- Being on the Main Sequence
Observationally, being on the main sequence means that a star is located on that band on the HR diagram that has been called the main sequence.
Cartoon of a Hertzsprung-Russell (HR) diagram with stars
(not accurate).
From a combination of observation and modeling, we now know a lot about main sequence stars which are, in fact, the relatively easy stars to understand.
Physically, being on the main sequence means that a star is stably fusing hydrogen nuclei to helium nuclei in its core and that its structure is very nearly in hydrostatic equilibrium (i.e., it is very nearly a fluid at rest).
The heat energy released by the nuclear fusion compensates for the energy the star loses by radiating electromagnetic radiation (EMR) continuously into space.
Main sequence stars are very nearly in STEADY STATE (i.e., an unchanging state) for long periods of time.
-
Question: The near STEADY-STATE condition is possible
because the hydrogen fuel:
- is used up relatively slowly.
- is never used up at all.
- is used up relatively rapidly.
Answer 1 is right.The hydrogen fuel is used up as the hydrogen is converted to helium.
- Nuclear Physics and Nuclear Fusion
determines the chemical species.
Atoms consist of negatively charged electrons in a swarm about a nucleus made of positively charged protons and neutral neutrons neutrons.
Atoms and nuclei.
One really needs quantum mechanics to understand atoms and nuclei in detail---that is well beyond the scope of these lectures.
The number of
Species with the same number of
and different
numbers of
- About 10 Gyr.
- About 20 Gyr.
- There is not enough information to say.
- Stellar Structure and Stellar Modeling
A star's structure can be specified by giving the ``RUNS'' of its thermodynamic and other physical quantities: i.e., as a function of radius, giving its density, pressure, temperature, interior luminosity (i.e., luminosity from within a given radius), interior mass (i.e., mass within a given radius), etc.These quantities cannot be directly observed.
They must be modeled using:
- the BASIC STELLAR PARAMETERS: luminosity,
photosphere temperature, radius, mass, composition;
- known physics: thermodynamics, nuclear burning rates
(which are functions of temperature, density, and composition),
radiative transfer, convection,
and
Newtonian physics;
- computers.
We will not detail of the equations of stellar structure, but one of them that students should know of is the equation of hydrostatic equilibrium.
This equation demands that the upward net pressure force on any layer in the star must equal the gravity force downward on all the matter above that layer. This balance of forces is necessary to prevent collapse or expansion.
-
Question: The requirement of
hydrostatic equilibrium means that pressure:
- increases with depth.
- stays constant with depth.
- decreases with depth.
Answer 1 is right.This is true in stars, in the Earth's atmosphere, in the oceans, and in a bottle of water sitting at rest on table.
The pressure must increase with depth to sustain the greater overlying mass with depth.
The radiation pressure component increases with importance with stellar mass: for a star of 1 M_Sun, it is almost negligible; for a star of 8 M_Sun, it is about 10 % of the central pressure and a smaller fraction as one moves away from the center (Cl-162--164).
The equations of stellar structure can ONLY be solved on a computer to determine star models.
People have been doing so since the 1950s and nowadays the models can be quite sophisticated and accurate.
Main sequence stars are particularly easy to model since they are time-independent or STEADY STATE to good approximation.
Pre- and post-main-sequence modeling is a lot tougher because the stars are evolving more rapidly and hydrostatic equilibrium may not be a good approximation at times.
As an example, we present a cartoon of a model of the Sun.
Cartoon of a model of the Sun
(CK-263).
Models of all main sequence stars are possible.
Models allow us
- Radiative Transfer and Convection
Well we know heat energy is created in stars by nuclear burning in their cores and we know that it is radiated away. But how does the heat energy get out? The ``how'' is, of course, part of the modeling process discussed above: here we just go into qualitatively how the energy is transported.Recall a basic fact of thermodynamics: heat always flows from hot to cold spontaneously.
Why this is so can be understood at the microscopic level, but we'll give it a pass.
The interiors of stars are hot because of nuclear burning brought about by high temperature (initially due to stellar contraction and later due to nuclear burning itself) and high density due to gravity and the huge mass of stars.
Space is cold. This is actually a profound observational fact.
The simplest way we know space is cold is that it is dark.
If you look at stars, you see intense EMR.
But between stars, EMR is obviously much lower.
The temperature of the main component of EMR that pervades space is the cosmic microwave background (CMB) that has T=2.725 K (FK-640; HI-481). We will discuss the CMB in IAWL Lecture 31: Cosmology.
There are parts of space that are hotter.
But the upshot is heat is going to flow spontaneously from stars to space.
In fact, according to our current understanding of cosmology, the stars will never succeed in heating space up.
There may be a very cold thermodynamic equilibrium in of order 10**100 years, but that is a very speculative extrapolation of what we know today (HI-477). This future is discussed in IAWL Lecture 31: Cosmology.
In space, the energy is mainly transported by free-streaming EMR.
The heat energy transport in stars is mainly by two mechanisms: radiative transfer (but NOT free-streaming) and convection.
In the interior of a star, radiative transfer is always occurring and the process is essentially a photon random walk process that we discussed in IAWL Lecture 6: Light and Electromagnetic Radiation (EMR): Photon Propagation.
Now convection. is a universally important, macroscopic heat transfer process. It occurs in:
- most stars (all stars?) including the Sun.
- the interior of the Earth. It is the driver of the plate tectonics of the crust as is discussed in IAWL Lecture 11: The Earth.
- in the Earth's atmosphere.
-
Question: What is a common, everyday, obvious example of
convection?
- The freezing of water.
- The boiling of water.
- The churning of butter.
Answer 2 is right.Boiling is actually the formation of bubbles of gas throughout a liquid and does not necessarily involve convection (KB-117). But when water boils convection is usually also happening.
Say you have a cold upper surface and hot lower surface to a fluid layer. Gravity points downward.
- A blob of fluid at the bottom absorbs heat from the bottom.
- The heat raises it's temperature and causes the blob to expand
and become less dense.
- When it becomes less dense than the surrounding it tries to
float upward.
This is just the buoyancy force effect (which is actually a fluid pressure effect). It is familiar from playing in the pool.
- If the blob rises and cools sufficiently it contracts, becomes more
dense and sinks. The fluid is then STABLE against convection.
- But if it doesn't cool sufficiently, it can keep expanding and
rising in a runaway fashion.
- Eventually the blob reaches the cool top and possibly breaks up
and releases its
heat to the top. The possibly broken-up, cooled blob sinks
back to the
bottom and starts over again.
- Convection forms a cycle of heat transfer from bottom to top.
A convection cycle.
Convection happens whenever the TEMPERATURE GRADIENT becomes sufficiently steep and the insulation is a fluid---or at least sufficiently fluid-like as inside the Earth's mantle as is discussed in IAWL Lecture 11: The Earth.
-
Question: Why can't convection happen in truly rigid solids?
- The parts of the solid arn't free to move relative to each other.
- There are no heat flows through solids.
- Solids are perfect insulators.
Answer 1 is right.-
[Answer 3 is wrong, but if true, implies answer 2, but answer 2 does not
imply answer 3 since there could be no heat flows in solids for
other reasons. Just a little exercise in logic.]
Convection is a chaotic, turbulent process. Thus it is very hard to calculate its behavior in detail.
The full calculation requires three-dimensional hydrodynamics which is still difficult even with supercomputers. Frequently, one gets the wrong answer.
Dealing with convection is one of the difficult and uncertain parts of astrophysics. For example, our understanding of stellar evolution is we think quite good, but uncertainty about convection is one of the weak links.
Convection happens in radial zones (or layers) in stars where for some reason the TEMPERATURE GRADIENT becomes too steep for stability against convection.
A cartoon of the radiative and convective zones of main sequence stars
(not accurate).
Remember radiative transfer happens in convective zones too: it is just NOT the dominant heat transport mechanism there.
In the Sun, the convective zone extends from about 0.71 R_Sun to the photosphere (Cox-342).
In the Sun, we actually see the tops of the CONVECTION CELLS. They are the solar granules.
Sunspots and granulation.
Between the granules the gas sinks.
Sunspots are somewhat colder (about 4000 K) than the surroundings (about 6000 K) and so appear dark. The are regions where magnetic field lines plunge or into the sun: thus they often come in pairs.
One can see the granulation off the spots. The granules are the tops of hot convective currents. They break up in about 10 minutes and have a size scale of about 1000 km (Cox-364; Se-148).
Credit: ?
- Nuclear Burning Stable on the Main Sequence
Nuclear burning is stable on the main sequence.By STABLE we mean the nuclear burning neither just turns off suddenly nor goes into a thermonuclear runaway and blows up the star.
Stability of a mechanical system.
A practical example of stability is all buildings. When kicked they don't fall down---one hopes not even when they are kicked really hard.
A RULER balanced on a finger is an example of an unstable system. Balance scales take advantage of instability: their instability allows a fine determination of mass.
A main sequence star's
(e.g., the Sun's) H burning stability.
.
Of course, down here on Earth we would like to have STABLE H BURNING or, as it is called, CONTROLLED FUSION.
The PPI chain is too slow for reactors (HRW-1109.
Instead, the main idea is to burn deuterons (a isotope of the hydrogen nucleus with one neutron: i.e., heavy hydrogen) and tritons (a isotope of the hydrogen nucleus with 2 neutrons: i.e., heavier hydrogen symbolized by T or H-3) to helium and create power (HRW-1109; KB-239).
Deuterons are essentially limitless since 1 in every 6700 hydrogens is a deuteron (HRW-1109). Tritons can be created in the fusion process itself (KB-240).
But it is very difficult to make plasmas hot enough (greater than about 10**7 K) and control them.
-
[A plasma is an ionized
gas.]
The fusion dream has been with us since circa 1950 and seems good for another 50 years.
Experimentally, CONTROLLED FUSION has been done, of course. But whether practical energy generation is possible is still uncertain.
If it works, then limitless, relatively clean energy.
CONTROLLED FUSION is safe because a fusion reactor has no chance of having becoming uncontrolled: its always on the verge of turning off: in fact, that's the problem so far: the fusion reactors are pretty much off.
Fusion reactor technology in itself is NOT a nuclear weapons proliferation concern: such reactors do NOT directly connect to bomb manufacture.
The energy is relatively clean in that the nuclear wastes it produces have relatively short half-lives, and so just burying the stuff in the ground for a few centuries is a reasonable procedure.
Can it work? Maybe, but I'm losing faith.
- Brown Dwarfs
Objects that form like stars or gas giant planets---it not certain whether the two processes are distinct (see IAWL Lecture 10: Solar System Formation) ---that are less than about 0.08 M_Sun (about 75 M_Jupiter) cannot burn ordinary hydrogen: i.e., hydrogen with a nucleus consisting of a single proton (CK-306).These objects are called brown dwarfs which is another misnomer: they are sufficiently cold that they radiate principally infrared light. In reflected light, they might have various hues depending on the chemistry of their atmospheres which like solar system gas giants might be complex.
Currently, there is no consensus on what the lower mass limit for brown dwarfs should be. But one popular choice is that a brown dwarf should have more than 13 M_Jupiter. Objects smaller than that are not planets, are sometimes called sub-brown dwarfs.
At mass greater than 13 M_Jupiter, brown dwarfs can burn deuterons to helium; at mass greater than 60 M_Jupiter, they can burn lithium (3 protons, 4 neutrons) to helium by some process.
During these burning phases, the brown dwarfs are fully convective and so they burn up all their deuterons and lithium in ???tens of millions of years????. (I need to check that time period).
After their brief burning phase brown dwarfs can only generate heat by contraction, and so cool off forever: ultimately they must become very cold, dim objects.
-
Question: The above discussion of
brown dwarfs is based
primarily on:
- modeling.
- observation.
Answer 1 is right.One constructs hydrostatic equilibrium models of objects of the mass of brown dwarfs with cosmic composition and sees what one gets.
These models are not definitive yet by any means.
The first discovered brown dwarf was Gliese 229B in 1994.
Gliese 229B: the first discovered brown dwarf: HST image from
1995nov17.
This is a false color, infrared image.
The objects are NOT resolved: the image size correlates with its brightness. The spike is an artifact of imaging process.
Gliese 229B is the small object.
The big object is Gliese 229A: it is the binary companion of Gliese 229B and real star at about 6 pc from Earth.
Gliese 229B orbits at about 32 AU from Gliese 229A, is between 30 and 55 M_Jupiter, has radius of about the size of Jupiter's, and has a surface temperature of about 1000 K.
229B sounded familiar, but Sherlock Holmes lived at 221B Baker St., London.
References: FK-180 and CK-306.
Credit: S. Kulkarni (Caltech), D.Golimowski (JHU) and NASA/ESA
Now over 100 brown dwarfs are known.
Brown dwarfs and sub-brown dwarfs together may be about as common as stars in the universe based on current statistics.
If this is so, then they do NOT contribute much mass to the universe and have little effect on cosmological evolution. Because they do NOT eject any material back into space as stars do, they are just quasi-eternal sinks for matter, and thus do NOT contribute to cosmic chemical evolution.
In the solar neighborhood, one estimate puts the net mass of brown dwarf at about 10 % of the luminous mass (in stars, brown dwarf, sub-brown dwarfs, gas, and dust).
- Evolution on the Main Sequence
Stars do evolve on the main sequence.-
Question: What obvious change must a star be undergoing
even during its slow main sequence lifetime?
- It's carbon is being converted to diamond.
- It's core hydrogen is being converted to helium.
- It's surface hydrogen is being converted to helium.
Answer 2 is right.The surface is much too cool and rarefied for hydrogen burning.
For example, let us consider the Sun.
A cartoon of the evolving composition of the Sun
(CM-315).
The decrease in hydrogen fuel in the core has the seemingly paradoxical result that stars will get more luminous over their main sequence life.
Pressure support depends on the number of particles for ordinary gases. In fact, pressure is linearly proportional to the number of particles. When hydrogen is fused to helium, 4 particles are converted to 1. This causes a tendency to loss in pressure which in turn causes a tendency to contraction due the gravitational force on the Sun's own mass. But contraction increases the core density and increases the temperature essentially due to gravitational potential energy being converted into thermal energy. The higher density and temperature, the higher the pressure and thus collapse is resisted. But higher density and temperature tend to increase the collision rate of hydrogens and other isotopes in hydrogen burning chains, and thus the hydrogen burning rate. This tendency more than compensates for the decrease in fuel, and so there is a higher rate of energy production and the star gets brighter. Main sequence stars burn brighter as they exhaust their fuel. Reference Se-246; FK-467.Again, the core of a star on the main sequence, thus gets denser and hotter.
The extra energy tends to make the outer layers expand: so as the core gets denser, the star actually tends to increase in size.
Stars larger than about 1 M_Sun get a little cooler on the surface during main sequence evolution. Since they are getting brighter too, they move upward and rightward on an HR diagram (CK-314).
The Sun and smaller stars??? will actually get slightly hotter on the surface for awhile anyway (FK-467). They would move upward to the left on the HR diagram.
-
[These results are probably only
understood in a computer modeling sense. This means
we put the right physical ingredients in a computer model and
we get the results. The results defy any simple explanation.
A lack of simple explanation is often the case in computer calculations: we simply say we understand in a computer modeling sense.]
Cartoon of a Hertzsprung-Russell (HR) diagram with stars
(not accurate).
The brightening of Sun-size main sequence stars is not a lot actually.
The Sun is now about 30 % brighter than when initially on the main sequence about 4.6 Gyr ago.
It will be about 30 % brighter than now in about 3.5 Gyr (WB-106; FK-493)
This gradual brightening of the Sun is known only theoretically. But we think we understand main sequence stars well in their main behavior.
So the brightening is about as certain a result as a purely theoretical result can be.
The brightening is a pretty modest change for the Sun.
But it probably means the doom of complex life on Earth within of order a gigayear or two by first eliminating the carbon dioxide from the atmosphere and then eliminating liquid water.
-
[Well one might imagine that life evolves in a way to
overcome its current physical limitations, and endure
or modify the brute physical changes that are coming.
But that's more than physical science can know.]
The unhappy fate of life on Earth is discussed in IAWL Lecture 11: The Earth.
The main sequence lifetime of a star begins when the star settles on the zero-age main sequence curve on the Hertzsprung-Russell diagram??? (CK-304).
It ends when the star's core hydrogen is sufficiently exhausted for the burning rate in the core to slow down or, more loosely, when the core hydrogen is exhausted (CK-311).
-
Question: How does main sequence lifetime vary with
star mass?
- It increases with mass.
- It stays constant with mass.
- It decreases with mass.
Answer 3 is right.The lifetimes fall very quickly with mass in fact:
lifetime t = approximately constant * M**(-2.5) . This is a faster decrease than an inverse-square law decrease.
Some simple function behaviors.
Here is a Table of Approximate Main Sequence Lifetimes.
According to the concordance model of cosmology, the age of the age of observable universe since the big bang is 13.7+/-0.2 Gyr.
-
[We discuss the
concordance model of cosmology
in
IAWL Lecture 31: Cosmology.]
-
Question: If 14 Gyr is about the age of
observable universe, are there star types none of
whose members have ever
left the
main sequence?
- No.
- Yes. All stars born less than about 0.75 M_Sun are still on the main sequence.
Answer 2 is right.
If we look back at the Table of Approximate Main Sequence Lifetimes, we that some low-mass stars will be on the main sequence for hundreds of gigayears.
Post-main sequence stars of that initially had masses less than about 0.75 M_Sun don't exist now and won't for some time to come.
The fate of very low-mass stars (and also stars of mass up to about 8 M_Sun) is to become white dwarfs whose nature is discussed in IAWL Lecture 24: Compact Remnants: White Dwarfs and Neutron Stars.
Here will just say white dwarfs are star-mass objects of Earth-size (and thus very dense) that are not burning any nuclear fuel and are just cooling off forever.
-
[White dwarfs
exist, of course: there are just none that originated from
stars of less than about 0.75 M_Sun when on the
main sequence.]
But it is is probably a comparatively gentle process??? particularly for red dwarfs which are main sequence stars in the mass range 0.08--0.4 M_Sun (CK-311).
Red dwarfs are convective throughout and will convert almost all there hydrogen to helium.
They will never burn helium unlike more massive stars, and so should just contract probably pretty slowly into white dwarfs.
They probably will just cool in the post-main-sequence phase (CK-311), but since contraction causes heating they might heat up for awhile???? but I've not been able to find out for sure.
IAWL Lecture 23: Late Star Evolution and Star Death takes up the subject of the post-main-sequence evolution of stars of about solar mass and larger.
- the BASIC STELLAR PARAMETERS: luminosity,
photosphere temperature, radius, mass, composition;
-
[Isotope is sort of a tricky word to define since it is a
relationship word like brother: all men are brothers,
but not all men are brothers to each other.
All atoms are isotopes, but not all atoms are isotopes to each other.]
-
[The electronic structure of atoms determines the chemistry.
Isotopes of the same atom have the same number of protons, and
thus their nuclei bind to electrons in almost identical manner.
Thus the electronic structure and chemistry of isotopes is
nearly identical.
The chemical behavior cannot be quite identical because the mass of the isotopes is somewhat different because of the differing number of neutrons. The mass of atom affects the reaction rates for chemical processes to some degree and even the electronic structure which has a small effect on the spectroscopy of the atom.]
A nucleus consisting of one proton and one neutron is a heavy hydrogen: a deuteron symbolized by D or H-2.
Nuclei are held together against the electrostatic repulsion of the protons by the strong nuclear force.
The strong nuclear force is a very strong force, but it is very short range. It acts only over a distance of 10**(-15) meters or less. This distance of 10**(-15) meters is 10**5 times smaller than an atom's size.
Thus, the protons and neutrons in a nucleus are compacted close together within about 10**(-15) meters of their nearest neighbors and nuclei have a size scale of order 10**(-15) to 10**(-14) meters depending on the number of protons and neutrons.
Now H nuclei strongly repel by the electrostatic force because they are like-charged particles.
In stars, only in the cores is it ordinarily sufficiently hot and dense that the electrostatic repulsion can be overcome and the H nuclei can collide closely enough that the strong nuclear force can bind them (i.e., fuse them).
-
[Astrophysicists, usually call fusion,
nuclear burning because
the fusion process is the nuclear analog of chemical burning.]
Hydrogen nuclear fusion or burning to deuterons.
The deuteron is a reactive nucleus compared to ordinary hydrogen and it burns He-3 (two protons and one neutron in the nucleus) relatively quickly (HRW-1106).
But the final product in stellar hydrogen burning is the very stable He-4 nucleus.
There are several H-to-He-4 burning processes in stars.
The dominant one in stars less massive than about 1.5 M_Sun is the proton-proton I (PPI) chain (HI-343; Cl-369; HRW-1106).
We will just look at the PPI chain in detail to illustrate a nuclear burning chain.
The PPI chain. (Correction: have ``more''.)
The dominant one in stars more massive than about 1.5 M_Sun is the carbon cycle (HI-343; Cl-390).
In the carbon cycle, the carbon nucleus acts as a catalyst: i.e., a reactant that facilitates the process without being destroyed in a net sense in the process. We will NOT go into details though.
-
[Hans Bethe
(1906jul02--2005mar06) discovered the carbon cycle
in 1938. In the early 1930s, he left Germany because of
the Nazis and in WWII he was the leader of the theory group
at Los Alamos in the Manhatten Project.
He's almost certainly the last of the great pre-war generation
physicists.]
4H + 2 electrons to He-4 + 2 neutrinos + heat energy
(HRW-1107;
Cl-390).
The heat energy is in the form of kinetic energy
of the particles and photons.
The neutrinos mostly just freely escape the Sun,
fly off into space, and have very little effect on
the universe it seems.
The rest mass
of the products is 0.7 % less than the
rest mass
of the reactants
(CK-262).
Now rest mass is a form of energy and energy is conserved overall.
The missing rest mass was transformed into the emitted heat energy (i.e., kinetic energy of the products and EMR).
The Einstein equation recall is
E = mc**2
and thus 1 kilogram of rest mass in energy terms is
1 kg * (3*10**8 m/s)**2 = 9*10**16 joules
= about 2.5*10**10 KW-hours
( 1 kW-hr = 3.6*10**6 J )
= the explosion energy yield
of about 20 megatons of TNT
( 1 megaton TNT = about 4*10**15 J ) .
For
main sequence stars one requires approximately
STEADY-STATE luminosity, and thus one requires that:
L = 0.007 * (dM/dt) * c**2 ,
where L is stellar luminosity,
and dM/dt is a calculus expression meaning
rate of change of hydrogen mass to helium mass.
Skipping the algebra, one solves to get
dM L
_____ = ______ * 10**(-11) M_sun/year , where I've used
used fiducial values
dt L_Sun
(FK-382, note).
If the rate of hydrogen mass burning is constant, then, as usual,
exhaustion time is:
Amount M/M_Sun
t = ___________ = f * _________ * 10**11 years ,
Rate L/L_Sun
f is the fraction of a star's hydrogen that can be burnt.
A star cannot burn all its hydrogen.
Calculations show that f is of order 0.1 at least
for stars greater than or equal to about 1 M_Sun
(HI-330;
FK-451).
Amount M/M_Sun
t = ___________ = _________ * 10 Gyr , where recall
Rate L/L_Sun 1 Gyr = 10**9 years.
-
Question: How long is the Sun's main sequence lifetime?
Answer 1 is right.
The Sun's current age is 4.6 Gyr (HI-330), and so it is about half done with its main sequence lifetime.
The main sequence lifetime of a star is about 90 % of its total life as a nuclear-burning star???? (HI-330), and so to 1st order the main sequence lifetime is the total nuclear-burning lifetime.
Thus, the Sun's is about half through its life.
