2 posts tagged “sun”
Jupiter is shining bright tonight, and with good cause. Three hundred and ninety-nine years ago today, Galileo Galilei changed the world, the nature of science, and the meaning of religion thanks to Jupiter. Using an improved telescope that he had invented, Galileo looked at Jupiter and saw that it had three companions; by the end of the week, he had found a fourth and proven that these small starry messengers revolved around Jupiter. Being a savvy sort, he published his findings in Sidereus Nuncius, a short treatise that was dedicated to Cosimo II de' Medici and called the four moons of Jupiter “Medicean stars”. We now know them as Europa, Ganymede, Callisto, and Io [1] and call them the Galilean satellites.
His discovery was first used as a method for keeping time [2], but it had even deeper implications. Under Aristotle’s view of the cosmos, the Earth was the center and everything revolved around it. Things in the heavens were perfect and pure, and were in heaven because they were pure and perfect. Because the ideology fit so well with the dogma of the Catholic Church, it was adopted as Church Law – to challenge it was to challenge the very essence of belief [3]. Though some troubling differences had arisen between the pure circles demanded by Aristotle and the observed paths of the planets, these were smoothed over by Ptolemy’s “epicycles” of circles on circles. Questioning these ideas was dangerous at best and heresy at worst [4, 5].
Galileo did worse than question them: he made it possible for anyone to see that he was right and the Church was wrong [6]. By simply looking through the telescope, people could see these new moons of another planet. They could see the “jug-ears” of Saturn [7]. They could see the phases of Venus [8]. They could see the spots on the face of the Sun and the scars on the face of the Moon [9]. And they could see that the Milky Way was neither food of the gods nor a nebula but thousands upon thousands of stars like our, scattered across the sky.
Galileo was first rewarded for his discoveries and then punished for his hubris. He became a superstar in Pisa, and other city-states wooed him, trying to get him to move and to bring his beautiful ideas with him. But his ego led him to clash with others, making enemies out of supporters. Eventually, he was brought before the Inquisition for heresy and threatened with torture. He renounced his views and spent the rest of his life under house arrest [10]. It would be 206 years before the Roman Catholic Church would take his works off of the banned list and 376 years before the Vatican would formally clear him of any wrongdoing.
In opening the heavens to us, Galileo laid the foundations of modern science. He showed that clear logic alone (Aristotle’s approach) is not enough. Logic must be backed with evidence and hypotheses must be checked against observations. He also started us on the road to discover who else is out there.
John
[1] Named after the lovers of Jove (Jupiter). Remember that every planet has a system for naming its moons:
In addition, each planet and other body has a unique system for naming its features (e.g., great lovers for features on Eros). Names for newly discovered moons, planets, and features must be approved by the IAU to become “official”.
- Mars the dogs of war Phobos (Fear) and Deimos (Panic)
- Saturn Titans
- Uranus Characters from a Midsummers’ Nights Dream and the Rape of the Lock
- Neptune Nymphs and children of the sea god
- Pluto Workers in Hades
[2] Why is time so important? Because it tells you where you are! From the angle that the sun makes with the horizon at noon, you can tell your latitude (how far north or south you are). But you need to know the time in order to determine your longitude; this is why locations are given in minutes and seconds. If you know the time to within one hour, you can determine your location to within 1700 km. If you know the time to within 1 minute, you can determine your location to within 27.8 km. If you know it to within one second, then you can determine your location with an error of less than 0.5 km. Until the creation of the first accurate, sea-worthy chronometer [a], the stars were the only way to determine time at sea. Using the Medicean stars, Pisan sailors were able to navigate more easily and more quickly across the oceans; it was this that made Pisa a naval power to rival England in the 1600’s. 300 years later, Lewis and Clark used the same method as they tracked across North America. And all of this came from “pure” research!
[3] Or so the Church scholars would have you believe.
[4] Copernicus published his heliocentric theory on his deathbed, and was still reviled in sermons sixty years later. Giordano Bruno held to a heliocentric universe and was burned at the stake for it (and other heresies).
[5] The Roman Catholic Church wasn’t the only group that wanted a geocentric universe. Nearly 1800 years earlier, Aristarchus had been threatened with expulsion from Samos for impeity because he had suggested that it was silly for a huge Sun to orbit a tiny Earth and wanted to have it be the other way around.
[6] While doing so, he also implied that the Pope was an imbecile. Many scholars believe that it was this, rather than his embrace of Copernican theory, that led to his troubles. Note to self: Don’t piss off the absolute ruler of the nation you live in when proposing a radical change to that nation’s beliefs…
[7] Or at least, most of the time, they could see them. This was one of the things that caused Galileo trouble – the rings are tilted and so change their apparent shape and width as Saturn moves in its orbit. When Galileo first saw them, Saturn was directly behind the Earth, so the rings stood out like the brim on a sombrero worn by a man standing behind you. When Galileo was trying to gather support, Saturn had moved in its orbit so the rings were nearly edge on (imagine that man and his sombrero moving over to your right – notice how the brim appears to get smaller?) and very difficult to see. As is the case with modern net trolls, Galileo’s enemies used this one change to argue that everything he did was a lie.
[8] Under the geocentric model, only the new and crescent phases were possible as Venus had to orbit between the Sun and the Earth. In the heliocentric model, all of the phases could be seen (and were).
[9] Not only did these allow Galileo to check Aristarchus’ estimate for the size of the Moon by comparing its mountains to those on Earth, it went directly against the belief that the Moon and Sun were perfect and pure bodies. The sunspots also allowed Galileo to measure the Sun’s rotation (another impossibility, according to Aristotle).
[10] Legends to the contrary, he is unlikely to ever have said “eppure si muove” (“And yet it moves”). To do so would have been foolhardy and needlessly brave – and Galileo was neither.
[a] Detailed in Sobel’s magnificent Longitude. England and France were locked in a battle to develop a way to determine time at sea (with Brussels a distant third); as a result, each had their own Prime Meridian. Britannia ruled the oceans because it was able to solve the problem before France.
[b] Galileo's finger, saved as a reliquary. Shown as requested by MadTante.
You gave me such good feedback on the last section that I'm posting this one, too.
10.2 The Sun
Just 149,000 km away, or about 390 times the distance to the Moon, the Sun (also known as Sol) is the nearest star and source of nearly all of the energy on Earth. With a surface temperature of 5780 K, it emits 3.827 × 1026 W of energy as light; for comparison, the US used an average of 3.3 x 1012 W in 2005. The Sun is primarily made up of hydrogen and helium gas with the electrons stripped from the atoms. This is called a plasma, and is sometimes referred to as the fourth state of matter[1]. Because the electrons and protons are not bound together, no chemical compounds can form and no chemical reactions can take place in its interior. Thus, only pure elements are found in the Sun. The strong electromagnetic forces can lead to unusual effects such as the creation of filaments [2] and help the Sun's outer layers propagate seismic waves; P-waves have been observed on the Sun's surface (Fig 10.2-1), caused by changes in its outer layers. Mapping the pattern of the ripples and their propagation speed through helioseismology allows us to detail the Sun's interior in much the same way that mapping the propagation of seismic waves allows us to understand Earth's interior.
The elements in the Sun were scattered by the explosions (supernovae) of stars. These earlier stars were born in turn from the supernovae of the earliest stars in the Universe; the Sun is thus a third-generation star. For reasons discussed in a later section, that makes it and the Solar System relatively rich in metals and other heavy elements. The Sun is 1,392,000 km in diameter, or about 110 times the width of Earth, making it the largest object in the Solar System. Yet it is slightly smaller than a typical star, which gives it a slightly longer lifetime than most stars. It contains 99.8% of the matter in the Solar System, holding 1.98 x 1030 kg (333,000 times the mass of Earth) [3].
Despite this, it is not very dense; its average density is just 1.4 gm/cm3. Measurements of planetary orbits and of the Sun's rotation show that its moment of inertia factor (I/Ma2) is 0.06. Remember that a sphere with constant density would have a moment of inertia factor of 0.40. Earth's mass is slightly concentrated toward its center, giving the planet a moment of inertia factor of 0.33. Thus, nearly all of the Sun's mass must be located very close to its center implying that the Sun's structure changes even more with depth than Earth's does.
10.2.1 The Sun's Interior
The interior of the Sun obeys the same laws that govern Earth's interior and can be modeled in the same way, following the methods used for the Adams-Williamson equation. As with the Earth, the Sun may be modeled as a series of layers which may be stripped away to reveal the conditions below. Each layer lies on the one beneath; its weight is balanced both by its internal pressure and by the energy flowing out from those below. Energy moves from the Sun's interior to its exterior, obeying the laws of thermodynamics. The total amount of energy and the total mass of the Sun may be found by adding up the contributions of each layer. As these are known from observations of the Sun's exterior, they provide a starting point for our model, just as Earth's interior may be modeled from observations of its seismic velocities, mass, and moment of inertia.
The first key to understanding the Sun's interior is finding out where its mass is located. For any object, its mass is simply the density times the volume. The volume of a sphere is well known; the volume of a spherical shell is simply the difference of the volumes for the inner sphere with radius r and the outer`sphere with radius r + dr. If the density does not change over the distance dr, then it can be represented as a single value rho(r). Thus, the
mass dM for each shell is given by:
When dr << r, this can be simplified to:1. dM = (4/3) pi ((r+dr)3 - r3) rho(r)
This is the continuity of mass constraint. As a check on equation 2, we can integrate both sides of equation 2 from the center (r=0) to the outer radius (r=r), giving the familiar formula for the mass of a sphere of constant density:2. dM = (4/3) pi (r3 + 3 dr r2 - r3) rho (r) = 4 pi r2 rho(r) dr
Just as the weight of a standing person creates pressure on the ground, the weight of Sun's outer layers create pressure on deeper material. The amount of pressure in both cases is equal to the weight (force) divided by the area it covers. A standing person exerts a greater pressure than one laying down because the same force is distributed over a smaller area. A large person exerts a greater pressure than a small one because more force is distributed over the same area. If either person stands on a nail, then the force is concentrated over a very small area giving high pressures and potentially painful results. Because the Sun is fluid, no such concentration can take place. If the pressure is higher in one location than elsewhere, then the material will flow from the higher pressure to the lower one [4]. As a result, the weight of the fluid at any place in a layer is balanced by the pressure surrounding it; this is known as hydrostatic pressure.3. M = (4/3) pi r3 rho
We can therefore find the pressure dP for any small area on the inner surface of the layer and know that it is the same for the entire inner surface. Consider an area dphi by dtheta on the inner surface (fig 10.2-2). If the layer is very thin (dr << r), then the volume above that area can be approximated by the area times the height:
The mass in this area is simply the volume times the density:4. V = dphi dtheta dr
The weight of this mass Fg is provided by Newton's Law of Gravity:5. dM = rho dphi dtheta dr
The pressure is thus:6. Fg = G M(r) dM /r2
The change in pressure with radius in the Sun is thus:7. dP = Fg/dxdy
= (G M(r) dM)/(dphi dtheta r2)
= (G M(r) rho dphi dtheta dr)/(dphi dtheta r2) = (G M(r) rho dr)/r2
This is the law of hydrostatic equilibrium. By this law, if the weight of a layer is greater than the pressure, then the layers beneath will compress until the two are balanced. If the pressure is greater than the weight, then the layers will expand until the two balance once more. These effects play an important part in the lives of stars.8. dP/dr = (G M(r) rho(r))/r2
Substituting equation 3 into equation 8 and assuming a constant density provides an estimate of the pressures within the Sun:
Integrating from the center (r = 0) to the surface (r = R) and remembering that the pressure at the surface P(R)= 0 gives:9. dP/dr = (G M(r) rho)/r2
=G(4/3) pi r3 rho2/r2 = (4/3) G pi (rho2) r
Using the values for the Sun’s radius and average density give a central pressure of ~1.3 x 1014 Pa; this is actually 100 times too low! That is because the Sun’s mass, and hence the pressure within it, is not evenly distributed. Gases compress easily, which means that the density increases rapidly with depth [5].This effect also explains why the Sun’s moment of inertia is so low. The highly compressed, dense plasma within 140,000 km of the center (20% of the radius) contains more than 50% of the mass, whereas the much less compressed material in the outer 200,000 km (30% of the radius) contains less than 10% of the Sun’s mass.10. Pr = (2/3) G pi rho2 R2
The increasing pressure also has an effect on the temperature of the material. As we discussed in chapter 5, temperature is simply a measure of the intensity of heat energy in a substance. This energy is responsible for the random motion of substances on an atomic level. Things that move more rapidly and collide more often have more energy and hence have a higher temperature. Increasing pressure increases the likelihood of a collision, and so increases the temperature.
This is the basis of the ideal gas law in chemistry, which relates pressure P, volume V, the number of atoms n, and the temperature T:
where R is the universal gas constant equal to 8.314472 J / (K mol). Because the number of atoms or molecules in even a small amount of any gas is large the number is typically given in moles,where one mole is equal to 6.02214 x 1023 atoms, molecules, or particles [6]. Just as the number of one-pound balls in a ton can be found by dividing the total weight by the weight of each ball (2,000 lbs / 1 lb = 2,000), the number of moles of gas in the Sun can be found by dividing the Sun’s mass by the average atomic mass of the things that make it up. The Sun is mostly hydrogen, so that it contains (2 x 1033 g)/(0.5 x 10-24 g) = 4 x 1057 moles of hydrogen plasma [7] .11. PV=nRT
Equation 11 can be re-arranged to give the temperature for any given pressure:
Using our values for P and n gives a central temperature of 5,600,000 K. This value is fairly close to the actual value of 15,000,000 K; the differences come about because the Sun is not composed purely of hydrogen and because the assumptions of the ideal gas law break down in the Sun’s interior.12. T = PV/(nR)
Because the Sun's interior is hotter than its exterior, heat flows from its center outward. As we saw in Chapter 5, heat may move in several ways. It can move through conduction; that is, the energy can move by exciting the bonds between the molecules. Though this is a common form of heat transport in planets such as Earth, it is neglible in the plasma of the Sun. Heat can also be transported by moving the material itself via convection; in its most extreme form, the hot material is ejected from the Sun's outermost layer creating the solar wind that pervades the solar system. The final method for moving heat is familiar to anyone who has gone to the beach for a tan; the light that streams from the Sun carries away energy derived from heat. This radiation is the dominant form of energy transport in the Sun as well as the main method for cooling its outer layers.
Pressure, and so temperature and density, vary throughout the Sun's interior, causing the type of heat transport to change with depth. These changes define the Sun's layers (Figure 10.2-3). The outermost well-defined layer [8] is the corona, which sheds heat by ejecting material. This outer layer is too thin for true convection; once lost, the material travels at a speed of 400 km/s and never returns. The Sun ejects over 1 million tons of matter each second from the corona, mainly as protons and electrons. When this matter impinges on Earth, it gives rise to phenomena such as the "Northern Lights" and radio static. As its base, this layer is cool enough for simple molecules to form; however, the temperature raises quickly to 3,000,000 K through a complex and poorly-understood process.
Lying directly beneath the corona is the visible face of the Sun, known as the photosphere. This layer cools at the surface by radiation, and emits the light that we see. It varies in thickness due to the action of the layer beneath; on average, it is ~400 km thick. At the top, it has a temperature of 4500 K, increasing to 7600 K at the bottom. The average temperature of the photosphere is 5800 K. The density and pressure also increase with depth. "Surface" pressure is 6.8 Pa, growing to 1600 Pa at the base. "Surface" density is 10-7 gm/cm3 and increases to 2 x 10-4 gm/cm3.
Below the photosphere convection zone reaches ~200,000 km deeper into the Sun; it should be no surprise that heat is transported through this zone by the motion of the material. Temperatures increase rapidly in this zone, reaching 200,000 K. The high temperature differential is what drives the convection, forming individual cells, called granules, that can be seen in images of the Sun made using special filters [9]. Density and pressure also increase rapidly in this zone, rising to 0.01gm/cm3 and 100,000 Pa.
The largest region in the Sun is the radiative zone. Extending nearly 350,000 km from the convective zone above to the core below, the radiative zone is a region of hot, thick plasma. The temperature increases throughout this zone to ~7,000,000 K; however, the temperature changes so slowly with depth that this layer cannot convect. Instead, it transports heat by emitting and absorbing photons. The photons move randomly between particles throughout the radiative zone, taking more than a million years to pass through. As the photons collide with the particles in the region, they support them against the weight of the layers above in much the same way that the air molecules in a balloon support the plastic against the surrounding atmosphere. If some of the air is let out of a balloon, it will shrink in size. Similarly, if the number of photons (and hence, the amount of heat flow) decreases, then the radiative zone will shrink, increasing the density and pressure on the layers beneath. This plays an important part in the feedback loop that keeps the Sun active. Density rises quickly in the radiative zone, reaching 20 gm/cm3. In stars just slightly larger than the Sun, a second convective zone exists at the base of the radiative zone, and mixes material back into the core with profound effects on the star's life cycle.
The Sun's final layer is the core. This is the region in which nearly all of the Sun's energy is produced. Though it has a radius of just 140,000 km, the core contains nearly half of the material in the Sun. Temperatures reach 15,000,000 K in the center, and densities go up to ~150 gm/cm3. Pressures in the core reach an astronomical 1016 Pa. These high pressures, densities, and temperatures in the Sun's core are essential to the reaction that generates its enormous energy output and intense magnetic field.
10.2.2 The Sun's Energy Producing Reaction
The Sun sends out light carrying more than 3.827 x 1026 W of energy. This energy spreads evenly throughout space, in much the same way that a candle's light will illuminate a large room; the closer you are to the candle, the more light it appears to give. The total amount of energy is still the same, it is merely spread over a larger area. Thus, Mercury receives nearly seven times the amount of energy per square meter (seven times the flux) that Earth does, and Neptune receives just one-thousandth of that flux.
Nearly all energy on Earth comes from the Sun. Wind energy is due to the uneven heating of Earth's surface by the Sun, just as hydropower comes from the movement of water that was evaporated by the Sun. Even fossil fuels began as solar power converted into sugars and starches by plants and then further transformed into hydrocarbons by burial and heating inside Earth.
The Sun also has a strong and variable magnetic field and gives off a constant
stream of charged particles. The interaction of those particles with Earth's
magnetic field creates effects such as the Aurora Borealis ("Northern Lights") [10].
It can disrupt broadcast signals for television and radio, and
even may even interfere with power transmission. The 1989 blackout in Canada was partly due to an unusually strong burst of energy from the Sun.
The power source for this incredible flow of energy has long
been a matter of supposition. Early astronomers knew very well what it could not be, but did not know what provided the Sun's energy. Burning coal would only provide the Sun with enough energy for ~6,000 years.
What about gravitational collapse? As we saw in Chapter 5,
a change in gravitational potential can release energy as heat [11]
; in 1856, Helmholtz calculated the length of time that gravitational contraction could power the Sun. He began with the amount of gravitational potential energy U in a spherical shell dr with density rho:
Integrating this gives the total gravitational potential energy omega:13. U = -4 pi G M(r) rho r dr
Some of the energy goes into heating up the material, and the rest is given off as light. Because heat energy essentially measures the motions of the individual particles, it can be found from the average kinetic energy KE per particle from :14. omega = - {int from 0 to R} {-4 pi G M(r) rho r dr} [A]
15. KE = (3/2) k T
10.2.3 The Sun's Composition
[1] After solids, liquids, and gases.
[2] Commonly seen in novelty "plasma lamps" and lightning strikes.
[3] Isaac Asimov once said that the Solar System consisted of "Sol, Jupiter, and assorted junk".
[4] It is this difference in pressure caused by local temperature differences that drives the motion of air in Earth's atmosphere.
[5] Anyone who has climbed a mountain has experienced this phenomenon in reverse. Going up two kilometers reduces the air pressure by nearly 80 kPa, so that on average the air pressure in Denver is only 83% of that
in Los Angeles.
[6] It may be useful to think of a mole as the "chemist’s dozen" - a set of things that makes counting the total
number easier.
[7] The average atomic mass of the plasma is less than that of hydrogen because it averages the atomic mass of
the protons (~1) and electrons (~ 0.00794).
[8] Other layers, including a tenuous atmosphere that extends past the orbit of Pluto, exist; though important, they are not germaine to this discussion.
[9] Granules should not be confused with sunspots, which are not directly related to convection in the Sun; instead, sunspots are caused by local "twists" in the Sun's magnetic field. In 1612, Galileo made the first recorded examination of the Sun's surface by charting sunspot patterns.
[10] The equivalent in the southern hemisphere is known as the Aurora Australis, or southern lights.
[11] In The Moon Is A Harsh Mistress, Heinlein suggested using rocks dropped from orbit as weapons.
[A] It is at this point that HTML can no longer support the math. Hope you've enjoyed this - from here on out, it gets weird, so it won't be posted (unless there is an overwhelming cry of "More! more!").
