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Lab 4 – Binary Stars
ASTR 1020
Name:
Overview
In this activity, you will explore the behavior of two stars in a binary system and use the light
curve of the system and the velocities of the two stars to derive physical stellar properties.
Objectives
After completing this activity students will be able to:
• Determine graphically the observed contact time for primary eclipse of an eclipsing
binary.
• Determine the spectroscopic and photometric orbital elements of a double-line
spectroscopic eclipsing binary star.
• Calculate the masses and radii of the two stars in the binary system from the orbital
elements.
Definitions
• Eclipsing Binary – a system of
2 stars in which, from the
viewpoint of Earth, one star
crosses over and eclipses the
other star, resulting in a dip in
brightness during the eclipse
• Contact – (1st contact, 2nd
contact, etc.); times in which
one star in an eclipsing binary
system is eclipsing the other. 1st contact refers to the beginning of an eclipse, 2
nd
contact refers to the beginning of total eclipse (one star is completely in front of
another), 3rd refers to the end of the total eclipse, and 4th refers to the end of the eclipse
• Orbital Period – (P); the amount of time an eclipsing binary system takes to complete 1
full orbit around each other
• Phase – 1 full orbit of an eclipsing binary system
• Double-Line Spectroscopic Binary – a binary system in which both stars exhibit
absorption lines which, over the course of the orbit, are redshifted to the blueshifted by
the relative motions of the stars
• Center of Mass – the point in space where two stars in a binary system orbit around
2
• Astronomical Unit – (AU) The average distance between the Earth and the Sun, equal to
1.5 x 1011 m.
• π – Equal to 3.14.
Part 1. Photometric Elements of the Totally Eclipsing Binary SS Bootis
Figure 1: An expanded view of preliminary eclipse for a hypothetical binary
Part A: Relative Luminosity
The brightness, or luminosity, of each star relative to its companion can be determined from
the light level along the bottom of the eclipse. From the spectroscopic data it has been
determined that the cooler star is a K-type subgiant and the hotter star is a G-type mainsequence star. Since the primary eclipse is the eclipse of the hotter star by the cooler star, and
since the cooler star is the larger, it can be seen from Figure 1 that all the light observed during
totality must originate only from the cooler star.
Therefore, the relative luminosity of the cooler star (Lc) in the SS Boo system can be directly
determined by extending the eclipse bottom in Figure 2 toward the left until it intersects the yaxis. Read off the light level of this intersection to determine the value of Lc. Record your
answer below.
- Lc =
3
Figure 2: Preliminary eclipses of SS Boo
Recall that the total light of the system, outside eclipse when both stars are fully visible, was
defined to be equal to 1. Therefore, the relative luminosity of the hotter star (Lh) must be given
by Equation 1:
Use
your value of Lc and equation 1 to calculate the value of Lh. Record your answer below. - Lh =
?ℎ = 1 − ?? (???????? 1) (1)
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Part B: Relative Stellar Radii
The location of the first, second, third, and fourth contacts are labeled in figure 1. Using these,
locate the first, second, and third contacts of the SS Boo eclipse shown in figure 2. From each
of these three contacts, drop a vertical line to the x-axis and read off the time each contact
occurred. Write your answer below. - J.D.(hel.) of 1st contact = t1 =
- J.D.(hel.) of 2nd contact = t2 =
- J.D.(hel.) of 3rd contact = t3 =
These contact times can be used to determine the radii of the two stars relative to the orbit’s
circumference. From Figure 1 it can be seen that during the time interval between the first and
second contact the smaller star has moved a distance equal to its own diameter. Similarly,
during the time interval between first and third contacts, the smaller star has moved a distance
equal to the diameter of the larger star. The ratio of these time intervals to the orbital period is
the same as the ratio of each star’s diameter to the circumference of its orbit. Therefore, the
relative radii of the cooler star, rc, and the hotter star, rh, can be calculated from Equations 2
and 3:
?? =
?(?3 − ?1)
?
(???????? 2)
?ℎ =
?(?2 − ?1)
?
(???????? 3)
where P = orbital period = 7.60614 days, t1 is the time of first contact, t2 is the time of second
contact, t3 is the time of third contact, and π = 3.14. Using the orbital period and your times of
contact, use Equations 2 & 3 to calculate the relative radii of the two stars in the SS Boo
system. - rc =
- rh =
5
Part 2. Spectroscopic Elements for SS Bootis
Figure 3: Spectroscopic data for a hypothetical eclipsing binary system
SS Boo is also a double-line spectroscopic binary as shown in figure 3. From 1934-1935, Roscoe
Sanford observed SS Boo spectroscopically at Mount Wilson Observatory. His data and a radial
velocity curve for SS Boo are shown in figure 4. The scatter in these data is caused by the
faintness of the star. The y-axis is the radial velocity of the system in km/s, and the x-axis is the
orbital phase, or the fractional orbital position. These phases begin with 0 at mid-primary
eclipse. Thus ¼ of the way around the orbit corresponds to 0.25 phase units, and ¾ of the way
around corresponds to 0.75 phase units.
Part A: Center of Mass Velocity
The radial velocity of the center of mass for SS Boo can be determined by drawing a line which
connects the intersections of the two curves in figure 4. Determine the γ-velocity for SS Boo
and record you answer below
- γ = km/s
Part B: Mass Ratio
The mass ratio of the two stars in the SS Boo system can be determined by measuring the semiamplitudes of the radial velocity curves in Figure 4. At phase 0.25 and 0.75, draw a vertical line from
the top curve to the bottom curve. Measure in units of km/s from the γ-line to the top of each curve at
phases 0.25 and 0.75, and record these lengths as αc and αh
- αc = km/s
- αh = km/s
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Figure 4: Radial velocity curve for SS Boo. Filled circles represent cooler star and open circles
represent hotter star
Repeat the above procedure, except this time measure down to the bottom of the curves and
record as βc and βh - βc = km/s
- βh = km/s
Image credit:
Eliot Halley Vrijmoet
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These measurements give us a method of determining the mass ratio of the two stars, but not
their actual masses. The mass ratio is given by Equation 4:
???? ????? =
?ℎ
??
?? + ??
?ℎ + ?ℎ
(???????? 4)
where Mh and Mc are the masses of the hotter and colder star, respectively, and the α and β
values are your answers to questions 9-12. Record your mass ratio below:
- Mass ratio =
Part 3. Dimensions of the SS Bootis System
By combining both the spectroscopic velocities with the photometric properties, the actual
dimensions of an eclipsing binary such as SS Boo can be calculated. Thus, it will be possible to
calculate the absolute masses, in solar mass units, for each star in the system, and to calculate
the absolute radius of each star in solar radius units.
Part A: Stellar Masses
Since SS Boo is a totally eclipsing binary, it can be assumed that the orbital inclination is 90°
(edge-on). Thus, the separation between the two stars can be calculated with Equation 5:
? =
(0.211?)(?/365.25)
2?
(???????? 5)
where a is the separation of the stars, V is the maximum relative velocity of the two stars, and
P is the orbital period in days (same as before, P = 7.60614 days). The factor of 0.211 converts
the velocity from units of km/s to units of AUs/year. It should be obvious that the factor 365.25
converts the orbital period from units of days into years. Therefore, a will be in AUs. Use
Equation 6 to combine previously determined values of αh (Question 10) and βc to solve for V:
? = ?ℎ + ?? (???????? 6) - Total relative velocity = V = km/s
Use this value of V and P = 7.60614 days to calculate a using Equation 5. Record your answers
for a and V below.
8 - Separation = a = AUs
The sum of the stellar masses can now be calculated from the values of a and P using Kepler’s Third
Law, shown in Equation 7, which states
?ℎ + ?? =
?
3
(?/365.25)
2
(???????? 7)
Record your answer for the sum of the masses below. Your mass units will be in Solar masses
(M☉). - Mh + Mc = M☉
Use the equations for the sum of the masses and the mass ratio, along with your answers
for each, to estimate the individual masses of each star in the SS Boo system. Record your
answers below - Mh = M☉
- Mc = M☉
Part B: Stellar Radii
We can now calculate the absolute radius for each star in terms of the Sun’s radius with
Equations 8 and 9:
?? = ???(214.95) (???????? 8)
?ℎ = ?ℎ?(214.95) (???????? 9)
where rc and rh are your answers to questions 6 & 7, and a is your answer to question 15. The
factor of 214.95 is the number of solar radii (R☉) in one AU and converts the radii obtained
from AU to solar units. Calculate the value for Rc and Rh for SS Boo, and record your answers
below: - Rc = R☉
- Rh = R☉
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Stop and think about your answers. Do they make sense? Recall that one star is a K-type
subgiant and the other star is a G-type main-sequence star. The above procedure is one of the
few methods by which astronomers can obtain information about the masses and sizes of the
stars. Thus, it is very important to modern astronomy.
To complete this assignment for grading:
• Save this file: File -> Save As… -> Rename the file ‘YourLastName – Binaries’
• Upload your file to ‘Lab 4 – Binary Stars assignment in iCollege (click Add
Attachments -> Upload -> upload renamed saved file -> Update).
• Complete the Reflection activity on iCollege
• Have a beverage of your choice in celebration!