Adiabatic Lapse Rate (Greenhouse v. Gravity, Part 1)

Take a box of gas*:

and crush it (increase pressure by a lot):

(*ideal) It's not really easy to visually show "pressure", but two things happen to the box of air: it will shrink (it's easy to show this), and its internal temperature will increase.

The first effect may seem obvious, but why the second one?  In the particular action we took of crushing the box, we performed work on the box (so we did something that changed its internal energy) but did not add or remove heat from the contents.  And, if we assume that the box is a thermal insulator, then we know that the gas cannot respond to being compressed by radiating away energy.

Thermodynamically, a change of a box of air's internal energy is related to the amount of heat that is transferred into or out of it, and the work performed on the box to change its volume.

 [1]

Where U is the internal energy of the box, Q is the heat that is added to the box, and W is a work term that is equal to PdV, such that an increase in volume is seen as a loss of internal energy (the box expends energy to push its walls out).  Conversely, pushing on the box to make it smaller is an addition to the internal energy.  Broadly speaking, if you do work on the box (shrink it) without letting it radiate, then its internal energy will increase; and internal energy is a function of the temperature of the gas.  This type of action on the box—this type of transformation—is called an adiabatic transformation.  "Adiabatic" means "without transfer of heat."

Why does this matter?

Consider the atmosphere: due to the weight of the air above you, there is higher atmospheric pressure at the surface of the planet than there is, say, 10 kilometers above us.  If we are very sloppy in our treatment of what we just learned, we might conclude that the very fact that pressure at the surface is higher means that the temperature at the surface is higher.

And believe it or not, this is something that is used (again, sloppily and erroneously) by some to deny that the greenhouse effect causes warming on the surface of our, and frankly any, planet.  Because if gravity can cause pressure change and high pressure is associated with high temperature, then who needs a greenhouse effect, right?

The problem comes from a fundamental misunderstanding of what the above equation represents.  The above equation (and all of the ones I will include soon derive the "adiabatic lapse rate" in the title) describes changes in variables of a box, or parcel, of air that is undergoing a transformation.  It does not describe a static system; but, the pressure gradient caused by gravity describes a static system (it actually holds in a variety of dynamic systems as well).

The article linked above tries to make an argument that the temperature profile in the atmosphere with height, which we will call the environmental temperature lapse rate (the rate at which temperature lapses, or falls, with height), can be described merely by gravity.  How?  By thinking that because adiabatic lapse rate, the rate at which a parcel of air will cool as it rises adiabatically (i.e. as it goes through a pressure change adiabatically), can be, then the environmental lapse rate can be.  But these are not the same thing!

To derive the adiabatic lapse rate (skip to equation [14] if you wish), consider our above thermodynamic equation, and plug in the work equivalence:

 [2]

Let's define a term that we'll call "enthalpy," H, as:

 [3]

so a small change in enthalpy is:

 [4]

 [5]

 [6]

 [7]

We can also define the heat capacity of our system as being the amount of heat we need to add (remove) to (from) a system in order to increase (decrease) its temperature by a certain amount.  Typically, we would have to restrain certain parameters of our system in order to measure such a heat capacity, for instance the pressure of the system.  If our system is at constant pressure (dP = 0), and we define constant-pressure heat capacity as below:

 [8]

then it follows

 [9]

And so, in an adiabatic lift of a parcel of air, where heat exchange is zero:

 [10]

If we divide by the mass of the system, we can obtain the specific heat capacity and the specific volume, which are, respectively, the amount of heat needed to cause a temperature change per unit of mass, and the amount of volume a unit of mass occupies (inverse of the density).  These variables will be in lowercase from the uppercase above.  And finally, we can use the hydrostatic equation to finish our derivation of the adiabatic lapse rate:

 [11]

 [12]

 [13]

 [14]

Here, g is a negative quantity (which I find exceedingly more appropriate than how it's treated as a positive variable with a negative sign attached to it, as in the Wikipedia link above).  On Earth, this adiabatic lapse rate value is about –9.8˚C per kilometer in height.  In other words, for every kilometer you adiabatically raise a parcel of air, it will cool by 9.8˚C.

Pay close attention to how the equations still describe the parcel of air, and are under the framework of literally moving a "piece" of air through a medium that has reached a pressure equilibrium with gravity.  We do not know anything about the temperature distribution in this medium, I never had to reference it.  We only know (or at least presumed) that it is stable.

It is also completely worth pointing out that if this was indeed the environmental lapse rate here on Earth, then our environmental lapse rate should equal –9.8˚C/km, no?  But it does not, the environmental lapse rate is instead roughly –6.5˚C/km, up until you hit the tropopause.  This is not a simple "well they're close, it's just an error between measurement and theory"—no, theory actually dictates that in our atmosphere the environmental lapse rate cannot be as negative as the adiabatic lapse rate.  While I will not go into that in particular in this part, allow me to show how a "shallow" environmental lapse rate is still completely compatible with a "steep" adiabatic lapse rate.

Stability

Equation [14] describes how the temperature of an air parcel will change when it rises to a particular height.  Consider what will happen to it if it does: once it reaches that height, it will have the same pressure as the pressure of the air around it, and one of three things will happen.

• The air parcel will wind up being colder than the surrounding air, which means it is denser, and thus will sink.  This is a condition where the environmental lapse rate is "shallower" (lower in magnitude) than the adiabatic lapse rate, a condition of stability where vertical motion is hampered.  A stable atmosphere will stay the way it is.

• The air parcel will reach the exact same temperature as the surrounding air, which means its density is equal, and thus it won't experience a force stopping its motion (but also not helping it).  The environmental lapse rate and the adiabatic lapse rate are equal, and this is a condition of neutrality.  Neutral atmospheres are "stable" in that when you move an air parcel adiabatically, you are not convecting heat from one location to another.  So, the environmental lapse rate will not change.

• The air parcel will be warmer than the surrounding air, which means it is less dense and will accelerate upward further.  The environmental lapse rate is "steeper" than the adiabatic lapse rate, and the atmosphere is unstable.  In an unstable atmosphere, these adiabatically rising air parcels are carrying hot air upward—this will lead to warming higher up, which makes the environmental lapse rate more "shallow".  It will work its way to a stable condition.

This implies an important point: an atmosphere with a very shallow environmental lapse rate is stable and can coexist with a steeper adiabatic lapse rate.  In fact, an atmosphere that has no greenhouse gases, or in other words does not have gases that can react with thermal radiation, will be isothermal with no environmental lapse rate at all.  This is again something in particular I will not explain in this part.

The next statement necessarily follows: the fact that the pressure is higher at the surface does not dictate that the temperature will be higher at the surface.  (You need to have a radiatively-interactive atmosphere, one with greenhouse gases, in order to have temperatures higher at the surface.)

If you're still not convinced, allow me to derive the temperature-dependent pressure profile of the atmosphere.  In other words, the pressure at a given height that has a given temperature.  That given temperature will depend on the environmental lapse rate.

Starting with the hydrostatic equation, and soon using the ideal gas law:

 [15]

 [16]

 [17]

 [18]

 [19]

 [20]

 [21]

[22]

 [23]

In these equations, in particular our final one, tau is our environmental lapse rate (see the substitution from equation [19] to equation [20]), and variables that have zero subscripts denote values at the surface of the planet (or any surface, so long as that surface remains the same in the problem).

The real question: if we give our tau variable a value very very close to zero, does that make our pressure profile very wonky?  In particular, does it imply that our pressure profile won't be "high at surface, low up above?"  The graph below shows that the answer is no.  In fact, the pressure profile corresponding to a near-zero temperature profile (–0.01˚C/km) is very close to the pressure profile corresponding to an environmental lapse rate close to our adiabatic lapse rate (–9.8˚C/km).  For this graph, the temperature and pressure at the surface for each scenario are the same, and are 14˚C and 100,000 Pa respectively.

Wrap-up

So, not only does the adiabatic lapse rate not describe a static system, instead the change that an individual air parcel experiences when you move it up or down, but the suggestion that a pressure gradient must cause a temperature gradient is unfounded as well and has many mathematical counterexamples.  In the next post, I will offer up a couple theoretical examples to direct how we should think about energy transfer in a simple atmosphere, and why radiative interaction (i.e. greenhouse gases) is needed for the convection that drives our actual environmental lapse rate.  I'll also briefly discuss some of the published science on many of these scenarios.

Philosophy of Religion, Part 3: Craig's Kalam Cosmological Argument

Readings: "The Existence of God and the Beginning of the Universe" by William Lane Craig, and a brief selection from J. L. Mackie's book The Miracle of Theism.

I imagine that anyone reading this has already encountered the famous Kalam cosmological argument, which goes like this:

1. Everything which begins to exist has a cause
2. The universe began to exist
3. Therefore, the universe has a cause

You'll notice that nowhere in the argument does it say "therefore, God exists." A person might see this argument out of context and never suspect that it's trying to support theism. This is perhaps the argument's greatest weakness - it proves nothing about God on its own (why can't the cause of the universe be something else - maybe a physical starting point that we can't perceive, or something incomprehensible?). Instead, as reported by a (theistic) classmate of mine, not even Craig thinks of this as a standalone argument for the existence of God. Rather, it is meant to be one piece of evidence in a cumulative case for the existence of God. If the universe does not have a cause, then certainly God does not exist. But if the universe does have a cause, then a creator is at least one possibility, one which must be justified by further means. At the very end of his article, Craig offers a brief supplemental argument that the cause of the universe must be personal. He writes:

"In fact, I think that it can be plausibly argued that the cause of the universe must be a personal Creator. For how else could a temporal effect arise from an eternal cause? If the cause were simply a mechanically operating set of necessary and sufficient conditions existing from eternity, then why would not the effect also exist from eternity?"

Philosophy of Religion, Part 2: Swinburne's Induction

This is the second post about my Philosophy of Religion class. The readings I will discuss today were theologian Richard Swinburne's "The Existence of God" and A. C. Grayling's "Why I Am Not a Believer."

I won't talk much about Grayling's paper, as it is the usual atheistic rant against religion in its many forms. Without context, I wouldn't have been able to distinguish the essay from a passage of "The God Delusion." I was disappointed that our instructor did not select a reading which responded directly to arguments of the sort made by Swinburne. But maybe this was the point. As Grayling writes,

"Part of the sleight of hand at work here becomes obvious when one notes the great difference between what ordinary votaries of a religion believe and what their theologians and high priests say."

 The God that Swinburne defends bears little or no resemblance to the God of religious dogma which people really seem to want to believe in. Without the additional filter of religious tradition, his conception of God is virtually irrelevant to daily life, and might as well lead to a humanistic ethical outlook.

But what is Swinburne's conception of God (as presented in the essay), and how does he arrive at his conclusion? He acknowledges that a deductive inference of God's existence, in which the conclusion is mandatory given the premises, will not hold water. "It is most implausible," he says, "to suppose that such a statement as 'there is a physical universe but no God' ... contains any internal contradiction." Instead, Swinburne uses what he calls an "inductive inference". (Well, "induction" seems to imply an inference to the continuation of a pattern that has happened in the past; it might be better to characterize his argument strategy as "inference to the best explanation," for it is hard to establish a pattern when you only have one universe to reason from). He lays out four criteria that make for good induction, given a hypothesis H and observed phenomena P:

1. If H is true, then one ought to expect P
2. If H is false, then P is very improbable
3. H is simple (i.e., postulates "the existence and operation of few entities, few kinds of entities, with few easily describable properties behaving in mathematically simple kinds of way.")
4. H fits with our background knowledge

Reasonable enough. Now, Swinburne constructs his argument from three pieces of evidence:

1. Why a physical universe exists at all
2. Why this universe obeys orderly physical laws, rather than seethes in chaos
3. Why this universe is fine-tuned to allow the evolution of complex life

Creation Museum: the Failure of Creationist Astrophysics

Last weekend, I went down to Petersburg, Kentucky with a few people from SSA to visit the Creation Museum. Like a lot of things Answers in Genesis does, the Creation Museum seems to offer surprisingly accurate information at first glance, but once they start trying derive their erroneous conclusions from the evidence, it all goes horribly wrong. Case in point, the planetarium show.

It began as a pretty neat presentation of the various constellations visible from our solar system, as well as the various types of stars in them, showing their different colors and relative sizes. So far, so good. Then the narrator of the show began talking about the different life spans of various types of stars. Red stars are relatively cool and burn their fuel slowly, while blue stars are relatively hot and burn through their fuel quickly. Blue stars, according to the narrator, are “a reminder of our universe’s young age.” Wait... what? Indeed, all blue stars are comparatively young, but in this context “young” potentially means several million years old, and the ones we’ve seen blow up were definitely pretty old when they died. Aside from that, let’s examine the logic of this statement: essentially what the narrator said was that “since some things are young, nothing can be old.” That statement was the first car derailed in the trainwreck to come.

Different types of main-sequence stars. Massive blue stars squander their hydrogen fuel in only a few million years, while tiny red stars can make their fuel last for over a trillion years. More info here.

The focus of the presentation then moved to the planets we've discovered orbiting these other stars. Since large planets with short orbital periods are the easiest to detect, a large number of the planets we've discovered thus far have been so-called "Hot Jupiters" - gas giants roughly the size of Jupiter, or bigger, orbiting absurdly close to their parent star (some of them orbit their star closer than Mercury orbits our Sun). Before Hot Jupiters were discovered, it was thought that they shouldn't exist. The area near a star is too hot for gases to condense and form planets. Clearly, they must be the work of God, right? Well, what the narrator neglected to mention was that planets are now known to migrate early on in solar system formation. Interactions with the dust in the star's accretion disk and with other planets can cause new planets to drastically change their orbits before they reach stable ones. Hot Jupiters must have formed farther away from their stars and then migrated inward.

Hot Jupiters: they're very hot. More info about this picture here.

After finishing up talking about our galaxy, the narrator begins talking about other galaxies. He answers a question frequently asked by people who know what a light-year is: “If the universe is only 6,000 years old, how is it that we can see galaxies millions and even billions of light-years away?” The answer: “gravitational time dilation.” That’s right, an effect of relativity - a theory partly built on the principle that the speed of light in a vacuum is always the same - causes the speed of light in a vacuum to be much faster than the speed of light in a vacuum. That’s how light from distant galaxies is able to reach us in such a short time. Obviously, gravitational time dilation has no effect whatsoever on the speed of light. Gravity warps both space and time in such a way that the speed of light remains the same. Besides, there’s not enough gravity in between galaxies to cause this effect to any appreciable extent anyway. Usually when gravitational time dilation comes up, it’s either in the context of extreme gravity (near a black hole or neutron star) or when we have a need for extremely precise clocks (like we do in GPS satellites).

Contrary to what evolutionists would have you believe, it does not take light a year to travel a Lightyear. The unit is named for the famous space explorer, and vanquisher of Zurg, Buzz Lightyear.

The narrator then pointed out his last bit of evidence for a young universe: spiral arms. The claim was that if spiral galaxies are billions of years old, their spiral arms should have wound up so tightly by now that they would be unrecognizable as arms (since inner stars take less time to go around the galaxy than outer stars). This argument reveals a complete failure on the narrator’s part to understand what spiral arms are. They are not groups of stars that orbit the galaxy along with their constituents, they are self-perpetuating density waves that stars pass through as they orbit the galaxy. The arms maintain themselves in the following way: as stars approach a spiral arm, the extra gravity from the arm causes the stars to speed up; as they begin to leave, they slow down and end up spending a disproportionate amount of time within the arm. The reason why we can see spiral arms so clearly in other galaxies is because of the presence of bright blue stars, newly formed from the comparatively dense interstellar medium in the arms (since blue stars only live for a few million years, they die before they leave the arms). Longer, more detailed explanation here.

Stars go in, stars go out. Astronomers have explained that.

As I got up to leave the planetarium at the end of the show, I immediately regretted not speaking up to correct the errors in the show more audibly. Not only because getting kicked out of the Creation Museum would have been funny, but also because I genuinely think that at least some small fraction of the other people in that planetarium actually wanted to learn about astrophysics, and were deprived of that because of their poor choice (or their parents' poor choice) of venue. If I could have gotten even one of these curious, but misinformed, people to question the creationist narrative and do their own investigation, I would have been much more satisfied with relinquishing my $38 to that museum.

Taylor Polynomials

When attempting to solve certain problems, it is often useful to approximate a function as a polynomial. Polynomials are generally easier to deal with in calculus problems than other functions, and many functions can be approximated as polynomials to arbitrary precision.

The general method for producing a polynomial approximation of a function is as follows. First, choose the point that you want your approximation to be centered around. (When you’re solving a problem, you should expand your approximation around a point within the region that you’re most interested in, because Taylor Polynomial approximations are most accurate near their center, but for simplicity’s sake I’ll just pick x = 0 as the center for now, and give a more general formula later.) Once you've chosen the point that you want to expand about (in our case, x = 0), you can start writing down polynomial approximations. The simplest (and least accurate) polynomial we could use to approximate a function would just be a constant, equal to the value of the function at x = 0 (i.e. P₀(x) = f(0), where "P" is a polynomial function and the subscript indicates the order of the polynomial).

Zero-Order Taylor Polynomial (magenta) for f(x) (blue)

As you can probably tell, a zero-order polynomial usually won't be sufficient to provide a useful approximation of a function. It is, of course, exactly equal to the function at x = 0, but it quickly diverges from the function as we move away from x = 0, since the function is changing its value while our approximation is not. We can do better by using a polynomial that not only has the same value as the function at x = 0, but also changes at the same rate (i.e. has the same value of its first derivative). We do this by setting P'(x) = f'(0) and then integrating:

*Remember that the derivative of f evaluated at a single
 point (x = 0) is a constant, not a function of x.

And our approximation gets a whole lot better:

First-Order Taylor Polynomial (magenta) for f(x) (blue)

Of course, f(x) isn't linear, so our first-order approximation still diverges from the function eventually. If we only need to consider small values of x, this approximation may be sufficient, but if the situation calls for accuracy at larger values of x, we need to go deeper...

To the second dream lev- ...err, derivative.

The linear approximation eventually diverges from f because, while P₁ has a constant slope, the slope of f is changing. To get a better approximation, we need a polynomial that not only has the same value and slope as f at x = 0, but also is changing its slope at the same rate as f(x) (i.e. P''(x) = f''(0)).

As before, we find the constants of integration by setting
the nth derivative of P equal to the nth derivative of f at x = 0.

Second-Order Taylor Polynomial (magenta) for f(x) (blue)

For increasingly better approximations, we repeat this process of setting the nth derivative of P equal to the nth derivative of f at x = 0 and then integrating, for ever-increasing values of n, so that:

In fact, if we let n go to infinity, and if f(x) is a continuous, analytic function, then the Taylor Series is exactly equivalent to the function:

Earlier I promised I'd give a generalized formula to allow for the Taylor Polynomial to be centered around any point. To do this, just replace "0" with "c," and "x" 

(= x - 0) with "x - c":

Obviously, you're never actually going to calculate an infinite number of terms, but luckily in many cases you can get a pretty good approximation of a function with just a few terms.


Now, you might be wondering when you'd ever want to approximate a function as a polynomial. It turns out that in certain problems, either the function that you're working with isn't explicitly known, or it's just more difficult to work with than a polynomial. For example, if you're only dealing with small angles, it'll often make it easier on you to approximate sin x as just x, and cos x as 1 - x² (It's this approximation that allows us to describe the motion of a pendulum bob as simple-harmonic, as long as the maximum displacement angle is small).

Perihelion

A little over a day ago, the Earth approached the point in its orbit closest to the Sun, called perihelion.  This means that the Earth was getting more sunlight than in any day of the year (and to close approximation, still is).  This is a peculiar fact for us northern hemisphere denizens, since this is also our winter, and we don't quite get the most sunlight at all right now thank you very much.

No, if we wanted the most intense sunlight, we would want to go to our opposing Tropic, the Tropic of Capricorn, at the 23.5° S latitude.  There, solar insolation (power per square meter) at the top of the atmosphere (TOA) reaches 1413 W/m^2 at its peak during the day, the highest of any other latitude on Earth.  This reduces some as the light goes through our atmosphere and is reflected by clouds and particulates, but otherwise Antofagasta in Chile, São Paulo (São Paulo) in Brazil, Rockhampton (Queensland) in Australia, Polokwane in South Africa – these are our tanning destinations!

Now I'm not much of a tanning person myself (for several reasons, one being more genetically-oriented), but I'm a huge fan of free energy like anyone else.  If I wanted a winter vacation home and cared only about how much energy I could produce using solar panels, where should I go?  While peak insolation is highest at the Southern Tropic, total solar insolation is not at this time of year.  If you thought that perihelion during the northern winter was peculiar, you probably don't know this: to gather the most sunlight I need to go to the South Pole:

Larger version here.

The above graph shows average 24-hour TOA insolation at different solar declinations ("delta"), for all latitudes, and is corrected for changing Earth-Sun distance throughout the year (perihelion and aphelion are roughly concomitant with the solstices, off by ~11 days).

The main reason that solar insolation increases toward the poles, after you hit the Polar Circles, is that there's no longer any nighttime.  The light that the poles get while the rest of the Earth is in shadow outweighs the difference in insolation during the day.

A winter vacation home here would have the added bonus of increased efficiency too, since solar panels tend to work better in cold temperatures.  If, however, you're not the type that likes winter, if you're instead one of those "tanning" types that likes to have as much Sun as possible, then you'd be better off just living closer to the equator.  In fact, what this graph would imply is that you should live somewhere south of the equator, around the 2/3° S latitudes.

In this respect, the above graph is actually wrong.  This is because it is a single-day snapshot, with an equal number of days during the summer as during the winter.  If we define the split between summer and winter at the days where the Earth-Sun distance is perpendicular to the distance when at perihelion or aphelion (i.e. the two days in the year the Earth receives the same insolation in both hemispheres), the northern hemisphere's winter is shorter than the southern hemisphere's.  The flip side of that coin is that our summer is longer – we have a longer summer while we're further away, and the southern hemisphere has a shorter summer while it is closer.  This is because the Earth moves faster in its orbit while it is closer to the Sun, during perihelion; slower while it's further away.

The countering time and distance effects cancel each other out more or less completely.  If you wanted to bask in the most sun throughout the year, you'd want to live right on the equator.