The greenhouse effect drives the temperature increase we have seen at the global level over the last 140 years. Below is a pdf of an power point presentation that I created for a local town hall on environmental issues. I look at sunspots, an ENSO index, and carbon dioxide levels. All three are important, but CO2 has the strongest correlation to temperature. I do some plots and fit an arima model.

# #climatechange

# Are Atlantic Storms Increasing?

I fit three simple linear regression models to the logarithm of the number of storms plus one for the number of tropical storms, hurricanes, and major hurricanes from 1851 to 2014. The data can be found at

https://www.nhc.noaa.gov/climo/images/AtlanticStormTotalsTable.pdf

The results of the regressions – done in R – are below, as well as a plot of the models and data, where the fitted lines are found by taking the exponential of the fit to the model and subtracting one from the result. The blue lines are the fit and the green lines are 96.6% prediction intervals.

The models fit (to the log of the number plus one) are below. The data indicate a 0.39% cumulative increase in the number of tropical storms per year, a 0.19% cumulative increase in the number of hurricanes per year, and a 0.47% cumulative increase in the number of major hurricanes per year over the time period 1851 to 2014.

#### The Three Models:

#### Tropical Storms over time:

Call: lm(formula = lds[[i]] ~ ds[[1]])

Residuals: Min 1Q Median 3Q Max -1.50491 -0.24411 0.02969 0.22772 0.90256

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -5.237098 1.117248 -4.687 5.83e-06 *** ds[[1]] 0.003885 0.000578 6.721 2.92e-10 *** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3504 on 162 degrees of freedom Multiple R-squared: 0.2181, Adjusted R-squared: 0.2132 F-statistic: 45.17 on 1 and 162 DF, p-value: 2.922e-10

#### Hurricanes over time:

Call: lm(formula = lds[[i]] ~ ds[[1]])

Residuals: Min 1Q Median 3Q Max -1.74129 -0.24892 0.01169 0.27125 0.85703

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.9239955 1.3290087 -1.448 0.14964 ds[[1]] 0.0019150 0.0006875 2.785 0.00598 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4168 on 162 degrees of freedom Multiple R-squared: 0.0457, Adjusted R-squared: 0.03981 F-statistic: 7.758 on 1 and 162 DF, p-value: 0.005984

#### Major Hurricanes over time:

Call: lm(formula = lds[[i]] ~ ds[[1]])

Residuals: Min 1Q Median 3Q Max -1.2755 -0.3994 0.0403 0.3947 1.0793

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -8.1698419 1.6829089 -4.855 2.82e-06 *** ds[[1]] 0.0046922 0.0008706 5.390 2.45e-07 *** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5278 on 162 degrees of freedom Multiple R-squared: 0.152, Adjusted R-squared: 0.1468 F-statistic: 29.05 on 1 and 162 DF, p-value: 2.454e-07

#### Comments:

It is known that we are doing a better job of counting storms than in the 19th century, since some storms never make landfall, so would not necessarily have been counted in the 1800’s or early 1900’s. However, the largest increase above is for major hurricanes – which should have been measured quite well over the full time period of the data.

#### The R Code:

atl.strm.plot.fun <- function(ds = atl.strm) { print(ds[1:10,]) lds = log(ds[,2:4]+1) mod.log.ds = list(1,2,3) pred.ds = list(1,2,3) for (i in 1:3) { mod.log.ds[[i]] = lm(lds[[i]]~ds[[1]]) pred.ds[[i]] = predict(mod.log.ds[[i]], interval="predict", level=0.966) } pred.ds = lapply(pred.ds,exp) pred.ds = lapply(pred.ds,"-",1) yl=c(35,20,12) par(mfrow=c(3,1), mar=c(4,4,2,1), oma=c(1,1,3,1)) for (i in 1:3) { plot(ds[[1]],ds[[i+1]], ylab=colnames(ds)[i+1], xlab="Year", cex=.75, ylim=c(0,yl[i])) lines(ds[[1]],pred.ds[[i]][,1], col="blue") lines(ds[[1]],pred.ds[[i]][,2], col="green") lines(ds[[1]],pred.ds[[i]][,3], col="green") } mtext("Atlantic Storms", side=3, font=1, outer=T, line=1) lapply(mod.log.ds,summary) }

#### The First Ten Lines of the Data Set:

Year Tropical_Storms Hurricanes Major_Hurricanes 1 1851 6 3 1 2 1852 5 5 1 3 1853 8 4 2 4 1854 5 3 1 5 1855 5 4 1 6 1856 6 4 2 7 1857 4 3 0 8 1858 6 6 0 9 1859 8 7 1 10 1860 7 6 1

# The Mechanics of Climate Change: Some Comments on a Sidebar Article in Amstat News

(The article and comments can be found at http://magazine.amstat.org/blog/2010/03/01/climatemar10/. The book **Storms of my Grandchildren**, by James Hansen, published in 2009, by Bloomsbury USA, is a good overview of the causes of climate change. Dr. Hansen is a climate researcher and much more knowledgeable and experienced than myself in the area of climate research. As a disclaimer, Dr. Hansen’s sister is married to my second cousin, who sent my husband and me the book.)

To the editor of the **Amstat News**,

I read the article, “The Paper that Convinced Me of the Connection Between Carbon Dioxide and Climate Change”, by Peter Guttorp, in the March 2010 issue of **Amstat News**, with interest. I have been interested in the greenhouse effect since reading the **Whole Earth Catalog** in the summer of 1973, going to the sources given in the catalog for information about solar energy and the digestion of waste to produce methane, and doing my senior thesis in physics at Reed College on the potential for using solar energy for space heating in western Oregon. I built two hot water solar absorbers and modeled the heat transfers involved in heating an antifreeze solution pumped through one of the absorbers. The heat transfers involved in a solar absorber are similar to the heat transfers involved in the greenhouse effect of climate change. The **Whole Earth Catalog** mentions the greenhouse effect, as well as giving many fine references for alternative energy research. Also, I am a student of astrology and, as such, have familiarized myself with the mechanics of the solar system, including the earth’s orbit and rotation. I, also, am a amateur radio operator, so I have practical experience working with electromagnetic waves. And, I am a statistician, having received a masters and PhD from Iowa State in statistics and having done a variety of work using my education over the last 27 years. I feel like I have a unique set of knowledge to comment on Guttorp’s article.

#### Some Astronomy

To start, what are the mechanics of the earth’s orbit and rotation? For the hard facts involving the astronomy, I used the book, **Astronomical Algorithms, 2nd Edition**, by Jean Meeus, published by Willmann-Bell, Inc., in 1998. I, also, took some information from Wikipedia and used some information from the physics textbook, **Classical Mechanics**, by Herbert Goldstein, Ph.D., published by Addison-Wesley, in 1950.

In the early 1600’s, Kepler found, by looking at the detailed and extensive observations of the astronomer, Tycho Brahe, that the orbits of the planets around the sun are essentially ellipses. Elliptical orbits have perihelia and an aphelia, which are of interest in climate change modeling. The perihelion is the point on the orbit where the planet is closest to the sun and the aphelion is the point on the orbit where the planet is farthest from the sun. Kepler also discovered that the orbits sweep out area at a constant rate, which we now know is due to the conservation of angular momentum. We, also, know the actual orbits of the planets are not perfect ellipses. Some terminology is in order. For orbits, deviations from a perfect ellipse are due mainly to the gravitational effect of celestial bodies other than the sun, called perturbations (see Meeus, p. 163). Astronomers use the term, mean, to describe values of variables that, in some sense, are not corrected for perturbations or nutation (to be described below), and, in the case of the sun’s mean longitude, the eccentricity of the earth’s orbit (see Meeus, p. 183). The ‘true’ positions of celestial objects are called geometric positions (see Meeus, p. 412). If one corrects for aberration, which is the correction for the fact that light travels at a finite speed, results are called apparent results.

The earth orbits around the sun in a plane, which is called the ecliptic. Looking down from the north, the earth orbits in a counter-clockwise direction around the sun. The earth also rotates, around the polar axis of the earth (the imaginary line that goes through the south and north poles of the earth). The earth rotates in a counter-clockwise direction, looking down from the north. However, the polar axis is not perpendicular to the ecliptic. The angle between the polar axis and a perpendicular to the ecliptic is called the obliquity of the ecliptic. The earth’s rotation is gradually slowing down, at a non-uniform rate. Since the rotation is slowing down, there is a difference between the time the earth is actually at a given point and the estimated mean time expected. The difference is called the equation of time. The value of the difference, called delta T, is found by observation, so is only known for the past and present, but can be estimated into the future. In Meeus in Chapter 10, the above is discussed and values for delta T are given for the years 1620 to 1998, for every other year. Universal time (UT) is not corrected for delta T and dynamical time (TD) is.

#### The Precession of the Equinoxes and the Obliquity of the Ecliptic

When a top spins, the top of the top makes slow circles. Similarly, the polar axis of the earth circles around the polar axis of the ecliptic – which can be thought of as an imaginary line through the center of the earth perpendicular to the plane of the ecliptic. A complete circle takes about 26,000 years (Meeus, p. 131). According to Meeus (p. 131) the circling is caused by “the gravitational attraction of the sun and the moon on the earth’s equatorial bulge.” Because of the circling, the polar axis of the earth is tipped from being perfectly perpendicular to the ecliptic by the obliquity of the ecliptic. Right now, the obliquity of the ecliptic is about 23.4 degrees from the vertical. The obliquity of the ecliptic is slowly decreasing over time, with a (almost linear at this time) decrease of about 1.75 degrees over 200 centuries (Meeus, p. 148). There are also periodic oscillations, called nutation, in the circling of the polar axis (see Meeus, p. 143). Nutation is taken into account in calculating geometric placements, but not in mean placements.

Because the polar axis is making a circle over a 26,000 year time period, the places on the ecliptic where the solstices and equinoxes occur precess (move with respect to the fixed stars, which we assume are fixed, though the stars move too, very slowly, see Meeus, p. 150), and the movement is in the opposite direction of the orbit of the earth (the precession is clockwise, looking down from the north). The precession is called the precession of the equinoxes and is about 50 seconds of arc per year (Meeus, p. 131).

During a year, the time it takes for the earth to come back to the point of the spring equinox, or any other point in the seasons, is called a tropical year, and varies with the seasonal point chosen. The mean time, averaged over all seasonal points, is called the mean tropical year and currently takes 365.2422 days, not the 365.2442 days described in Guttorp’s article. The tropical year is mentioned on p. 133 of Meeus and a discussion can be found on Wikipedia, under the entry for tropical year.

#### The Seasons of the Year

The seasons of the year result because the polar axis is tipped. As the earth orbits around the sun, since the earth always stays tipped in the same direction, at just two points in the orbit the polar axis will be in line with the imaginary line connecting the earth and the sun. When the polar axis is in line with the imaginary line, the solstices occur. For the winter solstice, the polar axis is pointed away from the sun and for the summer solstice, the polar axis is pointed toward the sun. There are, also, two points in the orbit where the polar axis is perpendicular to the earth/sun line. When the polar axes are perpendicular to the earth/sun line, the spring (vernal) and fall (autumnal) equinoxes occur, on which days, day and night have the same length.

A table of the dates and times of equinoxes and solstices for the years 1996 to 2005 can be found on p. 182 of Meeus. Using formulas from Meeus in Chapters 21 and 25 for the mean precession and using a formula found on Wikipedia (under the entry for tropical year) for the length of the mean tropical year, I calculated mean values for the precession and length of the mean tropical year for 1860 and 1980. According to my calculations, in 1860, the mean precession was 50.466 seconds of arc per year and the mean tropical year was 365.242198 days, and in 1980, the mean precession was 50.315 seconds of arc per year and the mean tropical year was 365.242191 days. I calculated the precession of the sun over the year using the mean longitude of the sun. So, by Meeus’s formulas, the mean precession is decreasing in time and the mean tropical year is also decreasing in time.

#### The Movement of the Perihelion and the Aphelion and the Anomalistic Year

The points of perihelion and aphelion also move, with respect to both the stars and the seasons. The points of perihelion and aphelion move in the direction of the orbit (counter-clockwise, looking down from the north) , currently at the rate of about 47 seconds of arc per century (Meeus, p. 131). According to Meeus, the movement is “due to the gravitational attraction of the planets to the Earth”. As the earth orbits the sun, the time period for the return of the earth to the point of perihelion (or aphelion) is the anomalistic year given in the article by Guttorp. The time period of 365.2596 days given by Guttorp agrees with Meeus, although from Meeus’s formulas the length of the anomalistic year is slowly increasing.

From the above, the solstices and equinoxes move in the opposite direction from the perihelion and aphelion, with respect to the stars. Since the solstices and equinoxes move backward and the perihelion and aphelion move forward, the day on which the perihelion or aphelion occurs is later in the year as time goes by. The perihelion and winter solstice were coincident in 1246 AD, according to Meeus, p. 181. Formulas for the day of mean perihelion and aphelion in a given year can be found in Meeus, on p. 269. Using the formulas in Meeus, the mean times of perihelion and aphelion in 1860 were January 1st at 15.80 hours universal time (UT) and July 2nd at 6.91 hours UT. For 1980, the times were January 3rd at 19.54 hours UT and July 4th at 10.66 hours UT.

On p. 274, Meeus gives the true times of perihelion and aphelion for the years 1991 to 2010. The times are given in dynamical time. Using the table in Chapter 10 of Meeus, I converted the dynamical times of perihelion to universal time for the years 1992, 1994, 1996, and 1998. Then, using the astrological program, Kepler, I calculated the longitudes of the sun at the times of the perihelia. Longitudes are measured in a counter-clockwise direction along the ecliptic from the spring equinox, looking down from the north. On 1/3/1992 at 15h 02m 38s UT, the perihelion was at 282 degrees 29 minutes longitude, on 1/2/1994 at 5h 54m 12s UT the perihelion was at 281 degrees 42 minutes longitude, on 1/4/1996 at 7h 24m 46s UT, the perihelion was at 283 degrees 13 minutes longitude, and on 1/4/1998 at 21h 15m 45s UT the perihelion was at 284 degrees 19 minutes longitude. We can see that the longitudes do not vary linearly with respect to time. There is quite a bit of variation even for the four years given. I am assuming that the times given in Meeus are correct. If the times are correct, the effect of nutation and perturbations can be quite large and swamp the gradual shift of the perihelion and aphelion eastward.

#### Combining the Cycles

Given the above, the discussion of the two close together cycles, as described by Guttorp, does not make sense. The actual two cycles are, first, the combination of the precession the perihelion/aphelion with the precession of the equinoxes, which cause the seasonal points to shift clockwise (looking down from the north) with regard to the perihelion and aphelion, and, second, the seasons, which result from the tilt of the earth’s polar axis. The precession cycle is very long (about 26,000 years) and the seasonal cycle is just a yearly cycle. (A third celestial mechanical phenomena that might be of interest is the gradual decrease of the obliquity of the ecliptic. A fourth is the changing eccentricity of the orbit of the earth. The earth’s eccentricity is currently decreasing.

See the Wikipedia article on orbital eccentricity.)

##### Solar Energy Incidence

The heating of the earth mainly depends on energy from the sun. If we are concerned about climate change, we are interested in the heating of the earth, which depends partly on the incidence of energy at the top of the earth’s atmosphere. The incidence of solar radiation at the top of the atmosphere of the earth depends on the distance of the earth from the sun and the solar flux, both of which can vary quite a bit.

First we look at the distance. In Goldstein, Chapter 3 is on the two body central force problem, and Section 3-6 is on Kepler’s results. Goldstein gives the result that the instantaneous area swept out by a planet in an elliptical orbit is equal to [one half] times [the distance between the sun and the planet squared] times [the instantaneous rate of change in the angle swept out by the orbit]. The [distance between the sun and a planet] times [the instantaneous rate of change in the angle] is the speed of the planet in the planet’s orbit, if angles are measured in radians. As stated above, since angular momentum is conserved, the rate at which area is swept out is constant for an elliptical orbit. So [the speed of a planet] times [the distance from the sun to the planet], which is the rate at which area is swept out, would be a constant if the planet’s orbit were perfectly elliptical. At closer distances, the planet moves faster, and at farther distances, the planet moves slower.

We will explore solar energy incident on the earth for a given solar flux. Radiation from the sun spreads out in a cone like fashion from the surface of the sun. The solar energy incident on the earth would be that incident on a flat silhouette of the earth at the distance of the center of the earth. To see how distance affects incident energy, let us define a right triangle, centered at the center of the earth, with one leg going to the center of the sun and the other leg to the surface of the earth. If the earth were a perfect sphere, the radius of the earth would not depend on location, which we will assume. Using the triangle described above, one can see quite easily the radiation incident on the earth varies inversely with the distance from the sun, with the distance raised to the power of two, since the area of a circle is proportional to the square of the radius of the circle. Radiation then should be proportional to the inverse of the distance squared. I do not think that the speed of the earth affects the amount of sunlight incident on the earth.

##### Insolation at Perihelion and Aphelion

Due to the precession of the equinoxes and the precession of the perihelion and aphelion, the distance from the earth to the sun by time of year is changing. Presently, for the northern hemisphere, the sun is closest to the earth in the winter and farthest in the summer. Meeus, on p. 274, gives a measure of the distance between the earth and the sun at perihelion and aphelion for the years 1991 to 2010. On January 3, 2010, which is the day that the earth went through perihelion this year, the distance between the center of the sun and the center of the earth was .983290 astronomical units. On July 6, 2010, which is the day that the earth went through aphelion this year, the distance between the sun and the earth was 1.016702 astronomical units. So, for a perfectly elliptical orbit and constant solar flux, there would be a 3.4 percent increase in the distance from perihelion to aphelion and, I would think, a 6.5 percent decrease in incident energy. An astronomical unit is the close to the mean distance between the sun and the earth (see Meeus, p. 411) and equals about 150 million kilometers (see Meeus, p. 407).

##### Changes in Seasonal Lengths

The lengths of the seasons are changing too. On p. 182, Meeus gives the lengths of the seasons, in the northern hemisphere, from -4000 to 6500, in 500 year increments. From

-1500 to 6500, the spring gets shorter. From -4000 to 3500, the summer gets longer. From -1500 to 6500, the autumn gets longer. From -4000 to 3500,the winter gets shorter. At the present time, the transit of the perihelion occurs a few weeks after the winter solstice. So, winter getting shorter in the northern hemisphere, as the day of the fastest speed of the earth moves toward the midpoint of the winter season, makes sense. Following the same logic, spring in the northern hemisphere is also getting shorter as the perihelion approaches the spring equinox and autumn in the northern hemisphere is getting longer as the perihelion moves away from the fall equinox. For the summer in the northern hemisphere, the time of slowest speed is approaching the midpoint of the summer season, so the summers are getting longer.

##### Celetial Phenomena and the Heating and Cooling of the Earth

What effect might the celestial phenomena have on the heating and cooling of the earth? Two long term major celestial phenomena that might affect weather are the shifting of the perihelion with respect to the seasons and the slowly decreasing size of the obliquity of the ecliptic. The changes in the eccentricity is of much longer duration. The shifting of the perihelion causes both changes in the distance from the earth to the sun by season and the lengths of the seasons.

At this time in geological time, I would think that the effect of the shifting perihelion on the seasons would be to cool winters and springs and to warm summers and falls in the northern hemisphere, since winters and springs are getting shorter and summers and falls are getting longer. In the southern hemisphere, I would expect shifting seasons to warm winters and springs and to cool summers and falls.

I would expect the effect of the shifting perihelion on the distance of the earth from the sun by time of year to warm winters and springs and cool summers and falls in the northern hemisphere, since the earth is getting less distant in the winters and springs and more distant in the summers and falls. For the southern hemisphere, I would expect the effect to cool winters and springs and warm summers and falls.

Looking next at the decrease in the obliquity of the ecliptic, if the angle the polar axis of the earth makes to the ecliptic increases (the obliquity of the ecliptic decreases), I would expect the to see more solar radiation at the higher latitudes in falls and winters and less in springs and summers for either hemisphere. The obliquity of the ecliptic is decreasing, at about 47 seconds of arc per century at the present (Meeus, p. 408).

Combining the two celestial influences, the precession and the change in the obliquity of the ecliptic, in the northern hemisphere changes in the lengths of seasons would make things cooler in the winters and springs and warmer in the falls and summers; changes in the distance from the sun to the earth by season would make things warmer in the winters and springs and cooler in the summers and falls; and changes in the obliquity would warm things in falls and winters and cool things in the springs and summers. So, for the all of the seasons there are both warming and cooling influences.

For the southern hemisphere, changes in the lengths of seasons would make things warmer in the winters and springs and cooler in the summers and falls; changes in the distance from the sun to the earth by season would make things cooler in the winters and springs and warmer in the summers and falls; and changes in the obliquity would warm things in falls and winters and cool things in the springs and summers. Once again, all of the seasons have influences for both cooling and warming.

#### The Solar Flux

Actual insolation depends on the solar flux as well as the orbital and rotational influences given above. There is a plot of recent levels of solar radiation incident on the earth’s atmosphere, made possible by satellite measurements, at the Wikipedia entry on the solar cycle. From the plot, the total solar radiation incident at the top of earth’s atmosphere is about 1366 watts per meter squared and varies daily and with the sunspot cycle of eleven years. The plot shows that the solar flux increases with increasing sunspots.

#### Electromagnetic Radiation

We are interested in the causes of climate change, so I will write a little about the greenhouse effect. What is the greenhouse effect? The greenhouse effect is the natural process by which the earth stays warm enough for life as we know life. The effect is a result of the absorption, and emission of electromagnetic waves, also called photons, by the atmosphere and the earth. Electromagnetic waves are oscillating electric and magnetic fields that propagate through space. Electromagnetic waves are characterized by the wavelengths of the waves. Some familiar types of electromagnetic waves are light, with wavelengths from 0.4 to 0.7 microns, the infrared, with wavelengths from 0.7 to 300 microns, microwaves, with wavelengths from 1 millimeter to 1 meter, FM radio, with wavelengths from 2.78 to 3.43 meters, and AM radio, with wavelengths from 175 to 570 meters. As wavelength increases, the energy of an electromagnetic wave decreases. For the greenhouse effect, wavelengths in the visible and infrared are of interest. (The sources for the wavelengths given above are the Wikipedia entries for visible spectrum, infrared, microwave, FM broadcasting, and AM broadcasting. Note that, for electromagnetic radiation, [wavelength in meters] is the same as [three hundred] divided by [frequency in megahertz], which I know from my amateur radio work.)

The mechanism of the greenhouse effect is the absorption and emission of electromagnetic thermal radiation by molecules in the atmosphere. Basically, the atmosphere passes light, which heats the earth, which emits in the infrared, some of which radiation is absorbed by the atmosphere, which emits infrared in all directions including back to the earth, which slows the cooling of the earth. When an electromagnetic wave of the right wavelength hits a molecule, the wave is absorbed and the energy of the molecule increases. However, a wave will be emitted by the molecule at a later time. Between the atmosphere and the ear and within the atmosphere, there is a constant exchange of energies through the absorption and emission of electromagnetic waves. To repeat, thermal radiation is emitted by the earth and some of the energy is absorbed by the atmosphere, and emitted by the atmosphere in all directions. If the greenhouse gases in the atmosphere increase, the process of cooling is slowed, since more thermal radiation is returned to the earth with the increased absorption and emission of waves by the atmosphere. Also, the heating of the atmosphere pushes the surface of the atmosphere into a cooler region, which means the amount of energy emitted into space from the earth decreases – by Planck’s Law.

#### Blackbody Radiation

What is described above is a quick description of what is going on physically with the greenhouse effect. On a deeper level, our understanding of quantum mechanics explains the effect. As was discovered by physicists in the last half of the nineteenth century, all materials radiate thermal electromagnetic radiation. With the development of quantum mechanics in the early part of the twentieth century, scientists understood why. For any material, the spectrum and intensity of the thermal radiation from the material depends on the type of material and the temperature of the material. The spectrum is the product of [the spectrum of the radiation from a blackbody at the given temperature] and [the emissivity of the material at the given temperature]. The emissivity of a material, which varies with wavelength and temperature, can take on values from zero to one, inclusive.

The radiation from a blackbody is described by Planck’s law, which was found empirically by Max Planck at the end of the nineteenth century and published in 1901. Planck’s law gives the power of the emissions of a blackbody at a given temperature per unit area per wavelength or frequency per solid angle for any given electromagnetic wavelength or frequency. From quantum mechanics, we know blackbody radiation is the result of electrons jumping between electron shells within atoms as atoms absorb and emit electromagnetic waves. Two formulas for Planck’s law are given in the Wikipedia entry for Planck’s law, one for frequencies and one for wavelengths. At first glance, the two formulas appear different, since we know frequency is just the product of [the speed of light] and [the inverse of the wavelength]. However, the formulas are densities and the difference between the formulas is a result of changing the variable of integration from frequency to wavelength.

The radiation from the sun is essentially blackbody radiation for material at the temperature of the surface of the sun, which is about 5800 degrees Kelvin. Using the formula for the spectral density by wavelength from Wikipedia and the computer program R, I plotted the spectrum for a blackbody at 5800 K. The radiation is most intense between 0.4 microns and 0.7 microns, the range of visible light, with the peak of the energy spectrum being in the wavelengths of green light, around 0.5 microns. Again using R and the formula for the spectral density by wavelength from Wikipedia, I plotted the spectrums for temperatures of -33 C, -8 C, 17 C, and 42 C, typical temperatures for the materials at the surface of the earth. The heights of the peaks of the power decrease fourfold over the four temperatures, and the wavelengths associated with the peaks vary from around ten to thirteen microns, which wavelengths are in the infrared. So, depending on the emissivity of the material at the surface, the thermal emissions from materials at the temperatures at the surface of the earth peak in the infrared.

#### Absorptivity, Reflectivity, and Transmissivity

For any material, with respect to electromagnetic waves, incident energy is absorbed, reflected, or transmitted. At any given wavelength and temperature, the sum of the absorptivity, reflectivity, and transmissivity is one. At the surface of the earth, electromagnetic waves in the visible and the infrared are either absorbed or reflected. Of the energy that is reflected by the earth, some is reflected back from the atmosphere, some is absorbed by the atmosphere and radiated in all directions, and some passes through the atmosphere back into space. Of the energy that is absorbed, some of the energy is converted to plant growth and some of the energy in converted to heat (or electricity with the use of solar cells). In the visible and the infrared, no energy is transmitted through the earth.

When solar radiation hits the earth, the rays are absorbed or reflected according to the albedo (reflective nature) of the material at the surface of the earth. The albedo (reflectivity) for some common surfaces are given at http://ice-glaces.ec.gc.ca/WsvPageDsp.cfm?ID=10165&Lang=eng. Looking at some common earth surfaces, the albedo for seawater is between 0.05 and 0.10 in the visible. For the surface of the earth, light that is not reflected is absorbed, so the absorptivity of seawater is from 0.90 to 0.95 in the visible. So, seawater absorbs most of the incident radiation from the sun. For soils the albedo is low in the visible and absorptivity high. Snow has a high albedo in the visible, so reflects much of incident sunlight. (In the infrared, snow has a low albedo and absorbs most of the energy incident.)

We are also interested in the emissivity of materials on the surface of the earth, since radiation emitted at the surface of the earth is absorbed by the atmosphere in the greenhouse effect. At http://www.omega.com/literature/transactions/volume1/emissivityb.html is a list of the emissivity of many materials for given temperatures. At everyday temperatures, polished metals have a low emissivity, metal oxides are quite high, water is quite high, as is snow. Soils are around the middle.

#### Insolation at the Surface of the Earth

Since the absorption of radiation at the surface heats the earth and determines the spectrum of thermal radiation emitted by the earth, we are interested in insolation at the surface of the earth. For a given solar flux, the amount of energy incident from the sun at the surface of the earth depends on the thickness and content of the atmosphere. The thickness of the atmosphere affects the amount of radiation incident, since light can be both absorbed and reflected in the atmosphere. The thickness depends on both elevation and the angle of the incident radiation. As elevation increases, the distance to the top of the atmosphere decreases. As the angle of incidence gets shallower, the amount of atmosphere solar radiation passes through gets larger. The content of the atmosphere, mainly the amount of water in the atmosphere, also has a large effect on what happens to solar radiation in the atmosphere. Water vapor can absorb sunlight. Water droplets can block and scatter sunlight. Other particles in the atmosphere also affect the transmission of light. Particles, say of soot or volcanic ash, can scatter and absorb sunlight. Energy that does not get through or stay in the atmosphere is either reflected by the atmosphere or absorbed in the atmosphere and then emitted into space.

At sea level, perpendicular to the sun’s rays, on a sunny day, the insolation averages about 1000 watts per square meter, according to the Wikipedia entry for insolation. After passing through the atmosphere, the density of the energy incident on the ground depends on the angle of incidence, which varies with sky cover, latitude, season, and time of day. The angle tends to be shallower at more extreme latitudes, the winter season, and away from true noon. Shallower angles mean less energy incident.

#### Gases in the Atmosphere and the Greenhouse Effect

From the Wikipedia entry for the Earth’s atmosphere, the main components of the atmosphere, for dry air, are nitrogen (about 78 percent), oxygen (about 21 percent), and argon (about 1 percent) by volume. On average, the about 1 percent of the atmosphere is water. Carbon dioxide makes up about 0.04 percent of the atmosphere. J. L. Schnoor, in **Environmental Modeling: Fate and Transport of Pollutants in Water, Air, and Soil**, published in 1996 by Wiley, in Chapter 11, gives scientific information about climate change and the greenhouse effect and the atmosphere. Water vapor is the main greenhouse gas. Schnoor states, on p. 612, that water vapor and preindustrial carbon dioxide, are responsible for about 98 percent of the heating resulting from the thermal radiation of the earth. We have life on earth because of the heating.

As stated above, the greenhouse effect is the result of the atmospheric absorption of the radiation emitted by the earth. Why does the atmosphere absorb in the infrared? For absorption, the amount of incident electromagnetic energy absorbed by a material depends on the absorptivity of the material. According to my excellent freshman year physics text, **Basic Physics**, by Kenneth Ford, published in 1968, by Blaisdell, on p. 596, molecular vibration and rotation is responsible for the emission of electromagnetic waves in the infrared, and good emitters are good absorbers. (Note, molecular bonds are the result of atoms sharing electrons in the outer shells of the atoms. The oscillation of the outermost electron in atoms is responsible for emissions in the visible.)

The earth emits in the infrared and, since greenhouse gases are molecules, greenhouse gases in the atmosphere absorb the earth’s radiation. There are greenhouse gases other than carbon dioxide and water, many of which absorb more radiation per molecule. In Schnoor, on p. 608, the concentrations in the atmosphere, as well as other information such as the relative absorptivity, of several greenhouse gases are given.

Some physical processes that both warm and cool the earth are described in Schnoor. Water droplets, in the visible, block energy from the sun, which has a cooling effect. Water vapour does transmits electromagnetic radiation in the visible. In the infrared, water vapor is a greenhouse gas and slows the cooling of the earth. One of the effects of a warmer earth is an increase in the evaporation of water into the atmosphere, which leads to increased clouds, which blocks sunlight, which cools the earth (Schnoor, p. 614). Also, because of sulfate aerosols generated by the burning of coal, clouds are brighter and reflect more sunlight back into space, which cools the earth. But the gaseous products of coal combustion are greenhouse gases and warm the earth (Schnoor, p. 614). Schnoor writes that weather patterns, such as El Nino, are important, too (Schnoor, p. 615).

#### Comments

So, do we need to worry about climate change? I think so. The loading of the humanly generated greenhouse gases is small, but steady, even though water vapor changes greatly with location and season. In statistics, we are trained to look at trends in data and to separate noise from signal. However, for observational data, we are trained to not assign cause based on correlation. When I took my methods course at Iowa State from Paul Hinz back in 1983, Dr. Hinz gave the example of cancer and smoking. While all the world wanted statisticians to say because there was a correlation between smoking and cancer, smoking causes cancer, we could not say the reason was causal. We had to wait until bio-chemists found a physical cause. Then, the observed correlation supported the physical theory. Dr. Hinz, in support of our hesitancy, gave the example of storks and babies in Copenhagen. There is a strong correlation between the number of storks and the number of babies in Copenhagen. Do storks bring babies? The physicists who model the greenhouse effect have reason to think we have a problem based on known physical theory. As statisticians, we can only say whether the data does or does not support the theory. As to the Guttorp article, I suspect the effect of the precession’s is too small and the noise too large to pull out a change in the climate over the short period of one hundred twenty years, but I am not familiar with the technique used. I do not know how sensitive such a technique would be.

#### Three Celestial Coordinate Systems

As an aside, I will describe the coordinate systems used in describing celestial phenomena. Three coordinate systems are commonly used to describe the position of any point of interest in the heavens with respect to the earth. The three systems are with respect to the ecliptic, the earth’s equator, and the horizon. In each system, there are two variables. The first variable measures the angle between some reference point on the circle of reference (a point on the ecliptic, equator, or horizon) and the intersection of the shortest line from the celestial point of interest (such as the sun or a star) to the circle of reference (perpendicular). The second variable measures the elevation, either above or below, of the point of interest from the circle of reference.

##### Ecliptic

The first system is longitude and latitude. Longitude and latitude are measured with respect to the ecliptic, as seen from the earth. Longitude is measured in degrees of arc east of the point on the ecliptic which the sun occupies at the spring equinox and is measured from spring equinox to spring equinox. The longitude of the sun is the point on the ecliptic behind the sun, looking out at the ecliptic from the earth, since the sun is essentially on the ecliptic. Latitude is the elevation north or south of the ecliptic and is always essentially zero for the sun.

##### Equator

The second coordinate system is right ascension and declination. Right ascension and declination are measured with respect to the equator. Right ascension is a measure of the angle around the equator from the point on the equator associated with the spring equinox. Right ascension is measured in sidereal time and increases as the point of interest is more east than the point of the spring equinox. The time is called sidereal time since the time is measured against the stars rather than the sun. A sidereal day is slightly shorter than a solar day, there being 366 sidereal days in a year, as opposed to 365 solar days, or 367 sidereal and 366 solar days in a leap year. (Another equivalent measure of angular distance around the equator is the hour angle, which is measured in degrees and is measured from an arbitrary meridian, either east or west. The relative values of the hour angle and the right ascension are the same if the right ascension is multiplied by 15 to convert hours to degrees.) The declination is the angle of the object of interest above or below the equator, as seen from the equator.

##### Horizon

The third coordinate system is azimuth and altitude. Azimuth and altitude are measured from the horizon. While the starting point and direction of azimuth varies by reference, Wikipedia, in the entry for azimuth, gives azimuth as being the angle along the horizon to the point of interest, measured east from true north. The altitude is the angle of the point of interest, above or below the horizon.

# Sunspot and Temperature Anomaly Data and their Bivariate Phase Spectrum

I start this blog post with a plot of the sunspot and temperature anomaly data that I have been using. The data is from 1850 to 2013.

The following plots are of the phase spectrum between the sunspot time series and the temperature anomaly time series. Also, the variable tau is plotted next to the spectrum. Tau would give the slope of the phase spectrum curve if the intercept were zero for a line fitting a stretch of curve. Tau is the phase spectrum divided by the frequency. The first set of plots uses all of the data. The second set of plots just uses the first 250 points. There are 1968 months in each time series.

From the plot, with frequencies up to around 275 cycles per 1968 periods (1968 periods divided by 275 cycles equals about 7 periods per cycle) have mainly an increasing phase relationship. From about 300 cycles per 1968 periods to 800 cycles per 1968 periods (about 7 periods per cycle to about 2.5 periods per cycle), the slope is mainly decreasing. For frequencies greater than 800 cycles per 1968 periods, the mainly slope is flat.

Constant positive slopes indicate that sunspots precede temperature anomaly with a constant lag. Constant negative slopes indicate than temperature anomaly precedes sunspots with a constant lag. A flat slope indicates no lagged relationship.

Looking at the plots of tau above, out to around 22 cycles per 1986 periods (about 90 periods per cycle), tau was changing quite a bit. Out beyond around 250 cycles per 1986 periods (about 7 periods per cycle), tau does not change much.

# Running Averages for a Sunspot Period of 129 Months and a Model Fit for the First 69 Years 7 Months

I have found the source of my temperature anomaly data – at Berkeley Earth – the data is the monthly global average temperature data for land and ocean, http://berkeleyearth.org/data. It took several hours of searching.

In my first graph of this blog I plot the graph of the last blog using a period of 129 months for the sunspot cycle. To choose the period, I used the F test from Fuller’s** Introduction to Time Series**, 1976, Wiley, p.282 to compare different periods close to 132. The period of 129 had the largest F value. The period is 10 years and 9 months.

The second graph I put up is of two plots. The first plot is of the temperature anomaly shown above along with a curve of predicted values, where the predictions are from the model generated by the regression of the 88th through 922th observations of the temperature anomaly vector on the 1st through the 835th observations of the sunspot vector. The model has the largest R squared of the several hundred single variable lag models at which I looked. The best fitting model was for a lag of 87 months fitting 835 observations.

The second plot is the difference between the two curves in the first plot.

This is a very simple model. I was surprised by the drop in the differences in the 1950’s and early 1960’s, but the trend is of an increasing difference as time goes on.

# Running Averages of Global Average Temperature and Sunspot Numbers

In this blog post, I have uploaded a graph of the running average over 132 months (11 years) of the average number of daily sunspots per month as found by the Royal Observatory of Belgium Av. Circulaire, 3 – B-1180 Brussels, Belgium and the running average over 132 months of the global average temperature anomaly taken from a source I have lost and cannot find. I believe the temperature anomalies are for air temperature over the ocean. The years are 1850 to 2013. I created the graph.

From 1855 to 1900, it appears that changes in the temperature anomaly follow the sunspot numbers by a year or two. And temperatures loosely increased as sunspots increased but not like from 1855 to 1900. By the average at 2008 there was no connection.

The source of just about all heat coming into this planet is the sun. Sunspots appear to be related to the solar flux, more sunspots correlate with a denser flow of energy, so one would expect the temperature of the earth to correlate with the sunspots.

I tried to take out the cycle of the sun spots and focus on the level of sunspot activity by taking a running average of the observations over the approximately 11 year sunspot cycle. The averages run from July of 1855 to June of 2008.

The plots indicate to me that there is more to global temperature than the sunspot cycle over the years for which we have data.