**Conservation of Energy & Sunspot Number Time-Integral**

**Conservation of Energy**

Earth is in the vacuum of space so the only way that the
planet can gain or lose significant energy is by
electromagnetic radiation. The energy equation applied to the planet is:

Energy IN minus
energy OUT = CHANGE in energy stored.

Over centuries, the average global temperature trend doesn’t
change much. This means there must be a breakeven for energy IN and energy OUT.
During a shorter period, changes to average global temperature are brought
about by deviations from breakeven.

Energy IN is the change, above or below breakeven, of energy
acquired by the planet. The planet can acquire (or lose) energy by two fundamentally different but complimentary mechanisms, both dependent on cloud changes as described in Reference 10. The hypothesis is made that this energy IN surplus (or
deficit) is proportional to the time-integral of the annual average sunspot
numbers. Since the average daily sunspot number is reported for each year, the
time-integral for a period, obtained by numerical integration (same as ‘running
total’), is simply the summation of annual, average daily values for the
period.

Radiant energy emitted spaceward from the planet is a
function of the time-integral of the fourth power of the global mean absolute
temperature. Energy OUT is the amount above or below breakeven.

The change in stored energy is the anomaly (temperature
minus a constant) times the effective thermal capacitance

^{ 1}. Thus the anomaly is found by dividing the IN-OUT difference by the effective thermal capacitance.**Influence of sunspot numbers**

Sunspot numbers have been regularly recorded since 1610.
NASA provides a fairly descriptive website

^{2}with some background information. The following graph^{ 3}, Figure 1, shows some recorded sunspot numbers.**Figure 1. Sunspots have been regularly recorded for over 400years.**

The Maunder Minimum (1645-1715), an era of extremely low
sunspot numbers, was associated with the Little Ice Age. The Dalton Minimum
(1790-1820) was a period of low sunspot numbers and low temperatures. An
unnamed period of low sunspot numbers (1880-1930) was also accompanied by
comparatively low temperatures.

An assessment of this is that sunspots are somehow related
to the net energy retained by the planet, as indicated by changes to average
global temperature. Fewer sunspots are associated with cooling, and more
sunspots are associated with warming. Thus the hypothesis is made that the rate
at which the planet accumulates (or loses) radiant energy over time is
dependent upon the sunspot number to some degree. Hence, changes in the amount
of radiant energy retained by the planet above or below breakeven are dependent
upon the time-integral of the sunspot numbers.

Also, a lower solar cycle over a longer period might
result in the same increase in radiant energy retained by the planet as a
higher solar cycle over a shorter period. Both magnitude and time are
accounted for by taking the time-integral of the sunspot numbers, which is simply
the sum of annual mean sunspot numbers over the period of study.

**Radiation from the planet**

The radiant energy emitted spaceward from the planet is a
function of the time-integral of the fourth power of the global mean absolute
temperature, here determined by adding 287.1 K to each reported anomaly as shown
in Figure 2 of Reference 4 for 1850-2012 and assuming 286 K prior to 1850. What is of interest here is the difference, above
or below breakeven, which is subtracted from the net IN, above or below
breakeven, to determine the energy change.

**Combined equation**

It is axiomatic that change to the energy retained by the
planet is indicated by change to the average temperature of the planet.

An equation that calculates average global temperature
anomalies (AGT) since 1895 with 90% accuracy has been reported

^{ 4}. Reference 4 also shows the influence of atmospheric CO_{2}to be insignificant so it can be removed from the equation by setting ‘C’ to zero. The influence of ocean oscillations can be removed from the equation by setting ‘A’ to zero. The offset, ‘D’ must be changed to 0.6519 to account for the different integration start point and setting ‘A’ to zero. The result, Equation (1), then calculates the trend resulting from just the sunspot number time-integral.Where:

Trendanom(y) = calculated temperature anomaly trend in year
y, K degrees.

0.004894 = the proxy factor, B, from Table 1 in Reference 4, W yr m

^{-2}.
17 = effective thermal capacitance of the planet

^{ 1}, W Yr m^{-2}K^{-1}
s(i) = average daily Brussels International sunspot number
in year i

43.97 = average
sunspot number for 1850-1940. (i.e. from the start of AGT reporting to the
start of the sustained run up)

286.8 = global mean surface temperature for 1850-1940, K.

T(i) = average global absolute temperature of year i, K,

0.6519 is
merely an offset that shifts the calculated trajectory vertically, without
changing its shape, so that the calculated temperature anomaly in 2005 is
0.3282 which is the calculated anomaly for 2005 if the ocean oscillation is
included minus half of the ocean oscillation range, K degrees.

Sunspot numbers since 1700 are provided numerically

^{ 5,6}on the web (until recently by NOAA). Recent SESC sunspot numbers (which use more sensitive technology and thus are not compatible here) are also reported^{ 7}. These numerical values are extended back to 1610 using Figure 1 and recorded in an EXCEL file. Figure 2 is a graph of this file.**Figure 2: Solar cycles 1610-2012.**

Applying Equation (1) to the sunspot numbers of Figure 2 produces
the trace shown in Figure 3.

**Figure 3: Anomaly trend from just the sunspot number time-integral using Equation (1).**

Although average global temperatures were not directly
measured in 1610 (no thermometers) or even estimated to sufficient accuracy
using proxies, the anomaly that Equation (1) calculates is higher than most
estimates for that time. Also, there is no way to determine for sure how much
and which way the ocean cycles would influence the value. If the period and
amplitude demonstrated to be valid after 1895 is assumed to maintain back to
1621, the temperature in 1621 and 2005 including the influence of ocean cycles
would both be 0.2 K higher than calculated by an equation considering only the
sunspot number time-integral.

Two changes to Equation (1) reduce the temperature
calculated for 1621 by 0.21 K without changing the value calculated for 2005 (42 replacing 43.97 and 0.4305
replacing 0.6519) are shown in Equation (2).

(2)
Applying Equation (2) to the sunspot numbers of Figure 2
produces the trend shown in Figure 4. Available measured average global temperatures are superimposed on the calculated values.

**Figure 4: Trend from just the sunspot number time-integral using Equation (2) with superimposed available measured data.**

The offset constants were determined so that the temperature
anomalies calculated for 2005 by the two equations are identical. The
difference in 1909 between Equations (1) and (2) is only 0.055 K which is a
small number compared to the accuracy of determination of average global
temperature in 1909. Figure 4 shows that temperature anomalies calculated using
Equation (2) appear to be somewhat closer to early reported measurements than
using Equation (1)

**Other assessments**

A similar assessment of the influence of the time integral
of sunspot numbers

^{9}was made by Jim Goodridge in 2007, using Shove's index of sunspot numbers (calculated from planetary synodic periods) starting before the Maunder Minimum. His findings of the trajectory of the net energy rate are shown in Figure 5. Although his graph states ‘solar irradiance’, it is actually constructed using the time-integral of sunspot numbers from Shove's index with an appropriate scale factor.**Figure 5: Prior assessments suggest a causative relation between the time-integral of sunspot numbers ("...accumulated departure from the average of all sunspot numbers for the entire 500 year index.") and planet energy gain.**

This assessment also used the time-integral of sunspot
numbers as a proxy. The energy rise starting in approximately 1940 that led to the Anthropogenic Global
Warming theory is observed.

Another relevant assessment has been made public at Hockey
Schtick

^{ 8}. Figure 6 was copied from that website by permission.

**Figure 6: Assessment**

^{ 8}of the sunspot time-integral overlaid with HadCRUT3 measurements.
Here also the energy rise starting in approximately 1940 that led to the Anthropogenic
Global Warming theory is observed.

**Discussion**

Others have looked at only amplitude or only time factors
for sunspot numbers and gotten poor correlations with average global
temperature. The good correlation comes by combining the two, which is what the
time-integral of sunspot numbers does.

Future temperature anomalies depend on future sunspot numbers
and future ocean temperature oscillation behavior, neither of which has been
confidently predicted for more than a decade or so in advance although assessments using planetary synodic periods appear to be relevant. As shown in Figure
3, the sunspot time-integral has experienced substantial change over the
recorded period.

The effective global sea surface temperature oscillation,
although dominated by the PDO, depends also on complex phase interaction with
lesser oscillations. Considering all this, the effective sea surface
oscillation can be expected to fade in and out in its contribution to AGT in
future decades. However, average global temperature should continue to
correlate with the time-integral of sunspot numbers, as it has ever since
sunspots have been regularly recorded.

**Conclusions**

The time-integral of sunspot numbers alone accurately
correlates with the estimated average global temperature trend for the entire
period that sunspot numbers have been regularly recorded.

The net effect of ocean oscillations is to cause the surface
temperature trend to oscillate above and below the trend calculated using only
the sunspot number time-integral. Equation (1) of reference 4 accounts for both
and also shows that rational change to the level of atmospheric carbon dioxide
has no significant influence.

**References:**

5. Average daily sunspot numbers by year (1700-2009) http://www.inference.phy.cam.ac.uk/sustainable/book/tex/GISS/spots.txt

8.
http://hockeyschtick.blogspot.com/2010/01/blog-post_23.html

9. http://wattsupwiththat.com/2007/11/04/guest-weblog-co2-variation-by-jim-goodridge-former-california-state-climatologist/

10. http://lowaltitudeclouds.blogspot.com/

9. http://wattsupwiththat.com/2007/11/04/guest-weblog-co2-variation-by-jim-goodridge-former-california-state-climatologist/

10. http://lowaltitudeclouds.blogspot.com/

**Tags:**

Climate change, temperature prediction, sunspot number
time-integral, Global Warming