Miras, Mass-Loss, and the Ultimate Fate of the Earth
L. A. Willson & G. H. Bowen, Iowa State University
Fire and Ice:
Some say the world will end in fire,
Some say in ice.
From what I've tasted of desire
I hold with those who favor fire.
But if it had to perish twice,
I think I know enough of hate
To know that for destruction ice
Is also great
And would suffice
Robert Frost
Whether the Earth ends in fire or ice depends how, when,
and how fast the Sun sheds mass.
Early or extended mass loss: Earth survives the red
giant stages of the Sun and ends "in ice".
Late, abrupt mass loss: Earth is engulfed in the bloated
Sun near the end of its red giant evolution.
1
1. Stages of Solar Evolution
Main Sequence
1010 years (=10 Gyr) of
4H
He in the core, then
Red Giant
Branch (RGB)
4H
He around a core of He
to maximum L~2000Lnow
Helium Core
Flash
abrupt start to He to C and O
core He to C and O at ~ 100LnowV
Horizontal Branch
Alternating H and He "burning"
until the C+O core ~ 0.6 MSun, when
mass loss removes the rest.
Asymptotic Giant
Branch (AGB)
The post-main sequences stages leading up to "the end" take
an additional 2x109 years.
10,000
1000
Mass loss
Post-AGB
Asymptotic Giant
Branch
Red Giant
Branch
100
L/Lnow
Pre-main
sequence
10
1
Main sequence
6000 5000
4000
Surface Temperature, Kelvins
2
3000
Increasing Solar Power => increasing T for the planets
Radiative equilibrium temperatures for the planets are
approximately given by
TRE = 290K (L/now)1/4 / (distance in AU)1/2
600
°C
Mercury
200
Mercury
400
Venus
100
Earth
T,
Kelvins
0
Mars
200
-100
-200
0
0
1
0.1
0.2
1.5
log(luminosity)
luminosity / luminosity now
0.3
2
The evolution of planetary temperatures during the main
sequence (now to about 5x109 years ahead)
Habitable zone (approximate):
3
Extending this to the end of the AGB:
Rocky planet vaporization zone
Mercury
2000
Venus
Earth
Mercury
Jupiter
Mars
Saturn
1000
Uranus
Neptune
Habitable zone
0
1
0
1 log(L/now)
10
2
100
log(luminosity)
L/now
3
1000
time->
Now
End of
Main
Sequence
4
10,000
RGB tip
Horizontal
Branch
shell
flashing
starts
Mass Loss
ends the
AGB
The maximum temperature a planet achieves depends on
its distance from the Sun and the maximum luminosity of
the Sun - both of these, in turn, depend on the mass loss
that ends the AGB.
4
Most astronomers have been using Reimers' Relation as a
recipe for mass loss
= -dM/dt = η 4x10-13 (L/LSun) (R/RSun)/(M/MSun) MSun/year
where LSun, RSun, and MSun refer to our present-day Sun.
This relation was derived from a fit to observations of mass
loss from red giants, with the adjustable parameter added
later when it was found that the original relation killed stars
too soon.
-4
-5
η=
1
0.6
logM
-6
-7
-8
5.8
6.0
6.2 6.4 6.6
logLR/M
6.8
Sackmann, Boothroyd and Kraemer (1993, Ap. J. 418, 457)
modeled future solar evolution using Reimers' Relation with
η=0.6.
5
Evolutionary stages in the Sackmann, Boothroyd and
Kraemer models:
10,000
shell flashing
and mass loss
1000
100
Asymptotic
Giant Branch
Horizontal
Branch
Red Giant
Branch
L/LSun
10
Pre-main
sequence
Now
1
6000
5000
4000
3000
Surface Temperature, Kelvins
6
Compared with the radius of Earth's orbit, the Sun is small
until it reaches nearly the tip of the red giant or asymptotic
giant branch:
10,000
100 RSun
AGB
RGB
1000
100 RSun
30 RSun
100
L/LSun
10 RSun
10
3 RSun
1
6000
5000
4000
3000
Surface Temperature, Kelvins
7
In their models Earth escapes its fiery fate mainly because
0.275 solar masses are removed on the first ascent of the red
giant branch.
Their RGB mass loss is entirely due to the Reimers' relation there is no mass loss "event" associated with the helium core
flash.
8
Given an evolutionary track of the form
R = function (M, L, Z),
any mass loss law (L, R, M, Z)
may be expressed as (R, M, Z).
-4
Bowen
0.7
-5
1.0
Vassiliadis &
Wood
-6
log ,
MSun/yr
0.7
-7
Reimers
1.0
-8
0.7
-9
Earth's
orbit if
MSun = 0.7
Earth's
orbit now
150
200
300
400
Radius / current solar radius
Evolution ends near dlogL/dt = -dlogM/dt - closer to this for
steeper mass loss laws, but always close. This is indicated
by the squares (0.7MSun) and dots (1MSun).
9
Using the Bowen model results, one finds that AGB evolution
ends with a "cliff"
logM = -10 -8
-6
4
-4
4
2.8
2
2
1.4
M
Chandrasekhar
limit
1
1
0.7
core mass
1000
10,000
luminosity
whose position depends on mass and metallicity.
0.6
0.4
logM
0.2
Chandrasekhar
limit
0.01
0.1
0.3
0.0
0.001
Z/Z = 1.0
core mass
-0.2
3.0
3.4
3.8
4.2
logL
10
4.6
How can the empirical relation (Reimers' relation)
and the theoretical mass loss law (Bowen's results)
be so very different? Is one of these wrong?
A steep mass loss law => severe selection effects:
Reimers' Reln. for
η=1 is shown as a
dashed red line
-4
-5
logM
-6
0.7
2
1.4
1
2.8
4
The cliff is the bold
black line, with
mass labels
-7
Evolution follows the
blue arrows
-8
5.8
6.0
6.2 6.4 6.6
logLR/M
6.8
Reimers' relation reinterpreted: It tells us which stars
are losing mass, not how a star loses mass.
(Analogous to: The main sequence is not an evolutionary
track but the location of stars "burning" hydrogen in their
cores.)
11
The "cliff" edge stars are the Miras:
4
5
logL
1
4
0.7
"Cliff" edge
stars
3
2
2.2
2.4
2.6
log(Period, days)
2.8
The Sun will end up between the 0.7 and the 1 solar mass
points in this plot, depending on how much mass it loses at
the helium core flash.
With a very steep mass loss relation, it is less likely than for
the "Reimers' mass loss formula" case that the Sun will lose
much mass on the first ascent of the red giant branch. The
most likely cases are "all" or "nothing" with the possible
exception of a finite M ejected at the core flash.
12
Averaged over the pulsation cycle,
10
density
10
The density is
increasing with time;
the green curve
describes the density
near the end of the
AGB.
-9
-11
10 -13
rho(170)
rho(180)
rho(190)
H
10 -15
10 -17
10 -19
0
1
2
3
4
5
6
distance (AU)
10 -8
Earth's orbit
density
10 -10
rho(170)
rho(180)
rho(190)
H
Mars's orbit
10 -12
10 -14
10 -16
10 -18
0.8
0.9
1
1.1
1.2
distance (AU)
13
1.3
1.4
1.5
1.6
Assuming the Sun starts the AGB with a mass about the
same as it is today, we find:
time-averaged
x
density = 10 gm/cm3, x = -15
The Sun's
mass
decreases, and
Mars escapes
-13
1.5
Mars
R,
AU
-11
1.0
-9
Earth
max R =
228 x now
Venus
R=200xnow
0.5
R=150xnow
2000
0
3000
1
2
3
time in millions of years
4000LSun
4
At the crash:
RSun = 180 Rnow (varying from 173 to 188 with P = 313 days)
LSun = 2790 Lnow
At the end:
LSun = 3950 is the maximum achieved.
14
Shell flashes modify L and R on a scale of 1000-105 years*:
3.6
3.5
~105 years
3.4
logL
3.3
3.2
3.1
3
2.9
2.8
1.647 10 8
1.648 10 8
1.649 10 8
1.65 10 8
1.651 10 8
1.652 10 8
t
time (years)
200
Radius/RSun
180
160
140
120
100
80
60
5
1.6 10
1.8 10
5
2 10
5
2.2 10
5
2.4 10
5
2.6 10
5
2.8 10
5
3 10
5
3.2 10
5
F
time (years)
200
Radius/RSun
180
~1000 years
160
140
120
100
80
60
5
1.6 10
1.65 10
5
1.7 10
5
1.75 10
5
1.8 10
5
F
time (years)
This will need to be included in future mass-loss calculations.
*
Models by S. Kawaler using ISUEVO, February 2000
15
The final fates of the planets, their moons, and the asteroids:
Object
distance peak T
Final fate
from Sun reached
(now)
(L<4000K)
Mercury
0.387
***
Into the Sun, RGB
Venus
0.723
***
Into the Sun, early
AGB
Earth
1.000
<2306>
Into the Sun,
late AGB,
unless ∆MRGB>0.2 MSun
and its Moon
Mars
1.524
Crashes into Earth
before Earth dies.
1868
asteroids
Jupiter
Escapes
unless ∆MRGB ≈ 0
Small ones (<<100km)
spiral in to Sun after
Earth dies.
5.203
1011
Saturn, Uranus, and Neptune
Baked but not
destroyed
Also baked but not
destroyed
16