Nuclear Astrophysics

Lecture 9 &10

Thurs. Dec. 16, 2010

Prof. Shawn Bishop, Office 2013,

Ex. 12437shawn.bishop@ph.tum.de

1

Alternative Rate Formula(s)

2

We had from last lecture:

With:

And:

And we have approximated the S-factor, at the astrophysical energies, as a simple constant function (units of keV barn).

And (pg 16, Lecture 6):

3

We have (again)

Set the argument of the exponential to be (dimensionless)

From the definitions of on the previous slide, you can work out that:

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First, an algebraic step of substituting into the equation for

We substitute this, along with Eeff into above. After some algebra:

A Power Law expression for the Rate

5

Let’s pull out the physics of this complicated formula by trying to write it as something simpler, like a power law function.

First, equate the last expression for the rate to something like a power law:

Take the natural logarithm of both sides:

Differentiate both sides wrt :

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Numerically,

Summarizing then:

Choose a value of temperature to evaluate the rate at:

Once we have this value for the rate at some temperature you have chosen, to get the rate at any other temperature, just make the trivial calculation:

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Finally, recall

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Summary of Reaction Rate

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Examples:

The core of our Sun has a central temperature of about 15 million degrees kelvin.

Reaction Z1 Z2 mu T6 Eeff n

p + p 1 1 0.5 15 5.88 3.89

p + 14N 1 7 0.933 15 26.53 19.90

4He + 12C 2 6 3 15 56.09 42.81

16O + 16O 8 8 8 15 237.44 183.40

Reaction rate becomes more sensitive to temperature as we “burn” heavier nuclei together.

Such a large sensitivity to the temperature suggests structural changes in star must occur at some point for certain reactions.

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Full Reaction Rates

p + p reaction14N + p12C + a

Solar abundances, density = 100 g cm-3:

Rate

Reaction Rate: Temperature Dependence

11

p + p reaction14N + p12C + a

Energy Generation & Standard Model

12

Let us consider a representation for the nuclear rate of energy production (in units of energy per unit time, per unit mass) to be given by the following mathematical form:

Then we can, quite generally, also write:

For a Polytrope model, once the solution to Lane-Emden equation is known, then we have the following results (for n = 3 Polytrope):

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Using the equations on previous slide, we can now write the scaled nuclear energy generation rate in the very compact and simple form:

The luminosity of the star (after settling into equilibrium) comes from the nuclear furnace in the core. Let’s try to write the luminosity in the following form:

Where we use the mass-averaged energy generation rate:

We use an averaged energy rate because it is clear that not all mass in the star is contributing to the nuclear energy rate: only the core of the star is contributing to the energy generation rate. So, we average over the mass of the star in the hope that the averaged rate times total mass (equation *) gives a reasonable result.

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We must now consider the integral:

Our differential mass element is, in the scaled radial coordinate (refer to Lecture 2):

And, from previous page,

Collecting everything into the integral:

On pages 29, 30 of Lecture 2 we had:

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Finally, then, we have that the mass-averaged nuclear energy rate is given by:

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Lane-Emden Function for n = 3

The integral we must consider is

We are raising the Lane-Emden function to the power of (3u+s). If this number is large, then we are multiplying by itself many times. The function is always smaller than one: take multiple products of it will cause it to become more sharply peaked, and its tail to grow smaller in magnitude.

We can exploit this feature to make a suitable approximation of the function in the integrand above.

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The power series expansion of , for the first few terms looks like this:

Let’s use the exponential approximation in the integral for

Mass-averaged nuclear energy rate:

A Polytrope Sun

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Exact eGaussian Approx.

We had the general energy generation rate expressed as:

For the proton-proton chain in the Sun, and (at ). Therefore, we have, in terms of that:

Integrating this over the star introduces another from . The integrand of the luminosity function then is something like:

Structure of Polytrope Sun

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Let us define the core to be that volume in which 95% of the nuclear energy is generated.

We find that this occurs around (next page). Recall the stellar radius in thevariable is .

So the core occupies about 23% of the total stellar radius.

So the fractional volume occupied by the core is

Mass occupied by the core: From plot on page 16, it is about 33% of stellar mass.

Question for you:

This sun is 1 solar mass, and 1 solar radius in size. Take

Use these numbers and the result on page 11 to determine the mass of hydrogen per second this Sun burns. Also use: u =2, and s = 4.6

Integrated Luminosity

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This is 95% at ; this radius corresponds to 1.6/6.89 = 23% of the stellar radius.

Let us take this as being an (arbitrary) criterion for defining the core.

As a fraction of the star’s volume, the core is only

Derivative of Luminosity

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Mass Interior to Radius

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Mass contained in the core is about 36%.

So, a star the size of our Sun, in purely radiative equilibrium (no convection) has 36% of its mass contained in only 1.2% of its volume!

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Arb

itra

ry U

nit

s

TemperatureDensityPressureEpsilon

Standard Solar Model: Radial Run of Temperature, Density and Nuclear Energy Rate

Remember:

HERTZSPRUNG-RUSSEL DIAGRAM & NUCLEAR BURNING

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PP-I

Qeff= 26.20 MeV

p + p d + e+ + n

p + d 3He + g

3He + 3He 4He + 2p

86% 14%

3He + 4He 7Be + g

2 4He

7Be + e-

7Li + n7Li + p 2 4He

7Be + p 8B + g8B 8Be + e+ + n

99.7% 0.3%

PP-II

Qeff= 25.66 MeVPP-III

Qeff= 19.17 MeV

Netto: 4p 4He + 2e+ + 2n + Qeff

Proton-Proton-Chain

25

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CNO burning cycles for hydrogen under non-explosive conditions.

Each cycle converts 4 protons into an alpha particle.

It is these cycles that nuclear burning enters once proton-proton and 4He burning (by the triple-alpha process) have built up enough 12C abundance.

Remember: These form cycles because, at some point along the path, we encounter a nucleus “Y” that has a binding energy for alpha particles that is lower than the minimum excitation energy of “Y” when it is made by proton capture. Example in CNO3:18O(p,g)19F*

a + 15N

CNO Nuclear Cycles

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The nuclear energy production in Main Sequence stars comes either from the proton-proton chain or from the CNO cycle. (C = Carbon, N = Nitrogen, O = Oxygen).

PP-Chain Parameters CNO-Cycle Parameters

28

From Lecture 2, we derived an expression for the central temperature of a Polytrope star as,

Central density:

In these formulae, the temperature is in kelvin, and density is grams/cm3.

Numerically, we find that:

PP-Chain Nuclear Energy Rate vs Core Temperature

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From page 17, we had:

Dropping the “pp” subscripts, and considering this formula at the centre of the star:

(Remembering )

Substitute in the previous result for :

CNO-Cycle Luminosity vs Core Temperature

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From page 17, we had:

Dropping the “CNO” subscripts, and considering this formula at the centre of the star:

Substitute in the previous result for :

Last lecture, we derived, for Polytrope (n = 3) the result:

Refer to Page 2 to understand that .

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The power per unit mass produced in the stellar core is, after all, what is responsible for the luminosity at the stellar surface. We now try to come up with a generalized formula for the nuclear energy generation rate for any Main Sequence star.

We do this by performing a mass-averaged energy generation rate. And this mass-averaged rate times the stellar mass should, if the model is anywhere close to reality, give us a quantitative trend that is similar to the observational data.

We now have all the ingredients we need to determine the relationship between surface temperature (effective temperature) and the stellar luminosity. Start substituting the results from previous slides. I will do the case for the pp-chain

Multiply and divide by unity: so that numerator will have

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Continuing:

Divide both sides by Solar Luminosity, , and use and and . Then, we have:

Similar steps will lead you to a corresponding formula for luminosity in terms of core temperature for stars with the nuclear energy generation rate dominated by CNO-Cycleburning. (And you should do this: you have all the formula from page 27 and 28 to do it).

This result is the relationship between the luminosity of the star in terms of its coretemperature, at the centre of the star, for proton-proton nuclear burning.

pp-chain burning

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From page 15, Lecture 6, the Main Sequence Mass-Luminosity relationship was derived. We found that:

And we have just found that the nuclear burning for pp-chains and CNO cycle give:

Solar abundances give, (refer also to page 5): , , . Note: The Eddington Quartic Equation (page 37, Lect. 2/3) lets us solve for

for a chosen stellar mass .

Once we have and we can equate ( ** ) and ( *** ) with ( * ) and get the stellar core temperature of Main Sequence stars in terms of the stellar mass.

Mass-Luminosity: Main Sequence

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Data are from: G. Torres et al., Astron. Astrophys. Rev. (2009)

We use this curve for the left hand sides of the equations on the previous page.

Main Sequence: Core Temperature vs Mass (for fixed Luminosity)

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CNO Cycle

PP Chain

At , we find that the pp-chain and CNO-Cycle give equal core temperature at the same luminosity.

We see that, for any stellar masses , the CNO cycle does not require as high a core temperature to produce the same luminosity as does the pp-chain. Less than this mass, the pp-chain can burn at lower temperatures than CNO to produce the same luminosity.

A Consistency Check on the Crossover Temperature

36

At the crossover temperature, the energy generation rates of both burning cycles must be equal. So, set

For you to show this result forfrom the equations on page 5 and 6

Many terms cancel on both sides, and we require the temperature to be the same. After cancelling density, mass, and so on, and solving for the common temperature:

Luminosity vs Effective Temperature: Hertzsprung-Russel Diagram

37

The previous plot tells us that: For a particular value of luminosity, the pp-chain requires a higher core temperature than does the CNO cycle for stellar masses .

We cannot, of course, measure the core temperature of stars, so the previous result would otherwise be only purely theoretical.

However, astronomers do measure the effective surface temperatures of stars. Our Main Sequence model should be able to at least quantitatively show the trend of stellar surface temperatures to confirm the previous theoretical result of how the two reaction cycles depend on core temperature and stellar mass.

In other words: can the stellar surface temperatures be used to definitively determine where in the Main Sequence stars the pp-chain dominates, and where the CNO-cycle dominates?

Let’s work toward the answer to this. Let the data tell us how the nuclear reactions in the core are connected to the temperature and luminosity we observe from the stars.

38

Back on page 28, we had:

Now, the surface temperature of the star is related to its luminosity by the Black Body relationship, with Stefan-Boltzmann constant:

This is how we will get the effective temperature related to the nuclear luminosity.

We need to relate the core temperature above to the stellar radius. We can do this with the Polytrope formalism because and

We use this now in the equation up top to express the core temperature in terms of the stellar radius (and mass). Using , gives the simple result:

Using:

Coupled to each other by structure constraints

We have a first “hint” here of things to come from the last equation:

(Core temp related to radius)

As stellar mass grows, we expect the core temperature to increase, which would seem to suggest that the pp-burning will dominate over CNO burning. However, as the core temperature increases, the star’s structure will adopt a new configuration; namely, the radius will grow (expansion) and it will be the competition of the mass versus the extent of expansion (growing r) that will regulate the core temperature and, therefore, the type of nuclear burning happening within the core.

Let us continue. At the stellar surface:

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Sub into equation up top:

Almost done. On page 32, we derived, for the pp-chain, the following result:

Sub in the last expression for , and after some algebra, we finally have:

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So, we choose a value for m, use EQE to get and use the mass-luminosity function of Lecture 5 on the LHS. Finally, solve for as a function of stellar luminosity. Result on next slide, with Main Sequence data.

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Main Sequence: Luminosity vs Effective Surface Temperature

PP Chain

CNO Cycle

Data from: G. Torres et al., Astron. Astrophys. Rev. (2009)

Sun

Data tells us the truth: The pp-chain cannot be the dominant burning source for high mass stars.

For stars with , the pp-chain dominates. Otherwise, it is the CNO cycle!

Summary: Main Sequence Stars• We have learned the mass-luminosity relationship based on the physics of the

stellar structure equations + a “simple” Polytrope model

• Using the mass-luminosity relationship (data), and nuclear reaction results (to be taught in the New Year), we have extended the above result to learn the behaviour of stellar core temperatures as a function of stellar mass (page 12)

• For high mass stars, the pp-chain requires a higher core temp. than the CNO cycle to produce the same luminosity. (page 12)

• Our Polytrope Main Sequence model, while not perfect, agrees with the trend of the observational Luminosity-Effective Temperature data. The data and model show that, for those stars with the luminosity is the result of the CNO nuclear reaction cycle; pp-chain nuclear reaction cycle is primary stellar energy source for masses less than .

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Hertzprung-Russel Diagram

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CNO Cycle

PP-Chain

Understanding how stars become Giants, and the nuclear processes within, will be our next step in the New Year.

APPENDIX: SCALED MAIN SEQUENCE STRUCTURE PLOTS

Polytrope model, for polytrope index n = 3

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Integrated Luminosity

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Let us take 95% of luminosity as being the (arbitrary) criterion for defining the core. Then core has a radius

As a fraction of the star’s volume, the core is only

Derivative of Luminosity

46

Mass Interior to Radius

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Mass contained in the core is about 36%.

So, a star the size of our Sun, in purely radiative equilibrium (no convection) has 36% of its mass contained in only 1.3% of its volume!

48

Arb

itra

ry U

nit

s

TemperatureDensityPressureEpsilon

Standard Solar Model: Radial Run of Temperature, Density and Nuclear Energy Rate

Solar Model Calculation

49

Nuclear Generation RateUsing the pp-chain , which is a power law in temperature, with exponent n = 4.6, we end up with the Polytropic energy generation rate expressed on in the integral on the LHS below, where is the Lane-Emden function of index = 3.

The L-E function, raised to such a high power, can be approximated to an excellent degree by a Gaussian (RHS), which can then be analytically integrated.