Lecture 17: Evolution of Low Mass Stars

Mass Dictates the Life of a Star after the Main Sequence Phase

• The life of stars of all masses during the main sequence phase is very similar.
• The main difference is that the higher the mass, the more luminous the star and the shorter the main sequence lifetime.

• What happens after the main sequence phase depends on the mass of the star.
• Define the following mass ranges:
  • Very Low Mass Stars: M < 0.4 M_Sun
  • Low Mass Stars: 0.4 M_Sun < M < 4 M_Sun
  • Medium Mass Stars: 4 M_Sun < M < 8 M_Sun
  • High Mass Stars: M > 8 M_Sun

• Most important processes for subsequent evolution are
  • Burning of heavier and heavier elements that require higher and higher temperatures at the center of stars.
  • New sources of pressure - degenerate (quantum mechanical!) behaviour of matter at high densities
  • Also convection plays role, since it mixes elements. Low mass stars are convective in the outer layers, high mass stars - in the core.
  • Mass loss during evolution - final masses are not the same as masses the stars had on while on main sequence.
• Must keep in mind
  • Evolution is often disbalance, at least thermal, but sometimes during violent stages, dynamical
  • Must distinguish central and surface (observed in HR) temperature. Central temperature parctically always increases, surface - experiences complcated changes.

• In this lecture we will focus on low and medium mass stars.
• The early stages of evolution for the medium and high mass stars are very similar to the low mass stars, but they occur faster.

The End of the Main Sequence Phase

The main sequence, hydrogen burning, stage for a star with the same mass as the Sun lasts for about 1010 years. Main sequence stage is finished when supply of Hydrogen in the inner 10% of the Sun runs out. At the end of this phase we have an inner core of Helium surrounded by Hydrogen. The temperature at the core is only about 15 million Kelvin, which is hot enough for Hydrogen fusion to occur, but too cool for Helium fusion. The temperature outside of the core is cooler and is not hot enough for Hydrogen fusion. Not possible for any nuclear reactions to occur!! No source of energy to create outward pressure to balance gravity. The core of the star will begin to slowly collapse. The collapse of the core will cause it to heat up.

The Red Giant Stage

As the contracting core heats up, a shell of hydrogen around the inert Helium core will heat up to 15 million K and begin to fuse. This begins the phase of shell-Hydrogen burning. The burning of Hydrogen in the shell actually produces more energy than in the main sequence phase (due to the higher T). However, the inert Hydrogen outside of the shell hinders the movement of the photons. When photons have trouble moving through a medium, they end up pushing outwards on the matter. This is called radiation pressure. The extra photons produced in the shell of hydrogen push outwards on the outer layers of the star. The expansion of the outer layers causes them to cool down. General Picture in this phase: Helium core contracts and heats up. Hydrogen shell around the core contracts, heats up and ignites. Outer Hydrogen expands and cools off. Since the star's surface temperature is lower, it will look redder than during the main sequence phase. The radius of the star increases by a large factor and becomes a giant. Final radius is 10 to 100 times the original size of the star. Final surface temperature is about 1/2 the original surface temperature. Luminosity given by blackbody equation increases. This stage is called the Red Giant Stage of a star's life. This stage lasts for about 2 billion years for a Sun-like star.

Degenerate Matter

• As the helium core contracts it becomes denser.
• When matter becomes very dense, a strange quantum mechanical effect can take place.
• The helium core is a mixture of helium nuclei and electrons.
• The electrons obey the Pauli Exclusion Principle which comes from quantum mechanics.
• Pauli Exclusion Principle: No two electrons are allowed to have exactly the same properties.
• One important property is position: If you try to squeeze together too many electrons into a small space they will repel each other.
• This repulsion is called Electron Degeneracy Pressure.
• This repulsion is much stronger (and different) than the usual repulsion which exists between charged particles.
• When degeneracy pressure is stronger than thermal pressure, we call the gas degenerate.
• In the case of the Sun's core, it becomes degenerate when
  • density is larger than 10⁶ kg/m³ (thousand times the density of water)
  • temperature is near 20 million K.
• The Sun's helium core becomes degenerate early during the red giant phase of its life.
• In all the low mass stars, the helium core becomes degenerate during the red giant phase.
• Medium and High mass stars are not degenerate while red giants.

Properties of Degenerate Gases

• A degenerate gas is very different from an ideal gas.
• An ideal gas's pressure depends on the density and the temperature.
• An degenerate gas's pressure only depends on density.
• As long as the temperature is cool enough that the gas is degenerate it doesn't matter how T changes, the pressure only depends on density.

Nuclear Reactions in a Degenerate Gas

• Nuclear reactions in a degenerate gas tend to be explosive.
• Imagine you turn on a nuclear reaction.
• The energy output heats up the gas.
• If the gas is ideal, its pressure increases and it expands and cools down.
• If the gas is degenerate, an increase in temperature doesn't increase the pressure, so it does not expand or cool down.
• Increasing the temperature makes it easier for nuclear reactions to take place, so the cycle is repeated and the reactions occur rapidly.
• This can continue until the temperature gets so high that the core is no longer degenerate.

The Helium Flash

• The red giant stage with an inert Helium core continues until the helium is hot enough for ignition through the triple-alpha process
  3 ⁴He &rarr ⁸Be^* + He &rarr ¹²C
• Since the Helium core is degenerate (in low-mass stars), the Helium-burning reaction is explosive: a large amount of the Helium fuses to Carbon in a few seconds.
• This is called the Helium Flash, but it is not observable, since the photons produced in the explosion are trapped in the Hydrogen layers.
• The flash does not last long, since it quickly gets hot enough the gas returns to the ideal gas state.
• After the flash, the helium burning reactions occur in a hot ideal gas, and the reactions occur at a slow, stable rate. (Difference between a nuclear reactor and a bomb.)
• After the flash, the luminosity decreases, and the outer layers of the star shrink.
• The period of stable helium fusion in the core is called the Horizontal Branch.

Final Stages of Life of a Low-Mass Star

Burning Helium leaves behind Carbon "ashes". Over time the core is depleted of Helium and the Carbon collects at the centre of the star. The Helium forms a shell around the Carbon and the Hydrogen forms a shell around the Helium. Carbon ignition requires hotter temperatures than are available in the core. But Oxygen is usually created in some quantities by 12C + 4He &rarr 16O As in the formation of a red giant, the star begins to contract and heat up. Helium in the shell begins to fuse to Carbon. Hydrogen fuses in a shell around the Helium. The burning in the shell causes the star to expand again, becoming redder. This is the beginning of a second red giant stage. (Also called an asymptotic giant branch, AGB.) This phase doesn't last very long, only one million years for the Sun.

The core during the last phase never get hot enough for the Carbon to ignite. However, the when the Helium shell ignites, it often does so in a series of flashes which tend to push the outer layers outwards. The star at this stage is gigantic, so the acceleration due to gravity at the surface of the star is very low. g = GM/R2 = acceleration due to gravity It is easy to launch the outer gas of the star so that it escapes from the star! The final stage of the Sun's life will be the ejection of the outer Helium and Hydrogen layers. These outer layers are hot and glow. The glowing hot ejected gas is called a planetary nebula. (Note: nothing to do with planets!)

Left over central core

• The Carbon (+Oxygen) core left behind is degenerate and supported by degeneracy pressure.
• No nuclear fusion needed to keep the left-over core from collapsing.
• Quantum mechanics won't allow the core to get too dense.
• The left-over Carbon core is called a White Dwarf.
• The White Dwarf's initial surface temperature could be as high as 200,000 K so it will glow blue-white.
• Over time, the White Dwarf cools, becoming redder and less luminous.
• Typical mass of a white dwarf is similar to the Sun, but its radius is similar in size to the Earth! Very dense!

The Ring Nebula This photo of the Ring Nebula shows the glowing ejected outer layers of the Sun-like star. The dot in the centre is the White Dwarf. Ring Nebula is about 1 light-year in diameter. (Compare with present size of Sun!) Ultraviolet photons from the hot central White Dwarf ionize the gas in the nebula, which glows when the electrons recombine with the nuclei. Blue = emission from Helium (hottest region) Green = emission from Oxygen Red = emission from Nitrogen (coolest region) The Sun may look like this one day. We can observe the Ring Nebula from our rooftop observatory in the Fall Semester.

Different Planetary Nebulae Planetary Nebulae aren't always spherical in shape. These photos were all taken by the Hubble Space Telescope.

Differences between Low and Medium Mass stars

• Low mass stars end up as White Dwarfs composed of mainly Carbon and Oxygen.
• Medium mass stars have higher temperatures in their cores.
• The higher T allows fusion reactions creating Oxygen, Neon, Sodium and Magnesium.
• Medium mass stars end up as White Dwarfs composed of the higher mass elements.

White Dwarf Stars

• White dwarfs are the exposed cores of Red Giant stars.
• The cores are typically composed of mainly Carbon, although Helium, Oxygen, Neon can also exist.
• The Carbon is ionized, so the composition of the star is a plasma of Carbon ions and electrons.
• The Carbon behaves as an ideal gas, so the gas pressure due to the Carbon decreases as the Carbon cools down.
• The electrons obey the Heisenberg Uncertainty principle which keeps the electrons from collapsing to the centre of the star.
• The electrons are negatively charged while the C ions are positively charged, so they are attracted to each other.
• The attractive electric forces between electrons and positive ions keep the C ions from collapsing to the centre of the the star.
• The force of electron degeneracy pressure balances the force of gravity in a White dwarf.
• Electron degeneracy pressure is independent of temperature, so as the star cools, the internal pressure stays constant, and the structure of the star stays constant.
• White Dwarfs cannot exist if the mass exceeds the Chandrasekhar limiting mass of M = 1.4 M_Sun

Life History of a White Dwarf Star

• When the outer layers of the Red Giant are expelled by the dying star, the inner White Dwarf core has a surface temperature over 100,000 K.
• Wein's law for a hot body with this temperature gives a peak wavelength of 2.9 x 10^-8m, corresponding to ultraviolet light.
• The photons emitted from the surface of a hot white dwarf will be very energetic and will easily have enough energy to ionize the gas surrounding the white dwarf.
• When the electrons recombine with the surrounding ions, they often enter an excited state and then jump down to the ground state emitting visible photons. This process is known as fluorescence.
• We call the glowing gas surrounding a hot white dwarf a planetary nebula.
• Planetary nebulae always have a hot white dwarf at their centre.

Cooling of the White Dwarf

• Nuclear reactions do not occur inside a white dwarf, so there is no source of energy.
• Over time the white dwarf cools.
• Since electron degeneracy pressure is independent of temperature, the star does not collapse and its radius stays constant.
• From the Stefan-Boltzman law, a star whose radius is contant, but has a surface temperature which changes with time has a luminosity which is proportional to T⁴.
• As the star cools, its luminosity will decrease.
• White dwarf stars slowly fade away. If they are born with a luminosity of 1/10 L_Sun, after about 5 billion years, their luminosity will be about 10^-4 L_Sun.
• As the star cools, the photons it emits have less energy and it is harder for the photons to ionize or excite the planetary nebula, so the nebula will fade with time. As well, the gas in the nebula will be moving outwards, so it will slowly mix with the surrounding interstellar medium.
• When the star becomes cool enough that the thermal kinetic energy of the Carbon ions is less than the electrostatic potential energy of the ions, the Carbon ions can "freeze" (i.e. form bonds) into a crystal lattice structure. After this freezing, the white dwarf is a solid.

Sirius B: a sample white dwarf

• The Sirius star system is a binary system consisting of Sirius A and B. Sirius A is a main sequence star and Sirus B is a white dwarf.
• When we look at the star system without a telescope, we only see the very bright Sirius A.
• With a good telescope, you can see the much fainter Sirius B.

• With an X-ray telescope, Sirius B is very bright, but Sirius A is very dim.

• By comparing the brightness at various wavelengths, we can find that the surface temperature of Sirius B is T = 27,000 K.
• Since we can measure a parallax angle for Sirius, we can find the distance to the star system.
• We can measure the intensity of light from Sirius B, so the luminosity can be calculated.
• By assuming that the luminosity is given by the Stefan-Boltzman equation, we can solve for the radius of Sirius B.
• The radius of Sirius B is calculated to be very close to the radius of the Earth!

• The orbital period of the Sirius binary system is close to 50 years.
• We can use Kepler's orbital law and the centre of mass equation to find the masses of the stars.
• Sirius B has a mass almost the same as the Sun's mass.

• The average density of a star is its mass divided by its volume.
• The volume of a sphere is V = 4/3 pi R³.
• The volume of Sirius B is
  V = 4/3 pi ( 7 x 10⁶ m)³ = 1.4 x 10²¹ m³.
• The density of Sirius B is
  density = 2 x 10³⁰ kg/V. = 1 x 10⁹ kg/m³.
• Remember: density of water is 1000 kg/m³.

• The escape velocity from the surface of a star is v_esc = (2GM/R)^1/2 .
• For Sirius B, this corresponds to a velocity of 6 x 10⁶ m/s, about 2/100 c.

Mass-Volume relation for White Dwarf Stars

• For normal matter, if you double the mass of an object, its volume will also double.
• For example in main sequence stars, the higher mass stars also have larger sizes.
• For degenerate matter, if you increase the mass of a star, its radius and volume decreases. Why?
  • For degenerate matter, the source of pressure is density. So it maintain higher mass, one has to increase the density inside.
  • But density = mass/volume, however, mass increase alone does not provide sufficient density (and pressure) increase so the volume has to decrease as well in order to balance gravity.
• For white dwarfs, the masses and volumes are related by Mass x Volume = constant.
• This is only approximate, since at the Chandrasekhar limiting mass of M = 1.4 M_Sun, the electrons would have to move at the speed of light, so the mass-volume relationship does not hold down to zero volume.
• The Chandrasekhar mass is the largest mass that a white dwarf can possibly have.