Download Solar evolution and the distant future of Earth

Solar evolution
stronomy textbooks tell us that
one day the Sun will expand to
become a red supergiant of enormous
size, finally swallowing its inner
planets, including Earth. However,
recent solar evolution models, which
account for a realistic amount of mass
loss, suggest a (slightly) less
catastrophic future for our planet.
A
I
Solar
evolution
and the distant
future of
Earth
Peter Schröder, Robert Smith
and Kevin Apps take a
speculative look at
what the future may
hold as the Sun
becomes a
supergiant.
6.26
t has long been known, and all recent stellar evolution computer models agree with
this, that solar-type stars at the end of their
“lives” expand to red supergiants of several
hundred times their initial diameter. This is a
reaction of the outer layers of the star to the
significant changes deep down in the stellar
core, which contracts significantly after it has
consumed its reservoir of hydrogen fuel. In
fact, any solar-type star goes through two such
phases of expansion. The first one ends with
the ignition of He-burning in the core, after
which the stellar core immediately goes into a
less condensed state and the outer layers
become less expanded. Stars in this first expansion phase are identified as RGB (Red Giant
Branch) stars in the Hertzsprung–Russell (HR)
diagram, the common astrophysicist’s plot of
stellar luminosity against temperature.
After the stellar core has consumed its helium,
the core must contract yet again, which leads to
the second expansion of the outer layers.
Supergiants in this phase are slightly less cool
and red, and they have been identified as AGB
(Asymptotic Giant Branch) stars in the HR diagram. RGB and AGB supergiants both become
not only much larger but also much more luminous than the original star used to be. These
supergiants obtain their energy mostly from a
hydrogen-burning shell around their dense core
of nuclear ashes. As the core becomes denser
and more massive with time, rising temperature
and pressure in the hydrogen-burning shell
increase the energy output.
Astronomy textbooks often have the following scenario for the end of the Earth (e.g. Prialnik 2000 p10, Hellier 2000 p45). In about
7.5 billion years’ time, our planet is expected to
be swallowed by the solar red supergiant when
it has expanded beyond the orbit of Earth. Figure 1 illustrates the classical evolution model of
the solar radius by comparison to the orbital
radii of the inner planets, according to our
computations with a well-tested evolution code
(see Pols et al. 1998 for a description of the
code and further citations). To make things
worse, already 2 billion years earlier, in about
5.5 billion years from today, the increasing
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Solar evolution
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solar luminosity will have doubled and turned
our planet into a no-go zone for life. Eventually, on the tip AGB, solar luminosity will peak at
several thousand times its present value.
The AGB expansion comes to an end only
when too little envelope mass is left to feed the
hydrogen shell-burning zone. At that point, the
star disperses the remains of its thin outer layers, ends its active energy production, and only
leaves behind its hot core, which would be
observed eventually as a “white dwarf”.
For a long time, computer models of solar
and stellar evolution have differed only in marginal (but labour-intensive) details. One of
these important details is the exact combined
opacity of metal and molecular lines, resulting
in slightly different radii for the extreme RGB
and AGB phases of solar evolution models. But
the general picture is mostly still agreed upon,
at least in most textbooks: an abrupt end and
complete disintegration of planet Earth by
merging with the solar supergiant seemed
inevitable. (For a somewhat more massive star
with a planetary system, the star might develop
a thick circumstellar envelope that would
affect a much larger fraction of the system:
there is tentative evidence from the presence of
water vapour [Melnick et al. 2001] that the
carbon star IRC+10216 is evaporating its
equivalent of the Kuiper belt, the source in the
solar system of the short-period comets.)
There has been one remarkable exception.
Sackmann et al. (1993) considered an important extra detail – mass loss – and their model
of the future Sun held good news: Earth may
survive after all, at least as a planet! But their
predicted escape is a narrow one and depends
on precisely the right amount of mass loss and
other details. So, before drawing any conclusions for the long-term prospects of planetary
real estate, we thought it was time to revisit
this point with our code, in a fresh and independent approach, and to elaborate the details
of the emerging picture of our distant future.
At the mercy of solar mass-loss
The main point, which indeed makes all the
difference, is stellar mass-loss. This is an onDecember 2001 Vol 42
1: The evolution of the solar radius without mass loss, compared to the orbits
of the inner planets, for the critical time during the RGB and AGB supergiant
stages of solar evolution. The yellow line shows the radius of the Sun in units
of its present radius, while the horizontal lines are the radii of the orbits of the
four inner planets in the same units. Mercury (red) and Venus (orange) are
swallowed by the Sun during the RGB phase, while the Earth (green) and Mars
(magenta) succumb at the later AGB phase about 120 million years later, when
the Sun becomes considerably larger. Note that the vertical scale gives the
logarithms of the various radii. Thus the radius of the Sun is already 10 times
its present radius at the start of this phase of evolution and increases to nearly
1000 times at the AGB tip.
going process while a star is a red supergiant.
The low gravity in combination with energy
deposition into the supergiant atmosphere by
complex hydrodynamic processes and by
absorption of stellar radiation leads to a “cool
wind” that steadily carries away mass from the
outer layers of the supergiant. The mass-loss
rate increases with the expansion of the supergiant as a result of the decreasing surface gravity and the final stages of the supergiant, which
come with significant mass loss, are shortlived. However, for a star with a mass like the
Sun, we find that the moderate loss during the
earlier giant stages on the RGB already
removes a considerable amount of mass, mainly during the final approach to the RGB tip.
This has long been known from modelling Heburning stars in the old and star-rich globular
clusters. Depending on the specific model
assumptions and estimated age, evolutionary
models of those He-burning stars in globular
clusters give them an initial mass of about
0.9 M (M is the symbol for the present solar
mass; we shall use M or M(t) for the mass of
the Sun at a general time) and a present mass
of just under 0.6 M.
Globular cluster stars have already been used
by Sackmann et al. (1993) to estimate the
expected mass loss M· of the late stages of the
Sun. They adopted the Reimers (1975) parameterization, assuming that mass-loss depends
only on the stellar luminosity L, mass M and
radius R (all taken in solar units), namely:
LR
M· = –η(4×10–13 M /yr) (1)
M
with a preferred η of 0.6, but they also tried
η = 0.4 and 0.8.
In our attempt to model stellar mass-loss in a
uniform, canonical way, we took an independent approach to several details which are relevant to the maximum radius of the supergiant
Sun. As described in more detail in Schröder et
al. (1999), apart from modern opacities and
neutrino loss rates, we use mixing parameters
that are carefully calibrated with precise properties of real stars. In particular, we made a
gravity-dependent adjustment of the mixing
length to match modern observations of
effective temperatures of tip-AGB stars. By
comparison, the Sackmann et al. models
appear to be too warm by about 10%, which
on its own would result in 20% smaller radii.
We then revisited mass-loss studies from
well-observed K supergiants. In fact, modern
work (Baade 1998) finds values about 5 times
lower (corresponding to η ≈ 0.2) than the original work of Reimers, three decades ago,
would suggest. With η = 0.2, our models of
stars like the Sun, including stars a little less
massive, reach He-burning after a mass lost on
the RGB of around 0.3 M. This is consistent
with globular cluster observations, but
remains a rather conservative estimate, since
giants with solar metallicity will be cooler and
may lose even more mass than the metal-poor
stars in globular clusters did. Our models also
consider dust-driven mass loss which, however, does not become important in the case of
the Sun.
Specifically for the peak of the Sun’s expansion (figure 2), the solar supergiant will have
already lost almost 0.2 M , similar to the predictions made by Sackmann et al. (1993). This
still conservative amount of mass-loss works in
favour of the Earth in two ways:
● Assuming that the Earth conserves its
orbital angular momentum, the orbit of the
Earth will expand proportional to 1/M(t), to
185 million km. This is comfortably enough to
escape the predicted maximum solar RGB
radius of 168 million km, although the logarithmic difference of only 0.04 is barely
detectable in figure 2. Interestingly, the Sun’s
increase in radius is actually larger in the RGB
phase than in the case without mass loss,
because of the decreased surface gravity after
mass loss. It is only the expansion of the
Earth’s orbit that saves it from being engulfed.
● There will be much less mass left in the outer
solar layers to feed the hydrogen shell-burning
zone during the second expansion of the Sun.
Our model computations for the solar AGB
supergiant suggest a maximum AGB solar
radius of 172 million km, much less than in the
case with no mass loss and barely larger than
in the RGB phase. At the same time, the Sun
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age (106 years)
2: The evolution of the solar radius, taking account of mass loss. The yellow
line shows the solar radius in units of its present value, while the green line
shows the Earth’s orbital radius in the same units; both of these curves are
plotted on a logarithmic scale, and can be directly compared to figure 1. The
blue line shows (on a linear scale) the mass of the Sun in units of its present
mass. The changes in the Sun’s mass can be seen to occur principally in two
brief phases near the RGB and AGB tips, and these rapid changes are
reflected in a rapid expansion of the Earth’s orbit at these phases, which just
keeps ahead of the expanding photosphere of the Sun.
Table 1: No place like home
The range of temperatures and times for which human life might
survive on various planetary bodies.
will have lost more mass and weigh only
0.68 M which enlarges the orbit of the Earth
to 220 million km.
Hence, we find that the inclusion of solar
mass-loss keeps planet Earth clear of the solar
photosphere by a small but crucial margin at all
phases of the Sun’s evolution. While the solar
luminosity will have increased to about 2800
times (tip RGB) and 4200 times (tip AGB) its
present value, the solar photospheric temperature will have dropped from 5800 K today to
only about 2700 K for the supergiant.
Compared to the earlier predictions of Sackmann et al. (1993), our results are quite similar, but there are differences in the details of the
picture. First of all, our models are a more realsitic match to modern observations – in particular, they predict a slightly cooler solar supergiant. As a result of all the differences, we find
that the future mass-loss is more concentrated
in the very tips of the RGB and the AGB than
is predicted by Sackmann et al. and so is the
expansion of the orbits. For Venus, this has
fatal consequences: while Sackmann et al. predicted that it would escape along with the
Earth, our models suggest otherwise.
The losers: Mercury and Venus
The inner planets Mercury and Venus will not
benefit from the moderation of our distant
future by solar mass-loss. Mercury will be consumed rapidly by the swelling Sun, long before
the Sun gets anywhere near the tip of the RGB,
and indeed before the orbit has a chance to
expand significantly.
Venus has pretty much the same size and mass
as Earth, and it is an intriguing example of
what was thought to happen to us in the future.
The current orbital distance of Venus, 108 million km, will expand due to the predicted solar
mass loss to 134 million km by the time the Sun
approaches the tip of the RGB. That is not
enough to escape the predicted maximum solar
RGB radius of 168 million km and Venus will
find itself some 30 million km below the Sun’s
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Planet (satellite)
Range of
Range o
photosphere.
ages
Once inside the stellar photoTplanet /Tplanet(0)
in 109 years
sphere, the orbit of Venus will
decay rapidly because of gas
Mars
1.29–1.43
11.6–11.7
drag. From our evolution model
Jupiter (Europa)
2.25–2.50
12.07–12.10
of the future RGB Sun, we find a
Saturn (Titan)
3.18–3.53
12.139–12.147
gas density in the outer layers of
Uranus (Oberon)
5.09–5.66
12.162–12.164
about 2.4×10–6 kg m–3. The drag
Neptune
(Triton)
8.44–9.38
never
force on a spherical body moving through a gaseous medium
can be expressed as (Weidenschilling 1977):
our moving out into interstellar space, to find
F = 0.5πCD ρv2 r 2
another star system? To answer this, we can
(2)
make a simple calculation that will enable us to
where CD is a dimensionless coefficient
predict the temperature at the surface of any
approximately equal to 0.44, ρ is the density of
planet or satellite as the Sun expands.
the gas, v is the velocity of the body relative to
Suppose that the luminosity of the Sun is L
the gas and r is its radius. If we assume v to be
and that the (black-body, or effective) temperequal to the orbital velocity of Venus, neglectature of a planet or satellite of radius R at a
ing any rotational velocity of the giant star
distance D from the centre of the Sun is T; L
which is likely to be much lower, and consider
and T are in general time-dependent as the Sun
Venus’s current mass and radius, we obtain a
evolves. If the planet has an albedo A, then it
deceleration of about 1.5×10–8 m s–2.
absorbs a fraction (1–A) of the incident radiaThis indicates that Venus’s orbit will decay
tion and in equilibrium between absorption
with a timescale of less than 105 years. In fact,
and reradiation we have:
the orbit will shrink by 0.01 AU in just about
L
1000 years, and the gas drag will then increase
2
(1–A)πR
= 4πR2 σT 4
(3)
4π D2
due to both the rising gas density and the rising orbital velocity. Hence, Venus clearly will
where σ is the Stefan–Boltzmann constant.
not survive the 5×105 years duration of the
This is more usefully written in terms of the
radius and effective temperature of the Sun,
remaining RGB phase. However, the orbital
RSun and TSun , as:
decay time of a planet like Venus or Earth is
longer than brief radius changes due to such
R 1/2
TSun
(4)
T = (1–A)1/4 Sun
short-lived events as thermal pulses.These can
2D
therefore be ignored in this respect: even if the
where the subscript “Sun” is used to avoid
tenuous outer envelope of the supergiant
confusion with the planetary values. Here we
briefly swept over the Earth, during a pulse,
have used the definition of the effective temthe encounter would be too brief to cause any
perature of the Sun, L = 4πRSun2 σTSun4 .
significant orbital change. In fact, we do not
For the present Sun, this gives a mean temresolve thermal pulses for reasons of simplicity
perature for the Earth (assuming an albedo of
and computing speed.
0.3, although the result is not very sensitive to
the exact value) of 255 K. This temperature
When it is time to move
needs some explanation. First of all, it is too
low by about 10%, because it takes no account
Is there anywhere in the solar system that
of the Earth’s greenhouse effect (or of its interwould be safe at the height of the Sun’s luminal radioactive heat sources), but we shall use
nosity, or does our survival as a race depend on
December 2001 Vol 42
Solar evolution
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(a)
Tplanet /Tplanet(0)
Tplanet / Tplanet (0)
1.6
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5
4
3
2
1.0
0.8
(b)
6
1.8
0
2000
4000
6000
age (106 years)
8000
10000
12 000
1
12 000
12 050
12 200
12 100
12 150
age (106 years)
12 250
12300
3: The surface temperature of any planet (or satellite) relative to its present temperature. (a) The predicted variation for the first 12 billion years of the Sun’s life, up
to the point where it starts to climb the RGB. The Sun is at about 4.5 billion years. (b) An expanded look at what happens on the RGB and AGB. The maximum ratio is
about 6.6. There is a period of about 100 million years between the RGB and the AGB when it would be possible (maybe even necessary) to survive if we moved
back closer to the Sun, but none of the planets in that region would by that time be worth inhabiting. Not only would there have been a greatly enhanced solar wind
during the ascent of the RGB but the atmospheres of the planets would have been evaporated long before that by the strong temperature rise (Jones 1999, p360).
it as a rough guide. The actual mean temperature of the Earth is about 288 K, or 15 °C, and
in what follows we shall assume that the
changes in temperature calculated from the
above equation are about right but that the
present Earth’s temperature is 288 K. Secondly,
whether we take it as 255 K or 288 K, the temperature is an average over the whole Earth,
from the very hot equatorial regions to the
freezing polar ones, a range of more than
40 °C. A 30° rise in the mean temperature
would make the equator uninhabitable, but
would leave habitable zones near the poles.
This immediately raises the question of how
soon the Earth will be uncomfortably hot
everywhere, and where we might go then. Is it
possible to hop outwards from one planet or
satellite to the next, always keeping ahead of
the Sun? To find out, we need to know how the
temperature of any planet varies with time as
the Sun evolves. Taking proper account of the
increasing orbital radius which results from the
mass loss from the Sun, it can be shown that:
L 1/4 M 1/2
Tplanet = Tplanet(0) (5)
L
M
where Tplanet(0) is the present temperature of
the planet and T, L and M are time-dependent.
This function is illustrated in figure 3.
As a first application, let us ask how long it
will take for the temperature of the Earth to
rise by 5° (the rise anticipated in the next century or so if the current human-induced greenhouse effect continues unchecked). The equation predicts that it will take the evolving Sun
about 0.8 billion years to produce this rise – so
human activity may be accelerating astronomical effects by a factor of about 10 million.
The observant reader will note from figure
3(a) that, on this model, the mean temperature
of the Earth when the solar system was formed
was substantially less than it is now. A detailed
calculation shows that it was only 233 K, cold
enough for the oceans to have been frozen
solid – how, then, did life form as early as
650 million years after the Earth itself? It is
December 2001 Vol 42
now generally believed that the Earth’s early
atmosphere contained much more carbon
dioxide than it does today and that a greatly
increased greenhouse effect kept the Earth’s
temperature high enough for liquid water to be
present. The carbon dioxide content declined
slowly, at a rate that kept pace with the slowly
increasing input of energy from the Sun, so
that the temperature of the Earth seems to have
stayed fairly constant over its entire life to date
(Jones 1999); this remarkable result led to the
hypothesis by James Lovelock that the whole
Earth and its biosphere is a self-sustaining system: the Gaia hypothesis. If correct, this could
lead to the Earth resisting, for longer than one
might naively expect, the gradual rise in heating by the Sun in the future.
However, it seems inconceivable that even
the Gaia effect could continue to be effective
for long after the Earth is hot enough for the
oceans to boil. Let us suppose that all forms of
life will be extinguished by the time the Earth’s
mean temperature has reached 380 K (a rise of
125°). Figure 3 predicts that this will happen at
an age of 11.8 billion years, as the Sun starts its
ascent of the red giant branch.
Of course, this means that humans will need
to leave much earlier in order to continue living in something like our present conditions –
probably before the model temperature reaches 300 K (a rise of 45° in the mean temperature
to about 60 °C for the real Earth, as opposed
to the too-cool model); this occurs at 10.2 billion years. Encouragingly, this gives us 5.7 billion years to figure out how to move house.
Let us suppose that we want to find a planet
or satellite with a mean temperature of 285 K
(12 °C) (here we neglect any greenhouse
effect, because Mars and the satellites of the
outer planets probably don’t have the right
conditions), and that human life can live there
for mean temperatures between 270 K and
300 K. Then figure 3 tells us the range of
times for which the conditions are right on
various bodies; the results are given in table 1,
assuming that the current albedo of the body
does not change. (In practice, the albedo is
very likely to change, but we do not attempt to
model that: table 1 is thus only illustrative.)
Table 1 shows our dilemma in a stark way –
there are periods when we could in principle
survive on one of the outer bodies, but there
are also long periods when none of these bodies is at a suitable temperature. In particular,
we shall have to leave the Earth before even
Mars is warm enough.
We had better get used to the idea that we
shall need to build our own survival capsules –
the planets are simply too far apart for planethopping to be a viable solution. Perhaps this is
the ultimate justification for developing the
International Space Station. ●
Klaus-Peter Schröder and Robert Connon Smith are
in the Astronomy Centre of the University of
Sussex, Brighton; Kevin Apps has recently
graduated from there with an MPhys in physics
with astrophysics.
Acknowledgement. We owe the idea for this look at
the distant future of Earth to a supposedly innocent
question posed by our colleague Andrew Liddle,
when preparing for a public lecture. We thank a
referee for drawing our attention to the article by
Sackmann, Boothroyd and Kraemer.
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