Victoria Yin | Honors Physics | 2012
Did you know that more than 50% of all star systems are binary stars? Half of the stars we see are actually multiple stars, revolving around a common center of mass.

# Distance Calculation

The period-luminosity relationship can be used to determine the distance to a Cepheid.

1. Apparent magnitudes can be taken through photometric observation and plotted to create light curves such as the one below. (Source)
2. Using this data, the average apparent magnitude m and the period in days can be determined. Here the data has an average apparent magnitude of 15.56 and a period of 4.76 days.
3. The absolute magnitude M can be determined by inserting it into the period-luminosity plot. M = -3.6 (Source)
4. Plug into d = 10 (m - M + 5)/5
d = 10 (15.57 - (-3.6) + 5)/5
d = 10 24.17/5
d = 10 4.834
d = 68,230 parsecs
5. This particular Cepheid is 68.2 kpc away.

# Period-Luminosity Relation

A Cepheid is an intrinsically variable star with a period of 1-70 days. Light curves of Cepheids reveal that they undergo a rapid rise in brightness and then a more gradual decline.

The period-luminosity relation was discovered by Henrietta Leavitt in the 1900s. She studied the variable stars in the Large and Small Magellanic Clouds. The stars with longer periods were brighter than the stars with shorter periods. Since all the stars were at approximately the same relatively distance from us on earth, she determined that more luminous Cepheids pulsated more slowly.

(Source)

Type I Cepheids, or classical Cepheids are about four times more luminous than Type II Cepheids. They follow the following pattern: the longer the period, the more luminous it is.

(Source)

Type II Cepheids, or W Virginis Cepheids, are less luminous than Type I Cepheids. They are older than Type I Cepheids. The decline side of their light curves have a characteristic bump.

(Source)

# Variable Stars

A variable star is a star whose physical properties (usually brightness) change over time.

There are two types of variables. Intrinsic variables affect the brightness through some change within the star itself, while extrinsic variables affect the brightness through some process outside of the star itself.

General Overview: (Source)

Some Examples of Intrinsic Variables:

Pulsating variables periodically expand and contract their outermost layers.

Changes caused by eruptive variables happen rapidly due to violent outbursts from processes within the star and do not exhibit periodic behavior.

A supernova is the stage near the end of a star’s lifetime in which there is a large sudden rise in brightness. It can be caused by a thermonuclear explosion which releases a lot of radioactive and heavy elements into space and destroys the star in the process or a core collapse. A supernova can experience a rise in brightness of up to 20 magnitudes (in other words… A LOT.)

A nova experiences a sudden and unpredictable rise in brightness from 7-16 magnitudes. After an eruptive event of a few days, the star returns to its original brightness. A nova does not destroy the star like other eruptive variables.

Some Examples of Extrinsic Variables:

Eclipsing binaries, found here, are one major class of extrinsic stars.

The second class are rotating variables. A star may have starspots that vary across the surface. As the star rotates, the brightness varies.

# Mass Calculations in Binaries

The mass of a binary system can be calculated using the equation M = 4π2r3/GT2.
Remember to convert to S.I. units before solving!

Example 1: (Source)

The α Centauri system is 1.338 pc distant with a period of 79.92 years. The A and B components have a mean separation of 23.7 AU (although the orbits are highly elliptical). What is the total mass of the system?
T= 79.92 years needs to be converted to 2.522 x 10seconds.
R= 23.7 AU needs to be converted to 3.5 5x 1012 meters.

M = 4π2 (3.55 × 1012)3/ ((6.672 × 10-11) × (2.522 × 109)2)
M= 4.162 × 1030 kg

The mass of the system is 4.16 x 1030 kg.

# Eclipsing Binaries

Eclipsing Binaries

• The detection of eclipsing binaries depends on photometric measurement.
• The orbital plane of these systems lies on edge to us so that the component stars periodically eclipse each other.
• Light curves are used to identify eclipsing binaries.
• Plot of apparent magnitude over time/phase
• Periodic dips in brightness occur as one star eclipses the other.
• The primary eclipse has a greater drop in brightness and occurs when the hotter star is eclipsed.
• The secondary eclipse has a lower drop in brightness and occurs when the cooler star is eclipsed.
• (Source)
• An example of an eclipsing binary is Algol, Beta Perseus.

The Algol Paradox says that masses can sometimes be transferred from one star to another.

# Spectroscopic Binaries

Spectroscopic Binaries

• The majority of binary systems are spectroscopic.
• Can be detected by Doppler shifts in their spectral lines.
• The spectrum of a binary system is a combination of the spectra from all the component stars.
• When stars are orbiting, A may be moving towards us and B away from us.
• Spectrum A will therefore be blue-shifted (higher frequencies)
• Spectrum B will be red-shifted (lower frequencies)
• (Source)
• Spectral line shifts vs. time show radial velocities.
• An example of a spectroscopic binary is Ursa Majoris, discovered in 1889.

# Visual Binaries

Visual Binaries

• Are usually widely separated- distance measured in AUs (1 astronomical unit = 149597870.7 kilometers)
• Long term observations are made to plot relative positions. This data is then used to calculate orbits.
• The brighter of the component stars has the suffix A. The dimmer component star has the suffix B.
• An example of a visual binary is Alpha Centauri. It consists of two stars separated by approximately 23 AU. It orbits in a period of about 80 years.

(Source)

# Overview

What are binary stars?

• Binary stars are two or more stars that revolve around a common center of mass.

What types of binary stars are there?

• Visual binaries- individual stars in the system can be seen through a telescope
• Spectroscopic binaries- stars of the system appear as a single star even when seen through a telescope
• Eclipsing binaries- if one component star passes directly in front of the other, it blocks out its light

Kepler’s Laws apply to binary stars

• Stars orbit in ellipses around a common center of mass
• Each star sweeps out areas (radius vectors) twice its own width
• Stars must orbit at different speeds because these areas offer differ greatly
• The period can be calculated using this equation

(Source)

Exoplanets by the hundreds, detailed observations becoming possible, and now planets around binary star systems seem to be normal.  We are living in a golden age of planetary discovery where our planetary catalogs are growing beyond expectations. (via Kepler Discovery Establishes New Class of Planetary System)

Ancient Egyptians Tracked Eclipsing Binary Star Algol

“Aldebaran’s great, okay
Algol’s pretty neat,
Betelgeuse’s pretty girls
Will knock you off your feet”.

Douglas Adams, Hitchhiker’s Guide to the Galaxy

Turn your telescope to the constellation of Perseus and you might note an unusual star called Algol, dubbed the “Demon Star” or the “Raging One.” You wouldn’t notice anything much different at first, unless you happened to be looking during a window of a few hours — every 2.867 days — when Algol’s brightness visibly dims.

This unusual feature was first noticed back in 1667 by an astronomer named Geminiano Montanari, and later confirmed — with a proposed possible mechanism — in 1783 by John Goodricke, who precisely measured the period of variability: it dims every 2.867 days.

But a new paper by researchers at the University of Helsinki, Finland, claims that the ancient Egyptians may have recorded Algol’s periodic variability 3000 years ago, based on their statistical analysis of a bit of papyrus known as the Cairo Calendar.

This isn’t the first time people have hypothesized that Algol’s variable nature was known prior to its discovery in the 17th century. Certainly it was a familiar object, prominent in mythology and lore. In the second century, Ptolemy referred to Algol as the “Gorgon of Perseus,” and associated it with death by decapitation. (In Greek mythology, the hero Perseus slays the snake-headed Gorgon, Medusa, by chopping off her head.)

Other cultures also associated the star with violence and bad fortune. It’s no coincidence that H.P. Lovecraft marked the onset of his final battle in the 1919 short story, “Beyond the Wall of Sleep,” with the appearance of a nova near Algol.

But the Helsinki researchers go beyond mythology and conjecture and provide a solid statistical analysis, based on historical documentation.

Goodricke proposed that Algol’s periodic variability was due to an eclipsing factor: namely, an orbiting dark body occasionally passed in front of the star, dimming its brightness temporarily.

Alternatively, he suggested that Algol itself had a darker side that turned toward the Earth every 2.687 days.

His hypothesis wouldn’t be confirmed until 1881, when Edward Charles Pickering discovered that Algol is actually a binary star system: there were two stars circling together, Algol A and Algol B.

Even more intriguing: it was an “eclipsing binary,” i.e., one in which the dimmer star in the system occasionally passes in front of its brighter sibling, dimming the latter according to predictable periods. Goodricke’s hypothesis was correct.

Actually, astronomers now know that Algol is a triple-star system, with a third star, Algol C, located a bit further out from the main pair, with a larger orbit.

All of this is necessary background for understanding the conclusions of the Helsinki scientists. The whole point of tracking the heavens so meticulously, for the Egyptians, was to make predictions about the future, dividing the calendar into “lucky” and “unlucky” days. The Cairo Calendar, while badly damaged, nonetheless contains a complete list of such days over a full year, circa 1200 B.C.

How did the Egyptians decide how to rate specific days? That’s a mystery. But the Finnish team took the raw data and reassembled it into a tie series, then used statistical techniques to determine the cycles within it. There were two significant periodic cycles. One was 29.6 days, very close to current estimates of a lunar month (29.53059 days).

The second periodic cycle was 2.85 days. Lead author Lauri Jetsu and her colleagues argue that this corresponds to Argol’s variable period. It’s suspiciously close to the 2.867 period Goodricke measured back in 1783.

Close, yes, but it’s not a precise match, which is problematic. The Egyptians weren’t known to be sloppy in their astronomical calculations. They should have been able to pinpoint a value much closer to Goodricke’s — unless, say, Algol’s period changes over time.

There is some evidence that this might be the case, possibly due to the presence of the third star in the Algol system. Calculating the behavior of a two-body system is one thing; grappling with the dreaded “three-body problem” is quite another, particularly since astronomers are only working with roughly 300 years of data. Algol looks like it’s living up to its “Demon Star” moniker.

That’s where Jetsu et al’s paper might prove to be more than just an intriguing historical oddity. It provides some missing data from 3000 years ago, which could help astronomers further constrain their models for Algol’s variable behavior.

Image: Canes Venatici constellation from Urnahographia by Johannes Hevelius.