Today in Astronomy 328 - physics.gmu.eduphysics.gmu.edu/~satyapal/astro328fall07/course... · 11 September 2007 Astronomy 328, Fall 2007 2 The Sun’s interior, on the average Since

11 September 2007 Astronomy 328, Fall 2007 1

Today in Astronomy 328

The Sun as a typical star:Central density, temperature, pressureThe spectrum of the surface (atmosphere) of the SunThe structure of the sun’s outer layers: convection, rotation, magnetism and sunspots

Figure: Solar eruptive prominence, seen in He II 30.4 nm from the EIT instrument on the NASA/ESA SOHO satellite (NASA/GSFC)


The Sun’s interior, on the average

Since we know its distance (from radar), mass (from Earth’s orbital period) and radius (from angular size and known distance):

we know the average mass density (mass per unit volume):

13

33

10

1 AU 1.4960 10 cm

1.98843 10 gm

6.9599 10 cm

r

M

R

= = ×

= ×

= ×

-33

31.41 g cm

4= only 26% of Earth's.

M MV R

ρπ

= = =


The Sun’s interior, on the average (continued)

For the average pressure, we use the equation of hydrostatic equilibrium, and assume (very crudely) that the gas pressure varies linearly from some central value to zero at the surface:

centersurface center

centersurface

2 2

P P PdP Pdr r r r R

GMGMr R

ρρ

−∆≈ = = −∆ −

= − = −

( )( )

center

8 33-2

10

15 -2 9

Then 2 2

6.67 10 1.99 10 1.41 dyne cm

2 6.96 101.34 10 dyne cm 1.3 10 atmospheres

GMPPRρ

−

= =

× ×=

× ×

= × = ×


The Sun’s interior, on the average (continued)

The Sun is an ideal gas:

Average particle mass is about the mass of the proton ( ), so

Now for some less-crude estimates….

PV NkT

P nkT kTm

=

= =ρ

(N= number of gas particles, n=number density: number of particles per unit volume, = avg. particle mass) m

( )( )( )( )

24 15

16

6

1.67 10 1.34 10K

1.41 1.38 10

=11.5 10 K .

mPT

kρ

−

−

× ×= =

×

×

241.67 10 gm−×


Central pressure in a star

The star’s center, being its densest and hottest spot, will turnout to be the site of virtually all of the star’s energy generation, so we will make somewhat more careful estimates of the conditions there.From hydrostatic equilibrium equation again:

But the pressure at the surface, P(R), better be zero because the star’s surface doesn’t move and there’s nothing outside the surface to push back, so

( ) ( ) ( ) ( )2 .

R R

r r

GM r rdPP R P r dr drdr r

ρ′ ′′ ′− = = −

′ ′∫ ∫

( ) ( ) ( )2 .

R

r

GM r rP r dr

rρ′ ′

′=′∫


Central pressure in a star (continued)

We can’t do the integral unless we know the density as a function of position, so instead we make a crude approximation:

and the integral becomes

3 M MV R

ρ ≈ ≈(ignoring dimensionless factors like 4π/3, because we’re just trying to get the order of magnitude right),

( ) ( ) 3

3 2 3 2 3

2

6

1R R

r r

R

r

M rGM GM rP r dr M drR r R r R

GM r drR

⎛ ⎞′ ′′ ′≈ ≈ ⎜ ⎟⎜ ⎟′ ′ ⎝ ⎠

′ ′≈

∫ ∫

∫



Central pressure (r = 0):

Lo and behold, a complete calculation for stars of moderate to low mass (Astronomy 553 style) yields

so we have derived a pretty good scaling relation for PC.

P GMR

r dr GMR

R

GMR

CR

≈ ′ ′ =

≈

z26 0

2

6

2

2

4

2

(still ignoring dimensionless factors)

2

419CGMP

R=



For the Sun:

So, for other main sequence stars, to adequate approximation,

33 101.99 10 g 6.96 10 cmM R= × = ×

217 -2

4

11

19 =2.1 10 dyne cm

10 atmospheres

CGM

PR

≅ ×

>

42 217 -2

4 2 419 =2.1 10 dyne cm .CRGM MP

R M R

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟≅ ×⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠


Central density and temperature of the Sun

Central pressure is a little more than a factor of 100 larger than average pressure. Guess: central density 100 times higher than average? (That’s equivalent to guessing that the internal temperature does vary much with radius.)

For the Sun, that’s not bad; the central density turns out to be 110 times the average density, Thus, since

As we will see in a couple of weeks, the average gas-particle mass in the center of the Sun, considering its composition and the fact that the center is completely ionized, is .

-3150 gm cm .Cρ =

241.5 10 gmCm −= ×

3 ,C M Rρ ∝

ρ⎛ ⎞⎛ ⎞

∝ = ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

3-3

3 150 gm cmCRM M

M RR


Central density and temperature of the Sun (continued)

But the material is still an ideal gas, so

Compare to : T doesn’t vary very much with radius.We can make a scaling relation out of this as well, to use in extrapolating to stars similar to the Sun but having different masses, sizes and composition:

615.7 10 K.C C C CC

C C C

P V P P mTN k n k kρ

= = = = ×

2 36

4

6

15.7 10 K for the Sun;

15.7 10 K .

C CC C

C

C

P m GM RT mk MR

RM mTM R m

ρ= ∝ = ×

⎛ ⎞ ⎛ ⎞⎛ ⎞= × ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠

T


Opacity and luminosity in stars

At the high densities and temperatures found on average in stellar interiors, matter is opaque. The mean free path, or average distance a photon can travel before being absorbed, is about = 0.5 cm for the Sun’s average density and temperature (given above).Photons produced in the center have to random-walk their way out. How many steps, or mean free paths, does it take for a photon to random-walk from center to surface? (PU problem 5.11.)

Suppose photon starts off at the center of the star, and has an equal chance to go right or left after each absorption and re-emission. Average value of position after N steps is

x x x x NN N= + + + =( )/1 2 0


Opacity and luminosity in stars (continued)

However, the average value of the square of the position is not zero. Consider step N+1, assuming the chances of going left or right are equal:

But if this is true for all N, then we can find by starting at zero and adding , N times (i.e. using induction):

x x x

x x x x

x

N N N

N N N N

N

+ = − + +

= − + + + +

= +

12 2 2

2 2 2 2

2 2

12

12

12

2 12

2

b g b g

.

xN2

2

x NN2 2= .



Thus to random-walk a distance , the photonneeds to take on the average

So far, we have discussed only one dimension of a three-dimensional random walk. Three times as many steps need to be taken in this case, so to travel a distance R, the photon on the average needs to take

x LN2 =

N L=

2

2 steps.

N R=

3 2

2 steps.



For the Sun, and for a constant mean free path of 0.5 cm,

Each step takes a time , so the average time it takes for a photon to diffuse from the center of the Sun to the surface is

Note that the same trip only takes for a photon travelling in a straight line.

N =×

= ×3 6 96 10

0 55 81 10

10 2

222

.

..

e ja f steps.

∆t c=

211 43

9.7 10 s 3.1 10 years.R

t N tc

= ∆ = = × = ×

/ 2.3 sR c =


The solar spectrum

By and large, the spectrum of the Sun resembles closely a blackbody.From the total energy flux at Earth (solar constant):

we get the Sun’s luminosity:

Setting solar luminosity equal to blackbody power gives Sun’s effective temperature:

f = ×1 36 106. erg s cm-1 -2

33 -13.826 10 erg sL = ×

2 44 5800 Ke eL R T Tπ σ= ⇒ =


The solar spectrum (continued)

In detail: absorption lines are also seen in the solar spectrum; they match up with many known transitions of atoms, ions and molecules. (See Lab #2.)

Figure: the ultimate high-resolution spectrum of the Sun (Nigel Sharp, from data by Bob Kurucz et al. ( NOAO/NSO/Kitt Peak FTS/AURA/NSF)


Formation of absorption lines in the solar atmosphere

Figu

re: C

hais

son

and

McM

illan

, Ast

rono

my

Toda

y


The outer layers of the sun

Convection:Opacity from atoms rises as one moves from center to surface (more atoms there; all ionized in the center).“Bubbles” can neutralize, cool and sink.Hot material rises to take its place.

Figure: Chaisson and McMillan, Astronomy Today


Solar “granulation:” the tops of convection cells



Further out: the corona and chromosphere

Corona: observed to be well over 1000000 K.Theory of corona: heated by acoustic noise from boiling top of convection zone; diffuse enough that it can’t cool very well, so it reaches very high temperatures.



Solar corona and atmosphere X-ray composite (NASA/SPARTAN)


Sunspots: solar magnetism

Sunspots appear dark because they’re slightly cooler than the rest of the solar surface.Zeeman effect measurements show that they are also maxima of magnetic field.Associated with other activity (e.g. prominences)



Sunspot progress, activity during solar cycle



11-year sunspot cycle, for the last 400 years



Sunspot formation and cycle: interaction of magnetism and differential rotation

The Sun rotates, but not as a solid body; this differential rotation wraps and distorts an initially “poloidal” solar magnetic field.

Occasionally, the field lines burst out of the surface and loop through the lower atmosphere, thereby creating a sunspot pair. The underlying pattern of the solar field lines explains the observed pattern of sunspot polarities. If the loop happens to occur on the limb of the Sun and is seen against dark space, a prominence is visible.The twisting and wrapping of the field lines eventually results in the production of a poloidal field again, but with north and south switched. Then the process repeats.22 years between identical field configurations, 11 years between sunspot-number maxima.


Sunspot formation and cycle: interaction of magnetism and differential rotation


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Today in Astronomy 328 - physics.gmu.eduphysics.gmu.edu/~satyapal/astro328fall07/course... · 11 September 2007 Astronomy 328, Fall 2007 2 The Sun’s interior, on the average Since