Obtaining Parameters of Double-Lined Spectroscopic Binary NSVS01031772

Introduction

Accurate descriptions of low mass stars are difficult to obtain and the most accurate data comes from eclipsing binary stars. Measuring these systems can allow us to calculate the radii, masses, and separation of the stars in the system. This lab focuses on double-lined spectroscopic binary NSVS01031772. 

The blue star is brighter than the yellow star in the binary system represented in Figure 1. Figure 1b is the secondary eclipse and Figure 1c is the primary eclipse. The primary eclipse causes a greater decrease in brightness, represented by the larger peak in the light wave curve and the secondary eclipse represents the smaller peak in the light wave curve (Figure 1d). Observing these stars through the Clay telescope and using data from the paper rvcurve.pdf (harvard.edu) will allow us to determine the mass, separation, radius, and period of these stars.

                                                                        Figure 1a

Figure 1b

Figure 1c

                                                                           Figure 1d

Eclipsing Binary Simulator (unl.edu)

Methods and Theory

Observations: we observed from about 7:30pm to about 10:30pm with clear skies the entire time.

• Point telescope at our binary system of interest located at RA, DEC = 13:45:35 +79:23:48
• The R-band filter was used which only allows orange/red light through (Figure 2).
• 
  
  
  
  
  Astrodon.jpg (360×300) (harvard.edu)
  
  
• A short exposure was taken to confirm image quality
• These reference stars are used
• Exposure time was set at about 120 seconds
• 65 photos were taken such as the one shown below.
• 
• AstroImageJ image software was used to calibrate the images and perform aperture photometry.
• Calibrate the files by flat fielding all your images to remove all the dust defects etc. 
• A light curve was generated

Analysis

These two stars are traveling on an ellipse. According to Kepler’s first law, the ellipse of each star has a focus at the center of mass of the system.

$a_1$ = semi-major axis of star 1

$a_2$ = semi major axis of star 2

The two stars have the same period (P)

According to Kepler’s third law, The two orbits have the same period P . Kepler’s Third Law states:

G = $6.67 \times 10 ́^{-8} cm^3  g^{-1}  s^{- 2}$
For simplicity, we can assume that the orbits are circular.

Therefore, star 1 has an orbit with radius $a_1$, mass $M_1$, and speed $v_1$ star 2 has an orbit with radius $a_2$, mass $M_2$, and speed $v_2$

The stars have orbits of the same period so 

From the data, we can create a light curve and a radial velocity curve.

We will assume that the plane of the stars’ orbit is in the plane of our line of sight.

The light curve measures the flux of the two star system over a period of time. An eclipse occurs when one star passes in front of the other. We can determine the period of orbit by measuring this change influx because we know that the flux of the system will decrease when one star moves in front of the other. This will also allow us to measure how long it takes an eclipse to occur.

 

The radial velocity curve measures the velocity of the star along the line of sight from out telescope. When the light is coming from a source that is moving towards us, the waves will be compressed and when the source is moving away from us, the waves will be elongated. Measuring this spectrum of light over lime will allow us to determine the radial velocity of the stars over the course of an orbit.

rvcurve.pdf (harvard.edu)

These graphs and our observations at the telescope will allow us to solve for several parameters including mass, radius, and period.

Mass

The masses of the stars are related to the stars' semi-major axes. To solve our equations we can imagine that the orbits are on an xy plane with the origin at the center of mass ($x_{com}$) with each star being one semi-major axis away from the origin in opposite directions ($x_1$ and $x_2$)

When we assume the orbits are circular, we get the ratio of the masses calculated from their orbital speeds:

We can then use Kepler's third law:

This allows us to solve for $M_1$ and $M_2$

Radius

Stellar Radii are determined using the radial velocities. Star 1 crosses in front of star 2 during an eclipse. The transit time $t_{tr1}$ is the time that it takes for star 1 to cross the diameter of star 2, which is a distance of $d_{tr1}=  2R^2$. Assume that star 1 is traveling at speed v1 and star 2 is traveling at speed v2 in the opposite direction. The eclipse velocity is the relative speed of star 1 and star 2. The transit time $t_{tr2}$ is the time that it takes for star 2 to cross the diameter of star 1.

Velocity is related to distance/time

Period

We can use the reduced mass $\mu$ to simplify the system to one object with a reduced mass of $\mu$

The new semi major axis of the condensed mass object is a = $a_1 + a_2$

Compiling data from all labs, we got the light curve graphs for multiple primary and secondary eclipses: 

Looking at the change in flux, we determined that graphs 0 and 3 are primary eclipses and 1, 2, and 4 are secondary eclipses.

In python we could use these together to find the period which, according to the python notebook came out to be 2.38 hours.

According to the paper by Lopez-Morales et al.,

$M_1= 0.5428 ± 0.0027M⊙, M_2= 0.4982 ± 0.0025M⊙, and radii R_1= 0.5260 ± 0.0028R⊙, R_2= 0.5088 ± 0.0030R⊙ P= 0.3681414$ days

Uncertainties

We had to assume that the orbits are circular and on edge.

Acknowledgements

I would like to thank Justina for teaching our lab section, and I would like to thank the weather for finally cooperating.

List of people worked with

I worked with Justina and the other members of lab class on 04/09/2023

List of resources consulted or used 

Python CoLab notebook: Copy of ASTRON16_LAB2.ipynb - Colaboratory (google.com)

Lopez-Morales, Mercedes & Orosz, Jerome & Shaw, J. & Havelka, Lauren & Arevalo, Maria & McIntyre, Travis. (2006). NSVS01031772: A New 0.50+0.54 Msun Detached Eclipsing Binary. 

rvcurve.pdf (harvard.edu)

Astrodon.jpg (360×300) (harvard.edu)

 reference stars 

AstroImageJ

Eclipsing Binary Simulator (unl.edu)