Evolution of Binary Stars
The Algol Paradox
Algol is a trinary system consisting of stars: Algol A, a B8 V star of mass 3.59 Mo, Algol B, a K0 IV of mass 0.79 Moand Algol C, a A5 V star of mass1.67 Mo. Algol A and B are in a tight orbit and are separated by 0.062 AU and together they orbit Algol C which is 2.69 AU away. Figure 10.19 shows a computer model of Algol A and B orbiting each other.
The fact that Algol is a multiple star system is not a paradox. However, consider the following example:
Example 10.7 Estimate the main-sequence lifetimes for the 3 members in the Algol system.
Solution: Use the main-sequence lifetime law () or use ageCalc to find the ages. They are:
Where's the paradox? You may have spotted it. Algol B is the most highly evolved of the system but it has already moved off the main sequence. Yet, a star of mass 0.79 Mo should take more than 70 times the age of a B8 star to do this. If we make the reasonable assumption that all three stars formed at the same time then something is odd in this system.
The resolution of the paradox is that Algol B was not always the smaller of the two stars! In close binary systems, the stars are in a constant "tug-of-war" and if the stars get too close together mass from one star can be pulled away and deposited on the other star. This process is called accretion. The 19th century French astronomer Edouard Roche worked out the complex mathematics of how this "tug-of-war" works. The applet provided below shows the region around the stars surrounded by a "dumb-bell" shaped loop. Each half of the "dumb-bell" is called a Roche Lobe and it represents both the maximum size a star can have before it must begin to accrete onto the other. The Roche Lobes also reflect the way in which the gravitational forces of each star distort their shapes. The point where the two lobes meet is called the Inner Lagrangian Point. This is the place where mass transfer occurs.
The Algol Paradox is resolved when we apply what we know about stellar evolution. As a star evolves it expands. If a star expands to the size of its Roche Lobe it will begin to lose mass to its companion. Consider the following example:
Example 10.8 In a binary star system a 10 solar mass star and 1 solar mass star are separated by 20 Ro. The radius of the 10 Mo star is 5 Ro and the radius of the 1 Mo is 1Ro. Assume that both stars formed at essentially the same time. Which star will fill its Roche Lobe first and approximately when will this take place?
Solution: To answer this question first use Roche to plot the Roche Lobes around the two stars and from this determine the maximum size either star can grow to before filling its Roche Lobe. Then use starEvolve to check which star will expand to this size first.
Step 1: In order to use Roche first set the mass-ratio. This value is defined as and the convention is to identify the most massive star as M1. In this case q = 1/10 = 0.1
Step 2: Use starEvolve to generate models of the stars and look at how the radii of the stars vary with time:
If the two stars in this example are coeval then after 23 million years the more massive of the two will have evolved off the main-sequence, filled its Roche Lobe and begun transferring matter onto the smaller companion.
Example 10.8 helps explain the Algol Paradox. Once the more massive star evolves to fill its Roche Lobe mass transfer occurs. Detailed models of this suggest that the process is remarkably "fast" with a substantial part of the formerly most massive star transferred in a few million years. Figures 10.19 and 10.18 are computer generated models of what we think the Algol system looks like.
Mass - Transfer and Angular Momentum
Another process is the tidal force that the gaining star exerts on the accretion disk. In the case of Algol the gaining star was able to remove large amounts of angular momentum from the in falling gas and was able to increase greatly its mass in the process.
Accretion and the formation of accretion disks is an extremely important process in the life of a star and we shall return to to idea in a future section when we consider how planetary systems form.
Novae and Cataclysmic Stars
Over time a heavy layer of hydrogen gas accumulates on the surface of the white dwarf . In the extreme gravitational field of the white dwarf the hydrogen layer is compressed to become a degenerate layer. As more and more gas rains onto the surface the layer steadily increases in temperature. As you saw previously, however, the temperature-pressure "thermostat" in a degenerate gas is ineffective and eventually the degenerate layer of hydrogen gas becomes hot enough for fusion to begin. Once this point is reached, and because degenerate gases conduct heat very effectively, the entire surface of the white dwarf ignites at once in a devastating thermonuclear conflagration. In effect, the surface of the white dwarf has become a hydrogen bomb!
Example 10.9 How much more energy did Nova Persei emit at its brightest compared to when it is in "quiescence" or in its normal state?
Solution Nova Persei changed from magnitude 13 when at quiescence to 0.2 at its brightest. This is a change of 12.8 magnitude units. Using the magnitude-luminosity
To understand the evolution of binary star systems
The Horizontal Branch is the region on the HR diagram in which stars are undergoing "shell burning"
Planetary Nebulae are one of the remnants of most medium mass stars and are produced by the ejected outer envelopes of these stars. They glow because they are illuminated by ultraviolet radiation from their parent star.
During the later stages of their evolution medium mass stars lose a considerable amount of mass through a greatly enhanced "stellar wind". This "wind" is enriched with Carbon and Oxygen which had been dredged up from the core through convection and is one of the ways in which stars "seed" the galaxy with heavier elements.
A White Dwarf is a compact object which represents the endpoint in the evolution of low and medium mass stars. A Black Dwarf is the ultimate end of a White Dwarf as it cools down. The time required for a White Dwarf to cool to be a Black Dwarf exceeds the present age of the universe.
The Chandrasekhar limit is the maximum mass that a White Dwarf can have