## April Fool's Day 2009

April 1 Jupiter vs Sun Size Comparison Matt Hamiltons take on DrBills "misspeaking" concerning mass of Jupiter. ("mis-speaking"? "Miss Peking"?)

Pressure vs. height in 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 8000 meters. As you can see, Mt. Everest is a little over 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.