Animated astronomy video

I just found this video in YouTube, created by the same people who makes PhD comics.

After watching it I had to share it in here. This is good stuff. Albeit not thorough, frenetically fast, deceptively messy, and beautifully executed. Really cute stuff.

Watch it now.

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Carolyn Porco: This is Saturn

I wanted to share this with you. A TED talk about the exploration of Saturn by Carolyn Porco.

I hope that you enjoy it!

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The Solar Neutrino Problem

The solar core emits neutrinos as a result of the nuclear reactions in the pp-chain.

The thing with neutrinos is, they have a very small cross-section, and therefore are very difficult to detect (they can go all the way through Earth without interacting with anything).

The Homestake experiment, Lead, South Dakota, had the collection and counting of neutrinos coming from the Sun as a purpose. It consisted on a tank with 100,000 gallons (~380,000 L) of C2Cl4.

Cl-37 can interact with neutrinos of sufficient energy to produce radioactive Ar-37, with a half-life of 35 days.

chlorine-to-argon

The counting rate of neutrinos was measured in solar neutrino units (SNU, 1 SNU = 1e-36 reactions per target atom per second). The standard solar model predicts a rate of 7.9 SNU, while the outcome of the experiment was 2.23 ± 0.26 SNU. This discrepancy is the solar neutrino problem (SNP).

The Super-Kamiokande: Consisted on a tank of 3,000 metric tons of water. Neutrinos are able to scatter electrons, and the Kamiokande detector is able to detect the Cherenkov radiation produced when this happens. Cherenkov radiation is produced when an electron travelling in a medium (e.g. in water) travels faster than the speed of light in that medium (which is not physically impossible, since the speed of light in any medium is always slower than in vacuum). This is somewhat analogous to breaking the sound barrier.

The Kamiokande experiment was the first able to experimentally verify that the neutrinos have a non-zero mass.

The explanation for the SNP is that, since neutrinos have non-zero mass, they can transform between different types or flavors. Thus, there are 3 types of neutrinos:

  • Electron neutrinosνe
  • Muon neutrinos, νµ
  • Tau neutrinosντ

The transformation between one flavor and the others (called neutrino oscillation) takes place when neutrinos interact with electrons. The experiments only detected the electron neutrinos, explaining why a smaller number of neutrinos than that predicted by the models was detected.

EDIT: The transformation between flavors seems to be of a more complex nature than simply “interact with electrons”, which I don’t fully understand. To get some clue, read the paragraph under the “Theory” heading in this wikipedia article.

Neutrino oscillation was evidenced by the Sudbury Neutrino Observatory.

See the text under the heading “Resolution” in the wikipedia article. The explanation is quite neat.

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Solar Model

The Sun, a typical main-sequence star of spectral class G2, with a surface of composition X = 0.73 (mass fraction of hydrogen) and Z = 0.02 (mass fraction of metals). Current estimated age: 4.52e9 years.

The standard solar model applies physical stellar modelling to describe the Sun (see Stellar Modelling).

  • The center of the Sun is rich in He-4 (product of pp-chain), and poor in hydrogen (low X, high Y).
  • The surface of the Sun is rich in hydrogen and poor in He-4. This is due to diffusive settling of heavier elements toward the center.
  • Since the birth of the Sun, its radius has increased by 10%, and its luminosity has increased by 40%.
  • The Sun’s primary energy source is the pp-chain.
  • 90% of the Sun’s mass is located within one half of its radius.
  • The luminosity suddenly increases at around 10% of the solar radius, which means that at this point the energy production is maximum. Two factors to take into account for this:
    (1) The energy production is higher as the mass within one shell increases, this is naturally happening as we get further away from the center.
    (2) The concentration of fuel available to generate energy (hydrogen) decreases as we move further away from the center.
    These two factors cause a maximum in the derivative of the Sun’s interior luminosity (dL/dr) at 10% of the solar radius (i.e. maximum of energy production).
  • At 71% of the solar radius, the energy transport regime changes from radiative transport (r < 0.71R) to convective transport (r > 0.71R).
Dependency of composition and luminosity on solar radius.

Dependency of composition and luminosity on solar radius.

Helioseismology = study of the oscillations of the Sun.

The Sun oscillates with roughly ten million vibration modes, typically with a very low amplitude (surface velocity of < 10cm/s, and luminosity variation  dL/L ~ 1e-6). Two kinds of modes identified.

  • p-modes, or five-minute oscillations, with periods between 3 and 8 minutes.
  • g-modes, with longer periods of about 160 minutes.

More about helioseismology.

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Stellar Modelling

Stellar Models require:

  • The fundamental stellar structure equations (FSSE)
  • The constitutive relations (CR): “equations of state” describing the physical properties of the matter of the star.

fsse

If the structure of the star is changing, the influence of gravitational energy must be included. This introduces a time dependence not present in the static (time-independent) case.

tot-e-grav-nuc

e-gravit

Also, if acceleration is non zero, the acceleration term must be added to the pressure differential (first equation in FSSE):

hydrost-eq-acc

The CR represent P, κ, and ϵ as a function of ρ, T, and composition.

cr

Boundary Conditions: specify physical constraints to our equations.

  • Interior mass Mr and interior luminosity Lr vanish in the center of the star (r=0).
  • Another set of boundary conditions are required at a r=R* somewhere at the surface of the star. The simplest conditions are assuming that the temperature T, the pressure P and the density ρ vanish at r=R*. Strictly speaking, these conditions are never obtained in reality, so more sophisticated conditions may be needed.

How is it all used? The volume of the star is imagined to be constructed of spherically symmetric shells of width Δr. These shells separate the volume of the star into discrete Δr increments (…, ri-2, ri-1, ri, ri+1, ri+2, …, thus i is a “label” for one given shell), and the different properties are calculated via numerical integration for each r, using the Stellar Structure Equations, and the Constitutive Relations.

This numerical integration can be done starting from one boundary (either r=0 or r=R*) or somewhere in the middle point of one star radius, and integrate in both directions (i.e. toward the center and toward the surface). By doing this, for each value of i we can find the respective Pi, Mi, Li, and Ti.

In the end, the values obtained should match with the defined boundary conditions in both ends with a desired accuracy. Usually, it requires several iterations, which means that if the properties at the boundaries do not match the desired values, we must change the initial conditions and repeat the numerical integration.

In sum:

  1. Decide a starting point, and a set initial conditions (guess).
  2. Perform numerical integration toward both boundaries (center and surface).
  3. Check accuracy of solution. If the solution is not accurate enough, return to step 1. Otherwise, you’re done.

star-model-numerical-int

Vogt-Russell Theorem: take as a general rule, more than a rigorous law.

The mass and composition of a star uniquely determine its radius, luminosity, and internal structure, as well as its subsequent evolution.

Changes of properties in Main sequence stars…

  • Larger M mean larger central P and T.
  • In low-M stars, the pp-chain dominates.
  • In high-M stars, the CNO cycle dominates.
  • Star lifetimes decrease with decreasing L.
  • Stars of the main sequence have masses which range between:
    - M < 0.08 M (no nuclear reactions taking place).
    - M > 90 M (energy pulsation mechanism and unstable stars).
  • A M change of 3 orders of magnitude corresponds to:
    - A L change of 9 orders of magnitude (i.e. a damn huge change)
    - Only moderate T change (factor of 20, 2,700K — 53,000K).
  • A lower T involves a higher opacity (κ), i.e. low T favors convection domination.
mseq-properties

Source (modified)

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Energy Transport

Energy transport inside a star takes place in three ways:

  • Radiation: dependent on opacityflux, density, and temperature. Larger opacity and flux involve steeper decrease in T with increasing r.
    radiative-transport
  • Convection: cool masses move “down” (in the direction of gravitational pull) and hot masses move “up” (against gravitational pull). If the T gradient is too steep, convection can play an important role.
  • Conduction: generally negligible.

We look more closely at convection.

  • Its treatment is very complex, and the theory is not yet fully developed.
  • It relies on Navier-Strokes equations in 3-D.
  • For the study of stellar structure, usually we use 1-D approximations (only spatial variable is r, i.e. we impose spherical symmetry).
  • The pressure scale height (a characteristic length scale for convection) is in the same order as the size of the star, thus convection is coupled to star’s behavior.

The model that we use is that of a hot bubble which (1) rises and expands adiabatically, and then after travelling some distance, (2) it thermalizes with the medium.

This is a thermodynamic model based on state functions (changes in thermodynamic state functions depend only on the starting and final states of the system, and not in the way followed).

The following equation describes how the temperature of the gas inside the bubble changes as the bubble rises and expands adiabatically:

convective-transport

heat-ratio

CP and CV are the specific heat capacities at constant pressure and constant volume respectively.

Superadiabaticity: Case that the star’s actual temperature gradient is steeper than the adiabatic temperature gradient (i.e. inside the bubble). It represents a condition for convection domination:

superadiabatic

 

If “act” is larger than “ad”, all luminosity is carried by convection.

It can be alternatively written as (assuming that the µ -mean molecular weight- does not vary):

superadiabatic-2

 

So, either radiation or convection dominate in different circumstances. When is convection favored?

  1. Conditions of large stellar opacity
  2. Ionization taking place, which means large specific heat, and low adiabatic T gradient.
  3. Low local gravitational acceleration
  4. Large T dependence of nuclear reactions (e.g. CNO cycle and triple alpha)
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Nuclear Reactions

What are the nuclear reactions taking place in the interior of the stars, and what are their elementary steps?

There are many. In these reactions, light elements fuse into heavier elements. The more massive is a star and the higher the temperature of its interior, the higher the number of possible reactions available, which transform atoms into heavier and heavier elements in a process known as nucleosynthesis.

The elementary steps of nuclear reactions need to fulfill some conditions:

  • The electric charge needs to be conserved.
  • The number of leptons (electrons, positrons, neutrinos, and antineutrinos) needs to be conserved.
    Conditions: matter leptons – antimatter leptons = constant
    Note: electron + positron = 2 photons (γ) (matter and antimatter annihilate each other producing photons)

elementary-particles

Neutrinos are somewhat special in that they have a very small mass, and a very small cross section, therefore they have very long mean free paths (of the order of 10e9 solar radii!).

In the following, I represent nuclei by the symbol:

symbol-nucleum

One possible reaction is the conversion of hydrogen into helium. This happens via the proton-proton chain or via the CNO cycle, where carbon, nitrogen, and oxygen act as catalysts.

proton-proton-chain

The proton-proton chain consists on 3 branches: PP-I, PP-II, PP-III.

PP-I. Source.

PP-I. Source.

The slowest step in the PP-I chain is the first one, which involves the weak nuclear force.

PP-II. Source.

PP-II. Source.

PP-III. Source.

PP-III. Source.

In an environment similar to the center of the Sun, 69% of the time the PP-I chain takes place, while 31% of the time the PP-II chain occurs. In such environment, the occurrence of the PP-III chain is more rare, with a probability of 0.3%. These probabilities change depending on the temperature on the star.

In summary, the three branches of the PP-chain:

The next pic represents the CNO cycle.

CNO cycle. Source.

CNO cycle. Source.

The cycle above culminates with the production of carbon-12 and helium-4. Another possibility, less probable (0.04%) is the production of oxygen-16 and a photon instead.

The effect of fusing hydrogen into helium is the increase of the mean molecular weight (μ). The star begins to collapse, and this is compensated by a gradual increase of T and ρ.

Therefore, the PP-chain is dominant in low mass stars with low T, while the CNO cycle dominates at higher T and more massive stars.

In the triple alpha process, helium is converted into carbon. The “alpha” comes from the historical mysterious alpha particles, which turned out to be helium-4 nuclei. This process may be thought of as a 3-body interaction, and it has a very dramatic T dependence.

Triple alpha process. Source.

Triple alpha process. Source.

The power laws for the reactions above are follow. Simply note the temperature dependence of each reaction. Each needs larger T than the previous, but the T-dependence is much stronger for the latter (indicated by larger powers). Thus a given increase in T causes a larger and larger increase in the reaction rates:

pp-chain-rate

CNO-cycle-rate

triple-alpha-rate

At sufficiently high temperatures, many more reactions become available, consisting on the formation of heavier and heavier nuclei. Examples are carbon burning reactions, yielding oxygen, neon, sodium, and magnesium, and oxygen burning reactions, yielding magnesium, silicium, phosphorous, and sulphur.

If we plot the binding energy per nucleon (nucleons = neutrons and protons), Eb/A, we obtain a maximum at iron, and a series of stability peaks corresponding to particularly stable nuclei (e.g. helium, carbon, oxygen). These “enhanced stability peaks” are caused by the shell structure of atomic energy levels (similar to the shell structure of the electrons around a nucleus).

Generally speaking, fusion reactions of elements with atomic mass larger than 56, tend to consume energy, while fusion reactions resulting in iron or elements lighter than iron tend to liberate energy.

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