Dynamical Parallax Examples

Quote of the Day

Be the change that you wish to see in the world.

— Mahatma Gandhi


Introduction

Figure 1: Book Cover of Parallax.

Figure 1: Book Cover of Parallax (Source).

Years ago, I read the book Parallax (Figure 1) and really enjoyed the tale of how 19th century astronomers measured the distance to the nearest stars. This measurement was critical to providing scientists some idea as to the scale of the universe.

The book Parallax describes how simple trigonometry, along with the introduction of  large telescopes coupled to precision measurement gear, could be used to determine the distance to a star by measuring the angular shift of that star as the Earth revolved around the Sun – a method called trigonometric or stellar parallax. During my recent perusing of the Wikipedia, I discovered that there was an alternative form of parallax measurement, called dynamical parallax, that allows one to estimate the distance to stars that are beyond the limits of trigonometric parallax.

In this post, I will verify my understanding of dynamical parallax by implementing the algorithm in Mathcad and applying it to nearby star systems for which we have accurate trigonometric data. I can then compare my dynamical estimates with the more accurate trigonometric measurements.

My Mathcad source and its PDF are here.

Background

Overview

Distant stars have an annual parallax shift that is so small that it cannot be accurately measured. However, the separation between binary stars, their individual brightness (i.e. magnitude), and orbital period frequently are measurable. We can use Kepler's 3rd law and a luminosity-based estimate of the total mass of the binary stars to compute the orbit's semi-major axis.  Given the size of the orbit, we can compute the distance to the star knowing the angle subtended by the orbit and our estimate of the orbit size.

This distance estimate can be be iteratively refined by comparing the measured angle width of the orbit to what our orbital radius and distance measurements would indicate.

The Wikipedia has a good description of the overall process, which I quote below.

With this technique, the masses of the two stars in a binary system are estimated, usually as the mass of the Sun. Then, using Kepler's laws of celestial mechanics, the distance between the stars is calculated. Once this distance is found, their distance from the observer can be found via the arc subtended in the sky, giving a preliminary distance measurement. From this measurement and the apparent magnitudes of both stars, the luminosities can be found, and from the mass–luminosity relationship, the masses of each star. These masses are used to re-calculate the separation distance, and the process is repeated. The process is iterated many times, and accuracies as high as 5% can be achieved.

Definitions

Stellar Parallax
Stellar parallax is parallax on an interstellar scale: the apparent shift of position of any nearby star (or other object) against the background of distant objects. Created by the different orbital positions of Earth, the extremely small observed shift is largest at time intervals of about six months, when Earth arrives at exactly opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units (AU) between observations. The parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit (Source).
Because of the exceeding small size of the angles being measured, accurate stellar parallax measurements are limited to distances of less than 30 parsecs using Earth-based telescopes and 300 parsecs using space-based telescopes. (Source)
Dynamical Parallax
In astronomy, the distance to a visual binary star may be estimated from the masses of its two components, the size of their orbit, and the period of their orbit about one another. A dynamical parallax is an (annual) parallax which is computed from such an estimated distance. (Source)
Dynamical parallax can be applied to stars that are closer than 500 parsecs to Earth. (Source)
Mass-Luminosity Relation
In astrophysics, the mass–luminosity relation is an equation giving the relationship between a star's mass and its luminosity (Source). I discussed a commonly used mass-luminosity relation in this post.

Analysis

Caveat

The following derivation will ignore the inclination of the binary star system. Figure 2 shows how orbits can be in any orientation relative to our line of sight. This nasty reality can be included in the model, but I first want to work the simple case of a binary system that perpendicular or "flat on" to our line of sight.

Figure 2: Image of Actual Solar Systems with Different Orientations.

Figure 2: Image of Actual Solar Systems with Different Orientations (JPL/NASA).

Algorithm

Figure 3 shows a flowchart for the dynamical parallax algorithm as described in the Wikipedia.

Figure M: Dynamical Parallax Algorithm.

Figure 3: Dynamical Parallax Algorithm.

Implementation

Algorithm Setup

Figure 4 shows the units and constants that I will use for evaluating my dynamical parallax implementation on some reference star systems.

Figure M: Algorithm Setup.

Figure 4: Algorithm Setup.

Dynamical Parallax Book Parallax Mass Luminosity Relation Alternate Binary Star Calculation Types of Binary Stars Finding the Orbit Inclination

Orbital Radius

Figure 5 shows how to use Kepler's 3rd law to determine the radius of an orbit assuming a total mass estimate and revolution period.

Figure M: Formula for Calculating the Radius of the Orbit.

Figure 5: Formula for Calculating the Radius of the Orbit.

Mass-Luminosity Relation

Luminosity-Mass Relationship

Figure 6 shows the Wikipedia's formula for the luminosity of star of a given mass. In fact, I actually need the inverse of the Wikipedia's function (i.e. mass given luminosity) – a function I call "f" in Figure 6. I computed the inverse using Mathcad's root function.

I also include a function that computes the luminosity of a star given the star's apparent magnitude.

Figure M: Mass-Luminosity Relation.

Figure 6: Mass-Luminosity Relation.

Dynamic Parallax Function

Figure 7 is a Mathcad function that implements my understanding of the dynamical parallax algorithm. I pass the data into the function as a vector – this allows me to apply the algorithm as a post-fix operator, which I often find convenient.

Figure M: Dynamical Parallax Algorithm.

Figure 7: Dynamical Parallax Algorithm.

Mass-Luminosity Relation

Test Cases

Figure 8 shows four stars that I chose binary star systems composed of stars that were about the size of the Sun. The mass-luminosity relationship is relatively accurate for stars of this size.

Figure M: Dynamical Parallax Test Cases.

Figure 8: Dynamical Parallax Test Cases.

Alpha Centauri Procyon 70 Ophiuchi Eta Cassiopeiae

Conclusion

I was able to use the dynamical parallax algorithm to determine the range to a number of nearby stars that have been analyzed using the more accurate trigonometric parallax approach. My results are reasonably close considering the accuracy of the luminosity-mass curve for stars.

The analysis shown here is just one way of determining the distance to a binary system using orbital information. You can also measure the orbital velocities of the component stars using the Doppler shift and applying a modified version of Kepler's 3rd law. That will be a subject for a later post.

 
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