Evolution of Binary Stars

275-275

So far you have gained a good grasp of how isolated stars work and how they change as they age. As you learned earlier, however, roughly half of all stars are members of binary or multiple star systems. This begs the question "Do binary stars evolve differently?"

The Algol Paradox

Every 2 days, 20 hours and 49 minutes the "Demon Star" winks at you. It's a rather "lazy wink" - taking about 10 hours to complete. The "Demon Star" is Algol or beta Persei. It is the second brightest star in the fall - winter constellation Perseus. In Greek mythology and star lore Perseus has just slain the Medusa and is holding her head. Algol is one of Medusa's eyes, and like clockwork every 2.86736 days Algol dims rapidly from magnitude 2.1 to 3.4. The reason for Algol's "wink" was correctly deduced by the young British amateur astronomer John Goodricke in 1783. Goodricke speculated that Algol was being eclipsed by a fainter, unseen companion star. In the next century Goodricke's conjecture was confirmed and today we know that

Figure 10.18 The constellation Perseus and Algol.

Algol is a trinary system consisting of stars: Algol A, a B8 V star of mass 3.59 M_o, Algol B, a K0 IV of mass 0.79 M_oand Algol C, a A5 V star of mass1.67 M_o. 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 M_o 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.

Roche allows you to draw the Roche Lobes around a binary system by adjusting the mass ratio. The green dot is the inner Lagrangian point - the "balance point" between the two stars.

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 R_o. The radius of the 10 M_o star is 5 R_o and the radius of the 1 M_o is 1R_o. 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

The figure on the right shows the Roche Lobes for this system. You can enter values for the radius of each star to see when the star fills its lobe and mass-transfer will begin. This defines the "critical" radius for each star: Star 1: Rcritical = 10.7 Ro Star 2:Rcritical = 3.8 Ro From your knowledge of stellar evolution you may already suspect that it will be the 10 Mo star that fills its Roche lobe first.

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.

Figure 10.19 How Algol A and B appear in relation to their Roche Lobes

Figure 10.20 Computer model of Algol (A) A B8V star and Algol (B) an evolved and tidally distorted KO IV star

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

In February 1901 a new star, as bright as Capella or Vega appeared in the constellation Perseus. In keeping with astronomical convention this new star was dubbed "Nova (for new star) Persei". At its peak its brightness was magnitude 0.2. Within a few days Nova Persei began to fade and within two weeks had faded below naked eye visibility. Within the year it had dropped back down to 13th magnitude. In the intervening years Nova Persei has undergone episodes of smaller scale brightening. Currently it brightens by 2 or 3 magnitudes for a few days roughly every three years. As the name says, Nova Persei (also known as GK Persei) is a member of a class of objects called novae. What are novae?The explanation came in the mid 1950's when another famous nova (DQ Herculis) was shown to be a close binary star system in which a sun-like star is orbiting a white dwarf and in which mass transfer if taking place from the solar-like star onto the white dwarf.

Matter from the donor star forms an accretion disk around the white dwarf and through the braking effect of magnetic fields as well as tidal interaction between the disk and the white dwarf, matter trickles down onto the surface of the white dwarf.

Figure 10.22 US Naval Observatory image of GK Persei or Nova Persei as it appears today.

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

relation this means that for a short period of time Nova Persei emitted more than 130 thousand times its "normal" energy output.

We now understand that Novae are particular members of a broader class of close binary systems known as Cataclysmic Variable Stars. While there is considerable variety within this group they all consist of close binary systems, most with orbital periods of only 1 - 4 hours and with a white dwarf as one of the members. Most have accretion disks and matter streaming across the inner Lagrangian point . The matter stream collides with the accretion disk and forms a "hot spot" on the disk. As the entire system revolves around the centre of mass the hot spot and disk can be eclipsed, as view from earth, which leads to a variation in the light intensity that we see. Figure 10.22 is a sketch of what we think a cataclysmic system would look like as viewed from above its orbital plane. Cataclysmic variables are one of our best "laboratories" in which to study the process of accretion.

Figure 10.22 What we think cataclysmic variables (including novae) look like.

Practice

1. In your own words explain what is meant by the "Algol Paradox" and how this paradox has been resolved.
2. Use Roche to create the appropriate Roche lobes for a star system with a mass ratio (m₂/m₁) = 0.2. Let m₁ have a mass of 5 solar masses and let the stars be separated by 10 solar radii. If the stars formed at the same time, which star will lose mass and when?
3. How will the mass ratio change over time for the example consider in question 2? Explain why this accelerates the rate at which mass transfer occurs once it begins in a system. (Hint - look at how the changing value for mass ratio will affect the sizes of the Roche lobes for each star.)
4. In your own words explain what a nova is and what conditions are necessary to produce a nova.

To understand the evolution of binary star systems

Chp 13.2

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.

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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