3 October 2011

Hydrostatic equilibrium Gravity is constantly trying to shrink all bodies- e.g. the Earth, Jupiter, the Sun. Why don't they contract? "Big" bodies (whether gas, solid, or liquid) are held up against gravity by a PRESSURE GRADIENT. The pressure increases as one goes into the body, to balance the increasing weight that has to be supported. (This slide was swiped from the "Stars" class, but THE SAME EXACT IDEA HOLDS FOR THE EARTH.) "Small" bodies can be held up by inter-atomic or inter-molecular forces. A metal bolt is NOT in hydrostatic equlibrium. The forces between the atoms are enough to conteract the weak self-gravity pull of one end of the bolt on the other. Where is the dividing line between "small" and "large"? Depending on the material, about a few hundred kilometers. The largest asteroid (Ceres- diameter= 1000 km) is big enough to be in hydrostatic equilibrium (which by the way is why it is the only asteroid to be classified as a "dwarf planet"). Most asteroids are "small" and not in hydrostatic equilibrium. This is why Ceres is approximately spherical and most asteroids are far from spherical.

Equation of hydrostatic equilibrium This equation gives dP/dr - the change of pressure with radius- for a body in hydrostatic equilibrium.

Barometric Equation For a (low mass) atmosphere around a planet, the EHE can be solved to give the barometric equation giving pressure as a function of height.

Pressure vs. height in Earth's atmosphere A graphical representation of the previous equation for the case of the Earth's atmosphere. Solution of the equation of hydrostatic equilibrium for an atmosphere around a massive planet results in the barometric equation, which shows that the pressure falls off exponentially with height. A characteristic size scale is the "scale height" which is the change in height to have the pressure change by a factor of 1/e (about 37%). The scale height for the Earth's atmosphere is about 7700 meters. As you can see, Mt. Everest is somewhat more than one scale height above sea level, and the pressure there is only about a third that at sea level.

Pressure vs height in air and oceans The top diagram is similar to the previous one (but note that X and Y axes are interchanged!). The bottom graph shows the change of pressure with depth in the ocean. The different form of pressure change in air and in oceans (exponential vs. linear) is due to the fact that air is compressible, while water is pretty much incompressible.

Pressure and temperature vs height This shows how the density and temperature changes with altitude for our atmosphere. This is a linear-log plot, so that the almost exponential falloff in density is almost a straight line. (Deviations from a straight line are due to temperature variations- if the blue line plotted *pressure*, rather than density, it would be straighter.) The temperature goes down, then up, then down, then up! See next graph for another view of temperature vs height.

Temperature vs. height in our atmosphere This shows the behavior of temperature with height. The temperature reversals are used as boundaries between the layers of the atmosphere- troposphere, stratosphere, mesosphere and thermosphere.

There is no "simple" explanation for this temperature structure. One must look in detail at the mechanisms which heat and cool the air at different heights.

Near the surface of the Earth (troposphere) temperature decreases with height, as you probably have experienced if you have ever hiked or driven up a mountain. To understand this, we must realize that the atmosphere is heated by radiation both from above (sunlight) AND FROM BELOW (thermal radiation from Earth). The surface of the Earth, at a temperature of 250 to 300 K, radiates in the 10 to 12 micron region *UP* from the surface. As the radiation heats the air near the ground, it is "used up", so there is less as one goes up, and so the air temperature decreases with increasing height where this heating mechanism dominates (troposphere).

Above the the tropopause, the air warms with increasing height, defining the stratosphere. This behavior is due to the presence of ozone in the stratosphere. As ozone absorbs solar UV photons (which of course come from ABOVE the layer) the air is heated more strongly at higher altitudes.

At the 100 km level, the air becomes hotter as absorption of solar UV and soft x-rays heats the air, and the very low density means that the air cools very inefficiently.

Above the thermosphere is the exosphere (not shown on diagram), extending up to 500 km altitude (and perhaps beyond). In this region, the particle density is so low that high speed particles can simply fly away from the Earth without bumping into any other particle. Lower in the atmosphere, even a very high speed particle (with speed higher than escape speed) could NOT fly away from the Earth because it would soon bump into another particle and give up some of its kinetic energy. The average distance a particle in a gas can move before colliding with another particle is called the mean free path. In the air near sea level, the mean free path is microscopic (about 300 nanometers) in the exosphere the mean free path it is many kilometers.

Temperature vs depth in ocean There is a common misconception that the decrease in temperature as one goes to higher altitude is somehow CAUSED by the decreased pressure. This is not correct. In the ocean, we see the exact opposite effect- as we go down (increasing depth) the pressure rises, but the temperature falls- opposite what happens in the troposphere.

So "higher pressure" does NOT automatically mean "hotter". In a lab where one has, say, gas in a thermally isolated (no external heat flow) piston, increasing the pressure WILL increase the temperature. (As can be seen from the Ideal Gas Law). But in the atmosphere, or inside a planet, or inside a star, one doesn't have a thermally isolated system, and the details of heat flows and cooling mechanisms must be understood to understand the temperature of a system.

P and S Earthquake waves There are 2 different types of waves produced by earthquakes. P (pressure) waves and S (shear) waves. P waves can travel through a liquid- S waves cannot.

Earthquake waves The chief tool to study the Earth's interior are waves from earthquakes.

Simplified internal structure of Earth Using earthquakes to make a type of "CAT scan" of the Earth, we surmise this structure of crust (thin layer of brittle rock), mantle (large volume of "plastic" rock which can flow VERY slowly), outer liquid iron core, and inner solid iron core.

Earth interior: density vs. depth This gives a schematic idea of the way the local density changes with depth as we go deeper and deeper into the Earth. The laboratory density of iron is about 7874 kg per cubic meter. Note that the density of the iron core of the Earth ranges from 10,000 to over 12,000 kg per cubic meter. This is because the iron in the core is greatly compressed by the gravity of the overlying material. The affect of the compression on the density of the material in a planet must be taken into account if we are to use the average planet density (which is easy to measure) as a guide to the amount of different materials that make up the planet.

The sharp discontinuity in density at about 3000 km depth marks the mantle-core (or rock - iron) interface. The Earth is clearly very well differentiated- almost all the high density iron sank to the center during formation of the Earth (except for the iron used to make SUVs).

Compressed vs. Uncompressed density It is easy to find the average observed or compressed density of a planet- just divide mass by volume. However, the observed density is NOT equal to the average "lab" density of the stuff that makes up the planet! This is because the gravity of a planet compresses that material and makes the density higher than it would be in the lab. For planets the size of Venus or Earth (but not so much for Mars or Mercury) there is a sizable difference between the compress and uncompressed density.

To find the uncompressed density, one could imagine pulling the planet apart, small block by small block. As you did this, the blocks would expand in volume (no longer being compressed by the planets gravity) so the total volume of the blocks would be significantly larger than the volume of the spherical planet!

Preliminary Reference Earth Model (PREM) Combining what we know of physics (equation of hydrostatic equilibrium, equation of state of materials at various pressure etc) with what we know from observations (mass and radius of Earth, behavior of seismic waves etc) we can try to produce a self-consistent model of physical conditions inside the Earth. (Stellar astronomers use very similar ideas to make models of stars.)

This table shows some results of one such model. One striking number is the central density of 13000 kg/m**3. This is about 1.7 the lab density of iron, and shows the effects of compression by the bulk of the Earth. Iron may seem incompressible in a lab, but with enough pressure, its density will increase, like air in a bicycle pump.

Note too the central pressure - 3.64E11 Pa. The atmospheric pressure at sea level- 1 atmosphere of pressure- is about 1.01E5 Pa. The central pressure is thus 3.6E6 atm!

Temperature vs depth in Earth The temperature structure reveals some interesting structure. Near the surface (top 100 km or so) there is a large temperature gradient, with T changing by 5-20 degrees every km you go down. At around 100 km depth, the gradient quickly drops to something like 0.5 degree per km, which holds more or less all the way to the mantle- outer core boundary.

The low temperature gradient in the mantle is due to tranport of heat by convection, which is an efficient heat transport mechanism. In the top 100 km, the rock is "brittle", and convection is not possible- rather, heat transport is by conduction.

Mercury core I won't say much about the internal structure of the other inner planets- we don't have much definite detail. However, even with the most basic info, Mercury stands out as being made (on average) of denser stuff than the other inner planets. On the previous slide about densities, note that the uncompressed density of Mercury is about 5300 compared to 4400 for the Earth. Mercury must have a significantly larger fraction of its mass in its iron core than does the Earth. In this simple cartoon, we see the iron core of Mercury extending almost 74% of the diameter of the planet, as opposed to Terra, where the iron core extends about 55% of the planets diameter. Why? Nobody knows for sure, but the leading idea is that Mercury was hit by a Giant Impact after it differentiated. This impact knocked off a significant fraction of the rocky mantle, but didn't knock off much of the iron core.

Mercury and Earth to scale

*****M**E**S**S**E**N**G**E**R***** to the Winged Messenger

MESSENGER: MErcury Surface, Space ENvironment, GEochemistry and Ranging. This spacecraft has been orbiting Mercury since March 2011. If all goes well, it will orbit Mercury for one Earth year. On its way to Mercury (via a complicated multiple-gravity assist orbit, as shown a few weeks ago) MESSENGER has already made several quick Mercury flybys and learned some interesting things. Just think what we will learn as the data from the orbital phase starts coming in! If you want to learn more about MESSENGER see http://messenger.jhuapl.edu

Caloris Basin A quick glance shows that the surface of Mercury appears somewhat like that our Moon- an old, geologically dead surface dominated by many many impact craters. This is one of the MESSENGER "flyby" images, taken on its way to orbiting Mercury. This image is made from several different colors of light. The orange spots at the edge of the large Caloris Basin are said to be volcanoes. So Mercury may be somewhat more active than the Moon. I will say no more about Mercury, as MESSENGER should return a wealth of new data after it starts orbiting the planet.