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Climate Hustle

Understanding Solar Evolution Pt. 1

Posted on 23 March 2011 by Chris Colose

When considering the evolution of planetary atmospheres, a critical factor that inevitably comes into the discussion is the gradual brightening of the sun over geological time.  Observations that Earth and even Mars likely had liquid water in their distant past, despite a less luminous sun, present a paradox (Sagan and Mullen, 1972): If the sun was so faint, how would the early planets not have been frozen over? Alternatively, if the planets were warm enough to support liquid water then, how are they not extremely hot now?  Examining this question is critical to paleoclimate and habitability issues in general, including understanding arguments on why greenhouse gas levels have been higher in the past while still being compatible with relatively cold conditions.

Our own sun is a G2V type star, the ‘V’ also indicating a ‘dwarf,’ or in other words a luminosity classification that tells us the star resides on the “main sequence” (i.e., a locus of points in a plot of stellar luminosity vs. temperature, known to astronomers as a Hertzprung-Russell diagram).  It is on the main sequence that the sun will spend most of its lifetime in a stable configuration with nuclear fusion as a power source.

 From http://casswww.ucsd.edu/public/tutorial/images/hr_local.gif

Stellar classifications (O, B, A, F, G, K, M), which have become even more finely tuned (for example with numbers, like A0, A1…A9, etc, as the field grows) are characterized according to their spectral properties.  Measurements of stellar energy distribution show many narrow wavelength bands with reduced fluxes (i.e., or spectral lines) due to absorption by atoms and ions in the surface layer of the star.  The strength of lines and the elemental abundances are in turn affected by temperature.  Hydrogen has the strongest lines for A, B, and F stars, while for ‘redder’ stars hydrogen lines are weaker, and heavy atoms may be seen.  M-dwarfs are low mass stars (between ~0.08-0.5 Msun) with temperature of about 2,400-3,900 K (compared to 5,800 K for our sun) and luminosities of 0.02-6% of our own sun.   Giants and White dwarfs are distinguished from main-sequence stars above.  Note that luminosity is a measure of the total power output (for example in Watts), so a star can still have a high luminosity even with a relatively low temperature if it is very big.  Likewise, white dwarfs are very hot, but are rather small and so have a weak luminosity.

In this post, we are interested in the question of why the sun actually evolves in the way it does. The implications for planetary atmospheres will be discussed in a follow up Part 2.

We cannot directly observe the evolution of a single star, since they evolve on timescales far longer than humans observe them, although how the luminosity can increase with time on the main-sequence is generally well-accepted.  This is through sound theoretical models, but also because we can observe clusters of stars elsewhere in the galaxy with hundreds of thousands of stars (and different masses) in order to build confidence in stellar evolution theory.

The key to understanding the evolution of stellar atmospheres relates directly to the principle of hydrostatic balance, the same fundamental force balance which allows Earth’s atmosphere to stand up as it does (rather than gravity collapsing all the air molecules into a thin layer near the surface).  In a hydrostatically balanced atmosphere, the downward force due to gravity is also compensated for by a force upward that results from the pressure of the underlying fluid.  In other words, hydrostatic equilibrium defines a balance between gravity and the vertical pressure gradient.  If the constraints imposed by hydrostatic balance were not obeyed to high accuracy, than even for a body as large as the sun, noticeable fluctuations in its radius would occur over a characteristic dynamical timescale of just many minutes to an hour.

The next step toward understanding stellar evolution is nuclear fusion, which occurs in the solar core.  It turns out that the easiest nuclear reaction is one between a proton and a deuteron.  The timescale for two protons to form a deuteron however is rather slow, on the order of 1010 years, but this slow rate helps to set the timescale over which the sun evolves on the main-sequence.  Once deuterium is formed, it can smash into another proton to make Helium-3, which can in turn react with He3 again, or for temperatures higher than about 1.4×107 K (achievable in the stellar interior), He3 prefers to react with He4 (various possible paths are shown here ).  In any case, what we’ve ended up doing is converted 4 hydrogen atoms into a Helium nucleus.  The decrease in hydrogen abundance as it is converted to Helium is critical to the sun’s evolution.  The important point for evolution on the main sequence is that this process leads to an increase in the mean molecular weight in regions where fusion is important. At still higher temperatures, other fusion chains like the CNO cycle or the triple-alpha process become important.  The ideal gas law gives a relation:

where P is the pressure at the center of the star, R is a constant, ρ is the density of the gas, μ is the mean molecular weight, and T is the temperature.  As μ changes, the temperature and pressure must also change to compensate, which in turns impacts the stellar luminosity.  How does this work?

Stars become helium rich over time only in the interior, while hydrogen remains abundant in the outer envelope, since the core is where nuclear fusion is most efficient.  In the core, there is a consequent reduction over time in the number of particles per unit mass.  From the ideal gas law, the decreased pressure of this sphere is no longer sufficient to support the overlying envelope, so if you were to imagine a hypothetical sphere drawn out in the sun, the increase in helium makes it impossible for the sphere to stay at the same radius. Contraction occurs and as the core density goes up, gravitational potential energy is released and (through something called the virial theorem) half of energy is radiated away and half increases temperature of the gas . The luminosity also increases, which is reflected in an increase in the solar irradiance striking a planet.  Gough (1981) proposed the following general equation to describe the luminosity as a function of time:

 

where L(t0) is the luminosity at the current age of the sun, t4.6 billion years, and L(t) is the luminosity at time t.  It follows for example that 3.5 billion years ago, the solar luminosity was at only ~76% of today's value, while during the Neoproterozoic ~700 mya near the last snowball episode, it was ~94% of today's value. 

Since there is still hydrogen in the core of the sun, slow evolution on the main sequence will occur for a few billion years still.  The central contraction will cause a hydrogen “shell” to get hotter and burn more strongly (note that eventually the CNO-cycle dominates, and energy generation becomes concentrated in a narrow region around a helium-rich core), and high interior temperatures and pressures are too high to be in equilibrium with gravitational forces. The extra energy output results in the dramatic envelope expansion that causes the sun to evolve onto the red giant phase.  This terminates the main-sequence phase of the star’s life, and by this time the surface of the sun will actually be somewhat cooler, but its radius will be extremely large (engulfing Mercury for example) and overall luminosity much higher.  Considerable mass loss can occur on the Red Giant Branch, in which case Venus and Earth's orbit will be moved outward (inversely related to the mass), potentially being saved from being completely engulfed.

The duration over which other stars will evolve on the main-sequence, as well as the rate at which end phases of its evolution cause  it to expand, contract, or change its surface gravity depend largely on its mass.  The evolution of stars will therefore differ depending on initial characteristics; for high mass stars for example, the details outlined above are modified somewhat in that they have a convective core, so the newly formed helium actually becomes well-mixed in the stellar interior. 

When considering stellar evolution and the prospect of life evolving, it is worth noting that the lifetime on the main sequence is inversely related to the mass, to a power generally between 3-4, a consequence of the efficiency at which they burn fuel.  This means large O-type stars may only last a couple million years on the main-sequence, while no one will ever find remnants of former main-sequence M-types, since their lifetime is longer than the current age of the universe.  M-types therefore are more stable than stars like our own with respect to luminosity variations.

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Comments

Comments 1 to 19:

  1. Chris

    Is the pressure exerted by the outward radiation flow in a main sequence star significant in holding it up against gravity? The text above suggest that it can be approximated as being a ball of gas with hydrostatic pressure only.
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  2. 1 perseus: quick estimate would be that the outward force from the Sun's luminosity is 1.3 x 1018 N or 0.2 Pa pressure at the surface.

    At the surface of the core, about 0.2 solar radius out, it would therefore be about 5 Pa. Looks pretty weak to me, considering Earth's atmosphere with Earth's gravity is about 100 kPa at the surface.


    Of course, perhaps I calc'd it wrong, but I think it's effectively ignorable.
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  3. I note that Gough's equation is linear; I have seen (somewhere) literature that graphs the time dependency relationship as logarithmic at ~ 7% per billion years. I'll see if I can find it, if nobody else knows.
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  4. Thanks Mark

    It seems as if Astronomy is not completely immune from myths either! There is a short discussion of the subject here

    Radiation pressure in stars
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  5. *blink*

    When 'skeptics' question the increase of solar output over time I've always just said, 'look... we have alot of stars to check it against'... but this works too. :]
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  6. OK, one reference I found is Guinan & Ribas 2002, ASP Conference series 269, 85, available here.

    The luminosity function (fig. 2) appears to be an exact match to the 7% per billion years log function (which isn't too much different from Gough's linear function). The equation would be

    L(t) = L(t0) [eln(.93)(t-t0)]

    ... where age t is expressed in Gyr.
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  7. In the interest of keeping a high standard here, I think I need to point out that why a star becomes a red giant is an unsettled question.

    "Despite all the investigation into
    the subject, the question has yet to receive an answer
    that is satisfyingly simple and sufficiently rigourous.
    There is still no consensus on why stars become red
    giants." (Stancliffe et. al. http://arxiv.org/PS_cache/arxiv/pdf/0902/0902.0406v1.pdf)

    I also think that showing ishochrones and/or evolutionary tracks would be more helpful than an HR diagram
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  8. perseus,

    In our sun, gas pressure is much more important, but radiation pressure can become significant in much larger stars.
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  9. #4 perseus: feel free to check my maths (please?! :P ), I did it by dividing the luminosity by the speed of light, since power/c = force for light. Then turning into pressure by dividing by the area.

    I assumed the radiation would be absorbed, I suppose it could be reflected and you'd have to multiply it.

    Iirc, luminosity grows as approximately M^4 but it's a long time since I did astrophysics! In that case some massive stars (50 times solar mass, say) would have significant radiation pressure at some levels.
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  10. Chris, thanks for the article. I enjoyed reading it - must be at least 20 years, maybe more, since I last read any detailed explanations of stellar evolution...
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  11. Great post Chris, thanks for the explanation on the topic. I had tried doing some superficial Googling on why stars increase in luminosity as they age, and the best source I could find was wikipedia's page which was actually very unspecific toward this aspect of a star's lifetime.

    Of course, when I say superficial, I'm talking about the hair of my chinny-chin chin superficial.

    I also think that quantum tunneling is a fascinating concept. I had always thought that the high internal density of our star forced fusion.
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  12. Great post - As an amateur astronomer (and I stress amateur), I really enjoyed this post and look forward to reading the sagan paper referenced in the first paragraph. I've read only the high level articles explaining the HR diagram, main sequence and stellar evolution but I enjoyed a more detail look at the science. For those of you, like myself, who couldn't access that link due to the pay fire wall, here is another location.
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  13. I just read that Sagan paper that was referenced in your first paragraph. There was an interesting portion that I'll quote here:

    "Major variation in the CO2 abundance will have only minor greenhouse effects because the strongest bands are nearly saturated. A change in the present CO2 abundance by a factor of 2 will produce directly a 2C variation in surface temperature."

    Granted that Dr. Sagan wrote that in 1972 and there have been many improvements in our understanding of how CO2 affects the Enhanced GHE. In 1972 the CO2 atmospheric concentration as measured by Mauna Loa was 326 ppm and the CRU temp data shows we have gained 0.5C since 1972 while increasing our CO2 concentration by roughly 70ppm or 21% over the 326 ppm level. So we are about 1/5th of our way to doubling CO2 from 1972 levels and we are on track to match the prediction made in this paper that was written almost 40 years ago.
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  14. garythompson,

    A good point, but I'd stress that CO2 is a big player in past climate, and there's no reason it cannot generate a large greenhouse effect. The key suggestion that the Sagan paper made was that Ammonia was a plausible greenhouse gas that could help rectify the faint sun problem, but it was later shown that this would be photochemically unstable in the atmosphere (e.g., Kuhn and Atreya, 1979). Although they were wrong, it is a great paper that paved way for an enormous amount of research...just how science should be.
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  15. In the past, the luminosity was less -- with a smaller diameter but a higher temperature.

    I was wondering what affect the change in temperature might make. While the overall energy would have been less in the past, the proportions in various bands would be different. Specifically, the proportion of IR would be less while the proportion of UV would be higher. In fact, I can easily imagine that the net amount of UV (and perhaps even visible) light would have have actually been greater from a dimmer yet hotter sun. This in turn would affect where within the atmosphere energy was absorbed. (It should be relatively easy to calculate the energy distributions, but I don't have numbers handy for the predicted temperature change over the last few billion years.)
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  16. tfolkerts,
    Actually, the sun was cooler in the past, and will continue getting warmer until it starts to leave the Main Sequence. For example, see fig.3 in Vandenberg et al. (http://www.astro.uu.se/~bg/Boundary.pdf) - this isn't the best reference for this purpose, but it does show a good example of a solar evolutionary track
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  17. Here is a good image for showing the evolution of stars at different masses, the movement for A to C being where our own sun is on the main sequence. Higher mass stars actually cool a bit as the luminosity grows on the main sequence, whereas the surface of our sun has gotten a bit hotter, but always bear in mind the luminosity depends on both the size AND temperature. A general rule of thumb is that if the core contracts (as I described in the post), the outer envelope often expands, and just like on Earth when you get rising air and enough expansion, it will cool.
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  18. Thanks for the links. I was apparently remembering some simplified evolutionary diagrams and/or diagrams for heavier stars.

    So I suppose my comment now works in reverse! Since the sun is WARMING, then proportionately MORE of the energy is currently in the UV. So UV is definitely stronger than before, while IR is staying closer to constant. In any case, the change in spectrum would be worth considering in addition to the simple change in overall luminosity. But I suspect that will some up in the next post. :-)
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  19. Actually the extreme ultraviolet was probably greater in the distant past, even with a fainter sun, so that does matter.
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