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NOTE: This page is a continuation of the notes and worksheets for topic 9.7 Astrophysics. Four separate pages were used for this topic because of the large volume of material in the topic. This will keep download time within acceptable limits.
9.7 OPTION - ASTROPHYSICS CONTINUED PAGE 3 PREPARED NOTES
BINARY
STARS
NOTE:
Numbers appearing in parentheses at the end of sentences or paragraphs refer to
the references provided in the Bibliography at the end of these notes. The only direct way to determine
the mass of a star is by studying its gravitational interaction
with other objects. Binary
stars, two stars in orbit around a common centre of mass, provide an ideal
opportunity to do just that. At
least half of all the “stars” in the sky are actually multiple star systems.
Analysis of the orbital parameters of such systems provides vital
information about a variety of stellar characteristics, including mass.
(1) The methods used to analyse the
orbital data depend on the geometry of the system, its distance from the
observer and the relative masses and luminosities of each component.
As a consequence, binary star systems are classified according to the
means by which they are detected. (1)
When astronomers can actually
see the two stars orbiting each other, the binary is called a visual binary.
The brighter star in the pair is called the primary star and is denoted
by the letter A after its name, while the other star is called the secondary and
is denoted by the letter B. For
example, the alpha star in the constellation of the Southern Cross is actually a
visual binary consisting of Alpha Crucis A and Alpha Crucis B, both of which are
easily discernable through a small telescope. Many years of careful observation are often necessary to ensure that the two stars are actually in orbit around each other. If the stars do form a binary system, each star will follow an elliptical orbit around the centre of mass of the system. The star with the larger mass will stay closer to the centre of mass of the system and will therefore have the smaller orbit. This makes sense – just think of two children on a seesaw. To balance, the child with the greater mass must sit closer to the fulcrum.
Some binary systems are
detected due to a periodic shift in spectral lines.
These are called spectroscopic binaries and are detected using the
Doppler Effect. If two stars in
a binary system have some component of their orbital motion along the
observer’s line of sight, then as the stars move around their orbits, they
periodically approach and recede from the observer.
Thus, the spectral lines of the two stars are alternately blueshifted and
then redshifted. (1 & 3) Through a telescope such a star
system may appear to be one single star. However,
an examination of the spectra coming from “the star” will reveal two sets of
spectral lines, one set from each star, each of which will display Doppler
shifts. The two overlapping spectra
shift relative to each other due to the Doppler Effect.
When the spectrum of each star in the binary system is visible, the
binaries are called double-line spectroscopic binaries.
If, however, one star in the pair is much more luminous than the other,
the spectrum of the less luminous companion may be overwhelmed and only a single
set of spectral lines will be seen. In
this case, the spectral lines shift back and forth due to the Doppler Effect and
therefore still reveal the binary system. This
type of binary is called a single-line spectroscopic binary.
(1 & 3) Consider the diagram below, which I have adapted from Reference 1 p.204.
The diagram above shows the
periodic shift in spectral features of a double-line spectroscopic binary.
For simplicity, only one spectral line from each star is shown.
Note that l0
is the non-Doppler shifted wavelength of the lines and wavelength increases to
the right of that position. The
relative wavelengths of the spectra of stars 1 and 2 are shown at four different
phases during orbit: (a) star 1 is moving towards Earth while star 2 is moving
away; (b) both stars have velocities perpendicular to the line of sight; (c)
star 1 is receding from Earth and star 2 is approaching Earth; and (d) again
both stars have velocities perpendicular to the line of sight of the observer on
Earth. Note that the spectroscopic detection of binaries is most likely if the period of the motion is short and the orbital speeds of the stars are high. So, most spectroscopic binaries are close binary systems. (3)
Some binaries have their orbital
planes oriented approximately along the line of sight of the observer.
In this case, one star may periodically pass in front of the other,
blocking light from the eclipsed star. Such
a system is called an eclipsing binary and is recognized by regular variations
in the amount of light received at the telescope (1).
Light curves (plots of apparent visual magnitude versus time or
orbital phase) can be recorded for eclipsing binaries using an appropriate
photometer. Such curves can reveal
enormous amounts of information about the binary system.
For example, light curves can confirm the presence of two stars.
The ratio of surface temperatures can be determined from how much their
combined light is diminished during eclipse.
The duration of the eclipse provides information on the relative radii of
the stars and their orbits. (3) Further to this, if the
eclipsing binary also happens to be a double-line spectroscopic binary, an
astronomer can calculate the actual mass and radius of each star in the system
from the light curves and the velocity curves. (3) Examine the light curve shown below. It is typical of a binary system consisting of small hot star and a large cool star with the orbital plane edge-on to our line of sight. Note that the smaller star is travelling from left to right across the front of the larger star. This diagram was adapted from Reference 12 p.323.
Note:
Try out this java applet, which allows you to model the light curves of
eclipsing binaries using a computer simulation (Syllabus point 9.7.5, column 3,
dot point 1). http://instruct1.cit.cornell.edu/courses/astro101/java/eclipse/eclipse.htm#example Sometimes one member of a binary is significantly brighter than the other and may be the only one of the pair visible through a telescope. In such a case the existence of the unseen member may be deduced by observing the oscillatory motion of the visible star (1). This reveals itself as a detectable “wobble” in the star’s proper motion (5). Since Newton’s First Law requires that a constant velocity be maintained by a mass unless an external unbalanced force is acting upon it, such oscillatory motion requires that another mass be present. Such a binary star system is called an astrometric binary. (1) DETERMINATION
OF STELLAR MASSES
As mentioned previously binary stars are very important in determining the masses of stars. Binary stars orbit each other in accordance with Kepler’s Third Law – the Law of Periods. For any binary system we can write:
where m1 = mass of star 1, m2
= mass of star 2, r
= average distance of separation of the two stars, T = orbital period of the binary system and G
= the Universal Gravitational Constant = 6.673 x 10-11 SI. This equation is known as Newton’s form of Kepler’s Third Law. Clearly, for any binary system if we can determine the orbital period and the average distance of separation of the stars in the system, we can determine the total mass of the system (m1 + m2). For visual binaries we can
usually determine the orbital period, although in some cases observations
over more than one lifetime may be needed.
If the distance from Earth to the binary can be calculated from
parallax measurements or using spectroscopic parallax, then the angular
separation between the stars in the binary can be translated into the physical
distance between the stars. Astronomers
must also take into account the angle of tilt of the binary’s orbit to our
line of sight. Substituting into
Kepler’s Third Law equation then allows us to calculate the total mass of
the binary system. (1 &
3) By plotting the individual
orbits of the two stars in the system, the centre of mass of the
system can be determined. Then,
comparing the relative sizes of the two orbits around the centre of mass enables
the ratio of the masses m1 / m2 of the two stars
to be determined. Having both the
sum of the masses and the ratio of the masses then allows the calculation of the
individual masses of the two stars to be achieved.
(1 & 3) See Ref. 1
pp.205-208 for a detailed analysis. For spectroscopic binaries
the wavelength shift of each star’s spectral lines is measured and then
this data is used to calculate the radial velocity of each star from the
relevant Doppler shift formula. Remember
that radial velocity is velocity along our line of sight.
This radial velocity data is then used to plot a radial velocity curve
(radial velocity versus time) for each star.
The ratio of the masses of the stars in the binary is obtained
from this curve, along with the orbital period of the binary. Kepler’s Third Law and Newtonian
mechanics are then used to relate the sum of the masses of the stars to
the orbital period of the binary, the orbital speeds of the stars and the angle
of tilt of the binary’s orbit to our line of sight. If all of these details are known, then again the individual
masses of the two stars can be determined.
Often, however, the angle of tilt is not known accurately and therefore
stellar mass determinations from spectroscopic binaries are often uncertain.
(1 & 3) See Ref. 1
pp.208-211 for a detailed analysis. There is one case in which the
angle of tilt is known and that is when the spectroscopic binary is also
an eclipsing binary. The
orbital plane of the binary must then lie edge-on to our line of sight.
Clearly, with this knowledge and with values for the orbital period of
the binary and the orbital speeds of the component stars, the sum of the
masses of the stars can be determined.
Then, since both the sum of the masses and the ratio of the masses are
known, the individual masses of the two stars can be determined.
(1 & 3) See Ref. 1
pp.211-214 for a detailed analysis. Note that in the case of astrometric binaries, it is usually possible to measure the orbital period of the binary and the average distance of separation of the two stars. Thus, using Newton’s form of Kepler’s Third Law we can get an estimate of the total mass of the binary system. It is usually not possible to determine the masses of the individual stars.
Example
Questions
1.
An astrometric binary is observed to have a period of 44.5 years and an
orbit with an average distance of separation between the component stars of 100
AU. Determine the sum of the masses
of the stars in this binary. (Data:
1AU = 1.5 x 1011 m) (Answer:
1.012 x 1033 kg) 2. An eclipsing binary has a period of 44.5 years and an average distance of separation between the component stars of 3.9 AU. Determine the combined mass of the binary system. (Answer: 6.0 x 1028 kg)
CLASSIFICATION
OF VARIABLE STARS
A variable star is one that varies in brightness (5). As at January 2004, the total number of designated variable stars was 38622 according to the “General Catalogue of Variable Stars”, 4th Edition, by P N Kholopov et al, Moscow (1985) and its published updates (also available on the Internet). There are two broad categories of variable star: (a) Extrinsic Variables, which vary in brightness for some reason external to the star; and (b) Intrinsic Variables, which vary in brightness due to changes in the star itself. Certain stars may vary in brightness due to both of these reasons. (5) EXTRINSIC
VARIABLES:
As mentioned above, Extrinsic
Variables vary in brightness for some reason external to the star.
Such reasons include rotation, orbital motion or obscuration.
Examples of common types of extrinsic variables include: (a)
Rotating variables, in which either the star’s ellipsoidal shape
or the presence of large cooler or hotter areas on the star’s surface change
the star’s brightness as it rotates. BY Draconis stars are examples of the latter. (b) Eclipsing Binaries, which as discussed previously, consist of orbiting stars that periodically eclipse each other. Algol (Beta Persei) is an example of an eclipsing binary system about 100 light years from Earth. INTRINSIC
VARIABLES:
Intrinsic Variables vary in brightness due to actual changes in the luminosity of the star itself not due to external processes such as rotation or eclipses. The majority of variable stars fall into this broad category. Intrinsic variables are further sub-divided into non-periodic or periodic variables.
NON-PERIODIC
VARIABLES:
These are intrinsic variables that
show irregular variations in brightness. Examples of such variables include: (a)
Novae – A nova is a star that undergoes a sudden, unpredictable
increase in brightness typically of 11 to 12 orders of magnitude (5).
The sudden increase in brightness is accompanied by an explosive loss of
a relatively small amount (about 10-5) of the mass of the star.
The outer atmosphere is blown away to form an expanding shell of gas
around the star that can often still be seen many years after the explosion.
Nova Cygni, which flared up in 1975 by about 19 magnitudes is an
example. (10) (b)
Supernovae – A supernova is a violently exploding star, which
may become over a billion times brighter than the Sun, and for many weeks may
outshine the entire galaxy in which it lies (5). The explosion completely destroys the star and throws most of
its matter into space at high speeds, leaving behind either a tiny, very dense,
collapsed remnant called a neutron star or in some cases a black hole.
SN1987A is a supernova observed in the Large Magellanic Cloud in
1987. The Crab Nebula is the
remnant of a supernova explosion in the constellation Taurus in 1054. (10) (c)
R Coronae Borealis Stars – These stars are sometimes referred to
as “reverse novae” since they decrease in brightness by as much as 10
magnitudes before returning to normal. They are supergiants with carbon-rich atmospheres.
The sudden minima are due to the accumulation of clouds of carbon dust,
which are then blown away and allow the star to return to its normal brightness.
The decreased brightness phase may last from months to years. (5 &
10) (d)
Flare Stars – These stars are red dwarfs that exhibit intense
outbursts of energy (flares) from small areas of their surface.
Examples include UV Ceti stars and certain BY Draconis stars. (5) (e) T-Tauri Stars – These are very young protostars. They are less than 10 million years old, with masses similar to or less than that of the Sun and with diameters several times that of the Sun. They are still contracting. They exhibit irregular variability ranging from ultraviolet flares on a time scale of minutes to optical variations on time scales of days to years. The prototype, T Tauri itself, lies within Hind’s Variable Nebula and varies irregularly between 8th and 13th magnitudes. (5 & 3) PERIODIC
VARIABLES:
These are intrinsic variables that
show regular variations in brightness.
These variables are also known as pulsating variables.
A pulsating variable undergoes a periodic cycle of contraction and
expansion as the star oscillates between a state where the inward acting
gravitational force is dominant and another where the outward acting radiation
pressure is dominant. The
periodic contraction and expansion produces the observed periodic variation in
brightness. As the star
contracts, the increase in luminosity caused by the temperature increase exceeds
the decrease caused by the contraction in radius, so the star becomes brighter.
As the star expands, the luminosity and therefore the brightness
decrease. (9 & 10). The types of periodic variables
can be categorized by the period and amplitude of their light curves.
The period is simply the amount of time between one occurrence of
maximum (or minimum) brightness and the next.
The amplitude is the difference in magnitude between maximum and
minimum brightness. Examples of
periodic variables include: (a)
Cepheid Variables – These are the best-known examples of
pulsating variables and are named after Delta Cephei, the first star of
this type to be discovered. Cepheids
are very luminous yellow giant or supergiant stars. Their brightness usually varies by about one magnitude with a
period of between 1 and 135 days, while their radius typically varies by 10 to
20 percent. We will say a little
more about these very important stars later.
They are extremely useful as a means of distance measurement.
(5 & 10) (b)
Mira Stars (also called Red Variables) – These are the most
common pulsating stars. They are
long period pulsating red giants and supergiants. They take their name from the star Mira (Omicron Ceti)
which lies about 250 ly from Earth. Mira
stars have periods of about 80 to 1000 days and have an amplitude of 2.0 to 10
magnitudes. (5 & 10) (c)
RR Lyrae – These are the second most common type of variable.
They are old yellow giants. Most
have periods between 0.2 and 1.2 days and amplitudes of 0.2 to 2.0 magnitudes. One of the two types of RR Lyrae star (the RRAB type) all
have approximately the same absolute magnitude of +0.5, making them valuable
distance indicators. (5 & 10) (d)
RV Tauri – These are highly luminous, yellow supergiants.
The light curves have overall amplitudes of 3 to 4 magnitudes.
The period lies in the range from 30 to 150 days. (5) NOTE:
The above classification of variable stars is
the one required by the current HSC Astrophysics Syllabus.
This classification scheme is based on the form of the light curve for
the star, its amplitude and periodicity or lack of it.
This classification scheme has been superseded. I
do not know why the Syllabus Committee decided to go back to the old system.
The new classification scheme adopted by the “General Catalogue of
Variable Stars”, 4th Edition, is based on the physical mechanisms
that underlie the different forms of variation or the physical structure of the
stars themselves. This new
classification scheme allocates variable stars to one of seven classes:
Eruptive, Pulsating, Rotating, Cataclysmic (explosive nova-like), Close Binary
Eclipsing Systems, Optically Variable X-Ray Sources and finally Unique Variables
(for mistakes and variables no-one has figured out yet).
SUMMARY OF CLASSIFICATION OF VARIABLE STARS:When trying to remember the classification scheme for variable stars, a simple flow chart like the following can be of great assistance. This particular flowchart appears in Ref.12 on p.326. A similar one is shown in Ref.9 on p.46. |
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