Everything in the universe is connected in one fashion or another. There are repeating patterns from the microscopic scale to the universal. Neuron connections in the brain look surprisingly like universal clusters of galaxies. Spiral shells of the Nautilus look hauntingly similar to other spirals in nature, including the shapes of spiral galaxies.
Throughout my life, my creativity has been infused and informed by a cross-pollenization of learning experiences. Mathematics has contributed to my art; my art has contributed to my research; and my research attitudes have aided in my science.
I’ve been developing long-range plans to build a university that includes such cross-pollenization of ideas. I hope it will stimulate a new profusion of “world builders” who will go on to helping humanity to the next level in its development.
Mathematics Can Be Fun
Today, I want to share my own fun with mathematics. Yes, you read that correctly. Putting “fun” and “mathematics” in the same sentence may seem to be an oxymoron to some people, but it’s not.
First of all, let me confess that my IQ is in the top 1%—139. I also realize that this gift from God can disappear at any time. By some scales of intelligence, I barely made the bottom run of “genius.” By other scales, I didn’t even make the grade. No big deal. We each have our value in life no matter what our IQ.
All 3 of my younger brothers had far higher IQs than mind. I suspect that both of my parents had higher IQs, too. The point here, is that I have been able to see some things other people have missed, partly because of my IQ, but more importantly, because of my occasional humility. Understand this, though, humility and confidence are highly compatible traits. With both of them, a great deal can be accomplished. In fact, humility is far more important than IQ, because without it a person tends to become blind with arrogance. People with the highest IQs in the world can become crippled with arrogance (confidence without humility).
Mathematics can be easy. For those who have already been through school, the first step is to toss out the arrogant belief that “I’m too dumb to understand mathematics.” You’re not! Mathematics is a tool. But it can also be treated as a game.
Most people love puzzles and games. Maybe not everyone, but the skills needed in mathematics are the same ones used in playing games and solving puzzles. If you know the language, the rules and the tools of mathematics, it can be just like that video game you love so much, or those puzzles that give you smiles when you solve them.
If you don’t like puzzles and games, there may be some other way of looking at mathematics that will work for you.
While teaching college algebra, I learned a number of things from my students. One of the more important lessons involved the fact that most students have gotten a raw deal in their education. Their teachers were awful! How can I know this? Because the students learned mathematics by rote instead of by understanding. This makes all the difference in the world. Some of my students were confused by how I was teaching algebra. “But we didn’t learn it that way!” said one student.
If you learn by understanding, then learning a new way of approaching the same problem will not cause confusion. Instead, there will be an “Aha!” moment that puts a smile on your face. I’ll give an example a little later to show you what I mean about balanced equations and some easy tricks you can use.
The confusion of that one student involved the procedures to keep equations balanced. Perhaps they had never learned the purpose of keeping equations balanced. But if you do something to one side of an equation and not do it also to the other side, then the expression will no longer be an equation. It will be unbalanced! Mathematics is full of simplicities like this.
A Lifetime Using Mathematics
In high school, in 1960s Montgomery County, Maryland, I aced all of my mathematics courses. I could easily see the patterns. But my questions stumped even the teachers, for they may not have learned mathematics as I would hope to teach it. Anyone can master this stuff, if they have enough persistence and the right guidance. Simple repetition will break down most of the barriers. But if a teacher can show how things fit together, then even people of high, below-average intelligence will be able to do algebra and trigonometry. They may even learn to create with them.
After high school, I didn’t go to college. However, I did study electronic engineering through a correspondence course. I also studied climate science. When writing one science fiction story, I wanted to know more about atmospheric physics, including planetary retention rates (how long a planet can hold onto its atmosphere). The discussions I had found on the topic involved calculus which I had not yet learned, so I bought some college calculus texts and studied on my own.
When Texas Instruments came out with a programmable calculator, I bought one right away and set about learning to program it to solve problems in 3D space—calculating the distances between stars.
That wasn’t easy, because the handheld calculator could only accommodate 50 programmable steps and there was no hard storage. In other words, I had to program the device each time I wanted to use it. Not only that, I had to key in the first half of my program, run it, and then input initial values while outputting interim answers. Then, I would have to key in the last half of my software code, then run it, and input the interim answers to generate my final result.
There were challenges and difficulties, but I let my enthusiasm push me past those barriers. I had the confidence that the problems could be solved, but the humility to let the solutions come to me in their own time. In fact, no matter what challenges you face in life, the right combination of confidence and humility can help you succeed.
Later, I would design and program 3D astronomy space software, Stars in the NeighborHood. After getting that to work flawlessly, I finally decided to go to college to study computer science. I earned my degree, summa cum laude. I worked for a few short years in the field and retired, moved to the Philippines and expanded my activities to include writing, producing videos and teaching.
Like most everyone on the planet, I care about the environment we live in. I was once a fan of Al Gore’s until I learned that he had the science all wrong. That prompted me to dust off some of my old climate science lessons and then to write 4 books on the topic—#1 Weather Bestseller (14 weeks), Climate Basics, and two other climate titles that remain in the Top 100 on Amazon’s Weather category—Thermophobia and Red Line – Carbon Dioxide.
I understood how the qualitative view of the Climate Change Alarmists was all wrong, but I wanted to dig into the quantitative side of things. That requires science and mathematics.
I had heard of an idea floating around that seemed to explain one of the supposed reasons for using the Greenhouse Effect, but solving that supposed problem without carbon dioxide or water vapor (both greenhouse gases). This was the infamous Effective Temperature problem. This problems had a formula that worked perfectly for airless bodies like the Moon or Mercury, but fell short with planets like Earth and Venus because of their relatively thick atmospheres. For Earth, the Effective Temperature equation yielded a temperature of about –18°C (very cold, sub-freezing), while the actual average surface temperature was closer to +15°C. That was a difference of 33°C—the difference between a comfortable planet on which to live and “Snowball Earth.”
For all these years, I had bought the idea that “optical thickness” (also known as the “greenhouse effect”) was the culprit behind this discrepancy.
The more I had learned about climate history, the more I realized that carbon dioxide fails to show any power over global temperature. That was for changes in carbon dioxide on a variety of time scales. But what about the older problem—the deficit found in the Effective Temperature formula?
The new idea I had encountered talked of pressure being the cause of the difference. Some of the enthusiasts of this new viewpoint mangled the arguments in favor of this approach, conflating temperature changes from atmospheric lapse rate with heat of compression. That was one problem. Another enthusiast thought he had proved the thesis by using a form of circular logic (a tricky logical fallacy).
I explain the qualitative overview of this in my article, “Ding Dong, The Greenhouse Effect is Dead.” A follow-up article which dives more deeply into the mathematics will also be linked here.
Here, I wanted to touch on a technique used in solving complex equations.
I started with a desire to find an equation that could tell me the average surface temperature, doing a better job than the equation for Effective Temperature. Don’t get me wrong; the Effective Temperature equation was pure genius. But again, it only works for worlds without substantial atmospheres.
The Effective Temperature equation states:
Te = 4√((L ∙ (1 – a))/(16 ∙ π ∙ d2 ∙ σ))
luminosity of a star, in this case, our sun = 3.846 ∙ 1026 W.
In cgs units, 1 W = 1 ∙ 107 erg ∙ s–1
a—the Bond albedo of the planet, in this case, Earth = 0.306.
d—the distance from the star. For Earth, 14,900,000,000,000 cm
σ—the Stefan-Boltzmann constant;
In cgs units ≈ 5.6704 ∙ 10–5 erg ∙ cm–2 ∙ s–1 ∙ K–4
Instead of Te (subscript “e” for “effective”), I wanted to find Tτ (subscript “τ” for tropopause). This is the temperature for a given planet at the tropopause—that part of the atmosphere where the predictable decrease in temperature with altitude stops and higher altitudes result in warming, instead of cooling. This tropopause is at the top of the troposphere. On Earth, this is from the surface up to about 11 kilometers altitude.
I also discovered that all planets with substantial atmospheres have roughly the same pressure at the tropopause—approximately 0.1 bar. On Earth, the figure is closer to 0.2 bar, but on all known planets the tropopause is amazingly close to the 0.1 bar figure, including Venus, Earth, Jupiter, Saturn, Titan (Saturn’s largest moon), Uranus and Neptune. Mars is not included because it doesn’t have sufficient atmosphere to have a troposphere or tropopause.
In order to find the desired surface temperature (T0), I first wanted to determine a method for calculating the tropopause temperature. I hoped that a formula like that for Effective Temperature could be used with a different constant—one based on our most studied planet—Earth. A number of assumptions were made in using this approach. I hoped this would give me a ballpark answer that could work for rough approximations.
To find my new constant, I took the above Effective Temperature equation, replaced Te with Tτ, and then solved the equation for the constant, instead of the temperature. I used “k” for the constant, instead of the Stefan-Boltzmann sigma (σ).
Two critical things to remember when adjusting an equation to solve for a different variable are,
- Keep the equation balanced by doing the same thing to both sides, and
- Use only compatible units.
This last one trips up a lot of people. If you’re using inches with one variable, don’t use miles, kilometers or light years with other variables. Or, if you use the cgs (centimeter-gram-second) system of units, don’t use lb/in2 for pressure and ft/sec2 for gravity. These are simple rules that take a little extra effort, but they mean all the difference between getting an answer you can use and trashing the entire effort in utter failure. We all used similar skills in life all the time. When driving, for instance, when we want to turn right, we turn the steering wheel clockwise. Simple. With practice, it become second nature to us.
Here’s the rewritten equation with our tropopause temperature and the embedded constant, “k,” which we want to find. We know all of the information in this equation except the elusive new constant.
Tτ = 4√((L ∙ (1 – a))/(16 ∙ π ∙ d2 ∙ k))
Since all of the terms on the right are under the fourth-root radicand (the little checkmark with the “4” next to it), we first want to raise both sides by the power of 4 in order to get rid of the radicand.
Tτ4 = ((L ∙ (1 – a))/(16 ∙ π ∙ d2 ∙ k))
Since k is in the denominator of the right-hand fraction, we can multiply both sides of the equation by k to move this variable to the left and eliminate it from the right side of the equation. Remember, when dealing with multiplication and fractions, k/k = 1, and multiplication by 1 is effectively the same as the value without the 1.
Tτ4 ∙ k = ((L ∙ (1 – a))/(16 ∙ π ∙ d2))
Similarly, to get rid of the temperature from the left side of the equation, we divide both sides by this temperature raised to the fourth power.
k = ((L ∙ (1 – a))/(16 ∙ π ∙ d2 ∙ Tτ4))
That wasn’t so hard. Carefully writing each symbol and only doing legal actions to both sides of the equation, and you never run into trouble.
With the figure –215°K, our new constant turns out to be 0.00011092 or 1.1092 ∙ 10–4. If you’re not familiar with degrees Kelvin, it uses the same size units as Celsius, but is temperature written in terms of absolute energy, where zero is the absolute zero of no thermal energy, or –273.15°C. When dealing with science, and equations which refer to the total thermal energy of a system, we always use an absolute temperature scale, like Kelvin.
A More ‘Scary’ Equation
If you’re up for it, let’s try out an equation that is a bit more complicated. Here, I’ll show you a trick I use quite often to save me time and headaches.
In order to convert the above information into the proper equation, I needed some way to find the height between Tτ (tropopause) and T0 (surface). One equation I found relates surface pressure to the pressure found at a given altitude, “h.” And I wanted to solve for “h.”
P = P0 ∙ (1 – ((L ∙ h)/(T0)))((g ∙ M)/(R0 ∙ L))
P—the pressure at an altitude above the surface. This will be the pressure at the tropopause.
P0—the pressure at the surface.
L—the lapse rate in °K per unit of altitude (the rate of cooling as you go up).
h—the desired altitude (tropopause).
T0—the surface temperature.
M—molar mass of the atmosphere.
R0—the universal gas constant (8.314462618 J/(mol·K).
Notice the terms of reduced size on the far right of the equation. All of that is an expression which is a single exponent of the preceding expression.
When solving for a different variable on an equation this gnarly, I will typically substitute other variables for expressions within the equation that will remain the same throughout the steps to find that new solution.
For example, the exponent I could set equal to x.
x = ((g ∙ M)/(R0 ∙ L))
The equation is then somewhat simplified with,
P = P0 ∙ (1 – ((L ∙ h)/(T0)))x
We can restore the truer value of x, later.
Again, we want to solve this equation for h. Since everything on the right side of the equation is multiplied by P, we will divide both sides by that amount to shift it to the left-hand side.
P/P0 = (1 – ((L ∙ h)/(T0)))x
Now, everything on the right side is raised to the power of x. To move this complication to the left, we have to take the xth root of both sides.
x√(P/P0) = (1 – ((L ∙ h)/(T0)))
To simplify things, we will multiply both sides by –1 (negative one). This will eliminate having a negative h further down the line. Remember to hit all terms with that multiplication; this changes the minus in the right-hand expression to a plus sign.
–x√(P/P0) = –1 + ((L ∙ h)/(T0))
Now, we add 1 to both sides of the equation.
1–x√(P/P0) = (L ∙ h)/T0
Then, we move temperature to the other side by multiplying both sides by that variable (remember, it’s in the denominator of the right-hand expression’s fraction.
T0 ∙ (1–x√(P/P0)) = L ∙ h
Now, all we need is to divide both sides by L, and reverse the positions of the left-hand expression and the right-hand expression to put the dependent variable, h, on the left-hand side of the equation, which is the standard convention.
h = (T0 ∙ (1–x√(P/P0)))/L
The last step is to restore the exponent, x, to its original form. Here, we have a radicand which is a little difficult to read because of all the parentheses, so we would like to use an exponent, instead. If you remember the lessons on exponents, you know that the square root of a number is the same thing as that number raised to the ½ power. In other words, to convert the xth root into an exponent, we take the inverse of the root, or 1/x. Remember,
x = ((g ∙ M)/(R0 ∙ L))
1/x = ((R0 ∙ L)/(g ∙ M))
See? Easy. But this only works for converting an expression from a root to an expression raised to a certain power.
√n = n½ = n0.5
So, our finished equation looks like this:
h = (T0 ∙ (1 – (P/P0)((R0 ∙ L)/(g ∙ M))))/L
Through similar mathematical magic, I should be able to calculate the height without T0, but using Tτ. And finally, I would be able to calculate T0 using a planet’s albedo, insolation (irradiance from the sun), molar mass from the atmospheric constituents, the planet’s gravitational acceleration, surface pressure, the approximate “constant” of 0.1 bar for tropopause pressure, lapse rate, and our new h formula which doesn’t depend on surface temperature.
My agenda is not really so hidden to those who have known me awhile. The end result of all these mathematical calculations is a quantitative confirmation of the hypothesis that the Greenhouse Effect (GHE) is unnecessary to explain either of the scientific problems of the Climate Alarmists:
- CO2 is not the 800-pound gorilla controlling global temperature, and
- CO2 is not the factor which keeps Earth from freezing, with the Effective Temperature equation deficit.
But there’s another reason I’m interested in finding this ultimate Surface Temperature equation: I want to find out if Mars can be made comfortably warm with the appropriate atmosphere. I would like to see if adding oceans to Mars, with this atmosphere is not going to freeze up and ruin the entire
This is the type of creative fun I live for. I was working on the “terraforming Mars” project as early as 1973.
Preliminary results look good. I still have to do more testing to confirm the results. Even I make mistakes! As a computer programmer, debugging was a significant portion of my professional activity.
Using levels of several gases up to about 10% of their toxic limit, I was able to get the temperature on Mars up to a toasty 299°K. That’s 11°C warmer than Earth is today.
The gases I used in my calculations were, argon (Ar), carbon dioxide (CO2), krypton (Kr), neon (Ne), nitrogen (N2), oxygen (O2) and xenon (Xe).
I fudged a bit on oxygen, though, bumping its value up roughly to Earth equivalent, which is about 40% of its toxic limit. So, this theoretical exercise proves that the surface of Mars can be made sufficiently warm, if we have the capability to terraform Mars at all. And that’s a comforting thought.
Imagine taking a short trip that might require 20 minutes at most—a short hop and a skip to Mars, to visit relatives, to do business or to settle into a nice long vacation on the Martian beaches.