The Sun

The Sun dominates the Solar System in many respects (e.g., mass, energy production), but luckily for astronomers, the Sun is a very average star and therefore useful as a basis for understanding all stars. The output characteristics of the Sun define the habitable zone in the Solar System. It is the only star whose surface we can study in any kind of detail!

Sun's Overall Properties

• produces 4x1026 watts of energy and based on the Earth's fossil record, has been doing so at virtually the same rate for over 3 billion years.
• has a mass of 2x1030 kg and an average density of 1400 kg/m3 (recall the computation of the density of Jupiter which illustrates that even an object comprised largely of H can be this dense; H is in a different state in the Sun than in Jupiter, however)
• has a surface temperature of 5800ºK (which implies that it must be a compressed gas because no materials can remain solid or even liquid at this temperature)
• rotates in about 25 days at its equator; it rotates differentially confirming its gaseous character
• has a magnetic field

How do we know the Sun's power output?

By measuring the radiation received just outside the Earth's atmosphere and applying our knowledge that the Sun's radiation will have been spread over a sphere whose radius is the Earth's distance from the Sun:

flux f = 1370 watts/m2 received from the Sun at the Earth, integrated over all wavelengths so

LSun = 4 d2f = 4 (1.50x1011 m)2 1370 w/m2

= 3.9x1026 watts

We can determine the Sun's temperature either by measuring the peak of its output or by appealing to use of the Stefan-Boltzmann law and knowledge of the Sun's radius Rsun:

(Lecture 4 also presented this calculation)

We can also make estimates of what conditions must be like in the Sun's interior:

Pressure: Keeps the Sun from collapsing under the force of its own gravity. Estimate by subdividing the Sun into two halves and computing the gravitational force of the two halves on each other:

The force from gas pressure must equal the gravitational force (this requirement is also call hydrostatic equilibrium):

This estimate really gives a value approximating the average pressure; the central pressure in the Sun is higher with a value of about 3x1011 atm!

From this pressure estimate we can derive an estimate of the average temperature in the interior of the Sun:

Again, this is an average of sorts of the interior temperature; the central temperature in the Sun is 1.5x10K.

How can the Sun produce this much energy for billions of years? The real issue is not the quantity of power but the length of time over which the Sun has been producing at this level.

-- at the end of the 19th century, this issue was a genuine puzzle.

-- chemical processes such as burning of coal would require combustion of 1500 pounds of coal per square foot of surface area so this would keep the Sun going for only a few 10s of 1000s of years

-- Kelvin and Helmholtz proposed that perhaps gravitational contraction of the Sun was the answer:

The gravitational potential energy between two particles is

This can be generalized to an sphere with uniform density (not quite correct for the Sun but this we give us a reasonable estimate of the gravitational self-energy of the Sun):

The length of time that the Sun could radiate by shrinking is the total energy available divided by the rate at which it is emitting:

Obviously this is not the answer for supplying energy for more than 3 billion years!

We now know that nuclear fusion supplies the energy for the Sun The basic principle is

One fusion of 4 H to He releases 4.x10-12 joules so

so computing a time scale as before:

What else can we learn about the Sun by observing its surface?

Photosphere = layer at which the Sun becomes opaque to radiation; "surface" of the Sun but is not solid! Note that the darkening apparent towards the limb in the visible light picture of the Sun above is the result of viewing through a longer path length of cooler material just above the photosphere.

Chromosphere = cooler region just about the photosphere, density is relatively low so this region does not emit strongly. During an eclipse, the chromosphere can be seen as a reddish layer in Ha light.

Corona = the Sun's outermost layer that merges with the solar wind. The gas is of very low density but very high temperature (1,000,000ºK). Magnetic disturbances may accelerate the gas particles to produce these high temperatures. Because of the very high temperature, this region is bright at x-ray wavelengths.

Transition zone = region between the chromosphere and corona where the gas temperature rises rapidly in a short (10,000km) distance.

A key issue in understanding the Sun is understanding how the energy produced in the core reaches the surface of the Sun where it can escape.

Energy Transport Mechanisms

Conduction - heat is transmitted by electrons moving in a medium

Radiation - heat is transmitted by photons

Convection - heat is transmitted by bulk motion of a gas or liquid

Which of these mechanisms is important inside the Sun?

• Conduction is unimportant because the density of the Sun is too low for energy to be moved as rapidly by this mechanism as by the other two. In stars with extreme densities such as white dwarfs, conduction can be important.
• The relative importance of radiation or convection will depend on several factors and varies with position within the Sun.

Radiative Transfer in the Sun's Interior

For regions close to the thermonuclear burning core, radiation dominates the outward movement of energy.

Photons do not just fly out of the interior of the Sun (what would the Sun's temperature be if they did?).

• in fact, a single visible light photon's worth of energy takes about 30,000 years to escape (as compared to the direct flight time of about 2 sec!).
• photons get emitted and re-absorbed many times in traveling from the core to the surface. A photon executes a random walk from the core to the surface (or you could say that it diffuses from the center to the surface).

This is not quite the whole story on how energy escapes from the Sun - the outer layers of the Sun just beneath the photosphere are quite opaque. This opacity essentially resists the flow of radiation and too little energy would be transported from the interior to match the amount being radiated away at the surface. Consequently convective regions develop where energy is transported upwards by underlying hot material rising - the material eventually cools and sinks back down. The convective layer in the Sun stretches from the photosphere to about 1/3 of RSun. Cooler stars have even higher opacity and hence larger convective zones.

The granules seen in pictures of the photosphere are direct evidence of convection.

Sunspots

• appear to be magnetic storms in the photosphere
• are ~1700° K cooler than the surrounding photosphere so they appear dark
• are depressed in elevation relative to the average level of the photosphere by ~700 km
• their magnetic fields are 1000s of times stronger than the surrounding field
• occur usually in pairs with one member having a N polarity and the other S
• they can persist for up to a couple of months
• prominences and flares erupt from the vicinity of sunspots

Solar Magnetic Field and Sunspots

We knew that the Sun had a strong magnetic field and that sunspots have even stronger fields than in the quiescent Sun without sending any space probes to measure the field.

How? -- By taking advantage of the Zeeman effect the magnetic field in a star can be measured. The Zeeman effect arises because an electron orbiting an atomic nucleus is a moving electrical charge which generates a magnetic field.

Electron-generated field's orientation with respect to external field:

 Energy Level Aligned lower Perpendicular unchanged Opposite higher

So the electron's energy levels are slightly altered by an external field and hence the emitted or absorbed wavelengths are slightly changed. Note that a spectrograph has to have good performance to detect Zeeman splitting as the wavelength shifts are rather small!

One model for sunspots is that the magnetic field in the Sun's surface becomes increasing tangled due to differential rotation.

Eventually the field breaks through the surface. This break allows the field to flow from one polarity of sunspot to the other. The field which has broken through the surface is aligned perpendicular to the surface at the break and suppresses upward convection and only cooler, downwards moving gas is present which accounts both for the lower temperature and depressed elevation.

Prominences are eruptions of ionized gas flowing along magnetic field lines near sunspots. Hot gas boils out of the Sun's surface. Cools, and falls back down. A prominence may last several weeks.

Flares happen in short time scales and last only a few minutes. Near a large sunspot, 100 flares a day may occur. These are very energetic events with large quantities of hot gas and x-rays being emitted - a flare may have an energy of 1025 Joules and a typical temperature of 2x10K.

Flares can be dangerous and/or annoying to humans:

• an astronaut caught outside the shuttle could get a lethal dose of radiation
• the x-rays can ionize the outer layers of the Earth's atmosphere causing disruptions to long distance radio communications and even power surges in power transmission lines.

Both flares and prominences arise due to kinks or disturbances in the Sun's magnetic field but a complete understanding of these phenomena is still lacking.

Solar Cycle

The numbers of sunspots follows an eleven year cycle called the solar cycle. This is actually only half of the complete pattern where for 11 years, the sunspot leading a group will have one magnetic polarity which reverses for the next 11 years for a complete cycle of 22 years.

The average location of sunspots in latitude also changes throughout the solar cycle with spots more likely to occur at higher latitudes right after sunspot minimum.

Weather on Earth shows some correlations with the solar cycle - the Maunder minimum in the 1600s when very few sunspots occurred and at least Europe suffered a "mini-ice age" is striking. Exactly what the physical connection is is not clear but must be related to how flares disturb the upper layers of the Earth's atmosphere.

To understand how the solar cycle might occur, it is important to understand that magnetic field lines are essentially tied to the ionized gas in the Sun. If the gas flows, the magnetic field is dragged along with it. A model for the solar cycles postulates that at sunspot minimum, the Sun's field is smoothly distributed.

As a result of the differential rotation of the Sun, the magnetic field lines become more and more distorted. Sunspots erupt where the field pokes out the surface. The field eventually gets so tangled that it cancels itself. The field then gets re-established in the smooth configuration but with opposite polarity.

-- Why the solar cycle occurs and how the field re-established itself is an almost complete mystery!