the eccentricity pictures below), the center of mass does not orbit or cycle. The distances of the two masses from the center of mass, r1 and r2, are related to the masses such that m1xr1=m2xr2, The center of mass is like the pivot point on a see saw. If the seesaw is properly balanced, the lower mass person is further from the pivot, moves more and faster. The figure in the eccentricity definition shows the motion.

 

Eccentricity A measure of the difference between the elliptical shape of the orbit and a circle. The value is always between 0 and 1 for an ellipse. Eccentricity has no units. The shapes of orbits can vary as shown in the following picture. The dot is the position of one of the stars, and the line is the path of the other star in comparison.

 

Realistically, both stars move. The paths are better represented as  the two ellipses below.  The masses of stars in a binary star system need not be the same. The more massive star might be at most 2 or 3 times as massive as the less massive star.  Since the masses are comparable, their mutual gravity causes substantial motion of both. The two stars both move around the center of mass. The amount of motion is in inverse relation to the masses of the stars. Similarly the velocities of the stars are in the same inverse relation

 

Inclination is the tilt of the orbit compared to the direction to Earth (and Sun). Eclipsing binaries MUST be nearly edge-on, 90o, or there would be no eclipse.   

 

 

 

Longitude of Periastron, w Periastron means closest (peri) to the star (astron), The orbits of stars are ellipses with the two stars moving around one another.  The longitude of periastron, w, tells the direction (angle) of this closest point compared to zero phase.

 

Longitude of periastron should be randomly distributed in space compared to the observer. But some longitudes produce more eclipses, so the eclipsing binaries don’t show a random distribution.

 

 

 

Mone, Mtwo These are the masses of the stars  in solar masses.  The radial velocity curve program calculates them based on the speeds in the orbit and the time the orbit takes. The light curve program uses them.

 

Orbital Period The time it takes for the stars to complete an orbit around one another. This is not negotiable. The period for each system is at the top of the page. The data we have is in days. Be sure to put it in and to tell the program that it is in days, not years.

 

Orbital Phase How far through the orbit. This is expressed as a decimal between 0 and 1 or as   degrees (0 to 360). Don’t enter this. It is the X axis on the plots.

 

Radius of Blue Star Radius of Red Star These are the radii in units of the average separation of the stars. So they are decimal  representations of numbers less than 1.

 

Semimajor Axis  is the name of the average distance between the two stars. The radial velocity program calculates it based on the speeds of the stars and the period of the orbit. The light curve program uses it.

 

Flat Bottomed or Slightly Rounded Bottom eclipses are due to a total eclipse (one star passing completely behind the other) and  (flat) and a transit (the other star moving across in front of the other).   Pointy or very rounded bottom eclipses are partial eclipses.

 

Since the same amount of area is covered at each eclipse, the size of the stars doesn’t affect the relative depth of the eclipses but the temperatures do. The spectral types (given) tell you the temperatures. Use the information. Don’t mess with it. Some of the programs allow you to change the temperatures, others allow only the letter  (OBAFGKM) but not the number.

 

If you cannot get the ratio of the depths of the eclipses to match the observations, it may be that the simulation is not modeling the temperatures precisely enough. If that is the case, the only thing that you can do is to use another computer simulation.

 

The depths and durations of the eclipses both increase as the system approaches edge-on, 90o inclination.  Increasing the radii of the stars   increases the duration of the eclipses, but decreases the depth. Normally, large stellar radii  (like 0.2 of the separation or more) cause the light curve to be rounded (rather than flat) even when not in eclipse. This is because the gravity of the two stars affect one another. The stars become elongated (rather like the shape of small balloons) in the direction to the other. As they spin, we sometimes see the smaller ends, sometimes see, the broader sides. The light curve simulation doesn’t model this effect, alas.  So be prepared to see a  data for a curved light curve and be unable to model it.

 

Eccentricity and Longitude of Line of Periastron Most close binaries (short period) have nearly circular orbits. That means that the eccentricity, e, is zero and the longitude of the line of nodes, w, doesn’t matter (it could be anything).    Nevertheless, there are non-circular orbits and you may encounter them.  

 

How can you tell whether the eccentricity is not zero?

In the radial velocity curve, the curve will not be symmetric if the eccentricity is not nearly zero,    In the light curve the eclipses may not be symmetrically spaced, and/or you may not be able to match the eclipses with any combination of the radii and inclination. In the radial velocity curve, the curve will not be symmetric if the eccentricity is not nearly zero,   

 

Normally the eclipses are 0.5 periods apart; one eclipse is centered at phase 0 and the other at 0.5. If this is NOT the case, then the orbit is eccentric, that is elongated, and the long direction, the major axis, is not toward the observer. Vary the eccentricity AND the longitude of the line of nodes (w, Greek lower case omega) to match the phases and depths of the eclipses.

 

Normally the two eclipses take the exact same fraction of the orbit.  So the time from beginning to end is the SAME for both of the eclipses. If not, the orbit is eccentric. If the eclipses are at 0 and 0.5 phase, then w, the longitude of the line of nodes is 0 or 180 degrees, but the eccentricity is not zero.

 

You can (and should) try to include the effect of eccentricity (e) and the longitude of the line of nodes (w)  in the light curve. But the effect is more easily seen in the radial velocity curve. It is normal to try to match the light curve, then work on the radial velocity curve to find eccentricity and w, then come back to the light curve to see the effects. Keep going back and forth until you get a good match.

 

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