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9.7 Option – Astrophysics: 5. Binary and variable stars

Syllabus reference (October 2002 version)
5. The study of binary and variable stars reveals vital information about stars	Students learn to: describe binary stars in terms of the means of their detection: visual, eclipsing, spectroscopic and astrometric explain the importance of binary stars in determining stellar masses classify variable stars as either intrinsic or extrinsic and periodic or non-periodic explain the importance of the period-luminosity relationship for determining the distance of cepheids	Students: perform an investigation to model the light curves of eclipsing binaries using computer simulation solve problems and analyse information by applying:

Extract from Physics Stage 6 Syllabus (Amended October 2002). © Board of Studies, NSW.
[Edit: 2 July 09]

Prior learning:
Preliminary modules 8.2 The World Communicates (subsections 3, 4 and 5).
Preliminary module 8.5 The Cosmic Engine (subsections 1, 2,3 and 4).
HSC module 9.2 (subsection 2).

perform an investigation to model the light curves of eclipsing binaries using computer simulation

Some excellent simulations modelling the light curves of eclipsing binaries are available on the Internet - see the web sites below. You can search for others yourself – try “eclipsing binary stars” + simulation. Alternatively you may be able to use a simulation included with a CD-based astronomy software package.
Carefully follow any on-screen instructions and information. Try manipulating any of the parameters in the simulation, one at a time. Observe and record the effect of any change you make and write a summary of your investigation.
The simulation models an eclipsing binary by varying the intensity of the light or by drawing a graph of brightness. Try to write a concise explanation of why these effects occur and how the simulation models an eclipsing binary star.

Modelling binary light curves Australia Telescope National Facility, Outreach, CSIRO, Australia. Scroll down till you see the heading. There are two exercises you can do.

Dan’s Astronomy Software Dr Dan Bruton, Stephen F Austin University, Texas, USA. This site has software to download – one program simulates eclipsing binary stars and displays the light curves.

describe binary stars in terms of the means of their detection: visual, eclipsing, spectroscopic and astrometric

A binary star, sometimes called a double star, consists of two stars orbiting their common centre of mass. Most binary stars are so far away from earth and so close together that the human eye cannot resolve them as separate light sources. Many of the stars in the universe occur in such pairs, or systems of more than two, gravitationally linked to each other.
A visual binary star can be resolved as two stars in a suitably large telescope. Over an extended period of time the two stars can be observed to orbit a common centre of mass.
An eclipsing binary system cannot be resolved by an optical telescope, but can be detected by regular fluctuations in the light output of the system. This happens because the orbital plane of the system is edge on (or nearly) when viewed from earth and the stars in the system continually eclipse each other (pass in front of each other), resulting in the variable light output.
Close binary systems are often unresolvable in any telescope but their consequent rapid orbital motion means that, unless the orbital plane is perpendicular to our line of sight, the stars making up the system are alternatively approaching and receding from us. If we observe the spectrum of the system, we will see the wavelengths of the spectral lines of the system regularly shifting as a function of time due to the Doppler effect. Such stars are called spectroscopic binaries.
Astrometric binary systems are again only visible as a single star because one member is too faint to observe. The stars are of course still rotating around their common centre of mass. The visible star in the system is observed to wobble, indicating the presence of the unseen companion.

Welcome to the visual binaries Richard Dibon-Smith, University of Toronto, Canada. Herschel's discovery of visual binary stars.

solve problems and analyse information by applying:

In attempting to solve problems by applying this equation, first identify the quantity that you need to calculate, then rearrange the equation to make this quantity the subject, and substitute for other known variables. This formula enables the determination of total mass of a binary star system if the period of rotation T and separation of the stars r are known. If one of the masses, say m₁, is known or if the distance of one star from the centre of mass is known, then the individual masses can also be calculated.

Note that in using this equation it may be necessary to convert some quantities to SI units:
- m₁ and m₂ are the masses of each of the binary stars. (M is often used to represent the combined mass of the two stars). Mass is given in kilograms.
- G is the Universal Gravitational Constant 6.67 x 10^-11 N m² kg^-2
- r is the distance between the two stars. It can be found by adding the radius of rotation for each of the stars around the centre of mass. Radius is given in metres.
- T is the period of orbit in seconds.
Analyse information by observing the relationship between any two variables when other variables are held constant. For instance, consider how the period of motion must vary with the combined masses of the components if the distance between them is held constant. Predict the effect of increasing the mass of one star on either the period or the average separation.

Sample problem 1

An imaginary binary star system has a parallax of 0.8” and the two stars have a separation of 0.01” and a period of rotation of 100 years. Find the mass of the system.

Solution:

The distance to the star system is (1/0.8) pc = km (parallax formula). From trigonometry their separation is m. The mass of the system is then given by

Sample problem 2

Observations have shown that star A orbits a larger star B with a period of 20 earth years. If the average distance of separation between these two stars is 5 x 10¹⁰ m. Determine the mass of this binary system.

Solution:

Period must be converted to seconds

20 years = 20 x 365.25 x 24 x 60 x 60 = 6.31 x 10⁸ seconds

Then the values are substituted into the given formula

explain the importance of binary stars in determining stellar masses

Astronomers need to know the mass of a star to be able to understand the processes that give a star its energy at different stages of its evolution.
An important significance of binary star systems is that they provide astronomers with the only means of calculating the masses of stars. Fortunately binary star systems are common and provide a simple measurement of the masses of stars.
To measure the mass of an object one needs to observe its gravitational interaction with some other body. As a binary star system is held together by gravity, that is, gravity provides the centripetal force, the orbits of the component stars can be analysed by Newtonian mechanics. The orbits must also obey Kepler’s third law. The radius of each orbit and the period of motion can be measured directly and thus the relationship between gravitational force, centripetal force and Kepler’s Law can be mathematically analysed to determine the combined mass of the two stars.

Determining stellar masses Davison E. Soper, University of Oregon., Oregon, USA. Relates the theory of Kepler’s and Newton’s laws to determination of mass with reference to different types of binaries.

classify variable stars as either intrinsic or extrinsic and periodic or non-periodic

Variable stars are stars whose brightness, colour or some other property varies with time. It is the change in brightness which is most often considered. Variable stars are classified according to both the nature of the cause of the variation and the way the property varies with time.
Intrinsic variable stars change their brightness because of changes in processes that go on inside the star. Stars in this group vary in brightness as they expand and contract, heat and cool, for example, supernovae, novae, and pulsating stars.

the brightness changes as one star passes in front of the other.
Periodic variable stars are those whose brightness varies in a regular, repeated way as a function of time. Intrinsic variables that are periodic include the pulsating stars such as Cepheid and RR Lyrae stars. Extrinsic variables would generally be expected to be periodic.
The brightness of non-periodic or irregular variable stars varies irregularly with time. Non-periodic variable stars include novae and supernovae, the eruptive variables.

Physics Tutorial Notes, Astrophysics Caresa Education Services, NSW . Scroll down until you come to 975.

Fact Monster: Intrinsic Variable Stars Columbia University Press. Gives details on intrinsic variable stars.

Fact Monster: Extrinsic Variable Stars Columbia University Press. Gives details on extrinsic variable stars.

explain the importance of the period-luminosity relationship for determining the distance of cepheids

Cepheid variable stars are periodic intrinsic variables with a characteristic light curve. The change in luminosity is related to the change in surface temperature as the outer layers of the star expand and contract.
The significant feature of Cepheids that makes them so useful is that the period of their brightness variation is directly related to their average luminosity (the period-luminosity relationship). The period varies for individual Cepheids from 2 to 60 days, with longer-period Cepheids being more luminous than those with shorter period.
This means that when we measure the period of variation of a Cepheid variable we can know its luminosity immediately. Comparing luminosity with the star’s apparent brightness, we can then calculate the distance to the star, either by using the inverse square law directly or by applying absolute and apparent magnitude in the distance modulus equation.
Cepheid variables are observed in our own galaxy and in other galaxies well beyond the limits of the usual stellar distance measuring techniques (e.g. parallax). If clusters and galaxies have Cepheids in them, the distance to them can be calculated accurately. Cepheid stars have been instrumental in determining one of the most fundamental observations in astronomy: the cosmic expansion of the universe.