# Analysis: Planetary Temperature – a Rigorous Formula

Note: This post is highly mathematical.

On other pages, I have derived rigorous formulas for a planet’s thermal radiation emissions to space and incoming and stored thermal energy. These analyses are expressed in terms of global average quantities or equivalently in terms of total energy summed over the globe over some period of time. Familiarity with those pages is likely be needed in order to fully understand the analysis presented here.

## Preliminaries

### Conservation of Energy

One of the most fundamental laws of physics is the Law of Conservation of Energy. One consequence of this law, is that there is a simple relationship between the rate of energy coming into the “climate zone” of a planet, , the rate of energy leaving the planet, , and the rate at which energy accumulates within that planetary climate zone, :

(1)

This equation states that any energy that enters the system must either leave the system or be retained inside the system. Since energy is neither created nor destroyed, this equation must hold at all times.

### Expanding the Terms of the Energy Conservation Equation

#### Expanding Energy Entering

The term for the rate of energy entering the system is given more explicitly by:

(2)

where is the incident solar irradiance at the top of the atmosphere (TOA); is the albedo, defined as the fraction of solar irradiance reflected back to space; and is the heat flow from non-solar sources entering the system. Mainstream science indicates that the term is negligibly small for Earth; but the term is included in this analysis because it is relevant to some other celestial bodies, and so that our result will be rigorously correct without any approximations.

#### Expanding Energy Leaving

The term for the rate of energy leaving the system is labeled , for outgoing longwave radiation. This is because the only way that any significant amount of energy can escape a planet into space is in the form of electromagnetic radiation. In practice, this happens in the form of longwave thermal radiation emitted by matter.

The only other way energy could escape the system would be if there was a net heat flow towards the core of the planet. However, in general, geothermal heat flows in the opposite direction, out of the interior of the planet, not into it. This leaves thermal radiation emissions to space as the only mechanism for cooling the system as a whole.

The power of outgoing longwave radiation may be expressed as:

(3)

(4)

In these equations, is the normalized Greenhouse effect (a dimensionless quantity less than 1); is the mean flux of surface-emitted longwave radiation; is the global mean surface temperature; is the Stefan-Boltzmann constant; is the global mean emissivity of the surface (weighted by so that warmer areas weigh more heavily in the average); and is a factor larger than 0 which I call the temperature-variation emissions boost factor (TVEB); and

The temperature-variation emissions boost factor (TVEB) is explicitly given by:

(5)

where denotes the average of the quantity over the surface of the globe and over some period of time; and denotes the sum (or integration) of the quantity over the surface of the globe and over some period of time. I find the expressions involving averages simpler, but include the form involving sums because some people prefer sums to averages. These quantities are simply related, insofar as where the constant is simply the surface area of the planet times the duration of the time period over which quantities (typically energies) are being summed.

The factor TVEB, or , emerges during the averaging or summing process, because an object whose temperature varies emits more thermal radiation than an object with the same average surface temperature but with a uniform temperature. In the case of the Moon, the TVEB factor contributes to the Moon being around 90℃ colder than the Earth, on average, despite the Moon absorbing more sunlight per unit area than the Earth does. (The effect of this TVEB factor on Earth is much smaller, since Earth experiences much small temperature variations.)

The normalized Greenhouse effect, , is explicitly given by:

(6)

where is the power of surface-emitted longwave radiation. The quantity, , indicates what fraction of thermal radiation power emitted by the surface is not reflected in the power of thermal radiation that escapes to space.

Note that if the atmosphere of a planet was transparent to thermal radiation emitted by the surface, this would inherently mean that the power of thermal radiation reaching space, would have to be identical to the power of thermal radiation leaving the surface, , so that . In other words, a non-zero Greenhouse effect is possible only because of the presence of materials in the atmosphere which absorb or scatter thermal radiation emitted by the surface.

For purposes of this formula, the quantity can simply be treated as a measurable quantity which plays an essential role in the formula we are deriving. We also know that it has a non-zero value only due the presence of materials that absorb or scatter thermal radiation. Although more can be said about this factor, knowing that much is sufficient for purposes of this analysis.

#### Expanding Energy Storage

It can be useful to decompose the energy storage rate, , into a sum of thermal energy storage and non-thermal energy storage, rates:

(7)

The thermal energy storage rate, , relates to increases (or decreases) in the amount of energy reflected in the temperature and phase (solid/liquid/gas) of matter in the “climate zone.”

The non-thermal energy storage rate, , relates to increases (or decreases) in other type of energy within the “climate zone.” Examples of other types of energy that might change include chemical energy stored in biomass (which increases via photosynthesis and decreases via metabolism and biomass decay), or gravitational potential energy (which increases when the atmosphere warms and grows taller), or kinetic energy in moving water or air (which increases when circulation intensifies). The non-thermal energy storage rate is likely to be quite small in practice, but is included for completeness.

### Expanded Conservation of Energy Equation

Now we can bring the various terms together into an expanded version of the energy conservation equation:

(8)

(9)

We can rearrange the above equations to solve for the surface-emitted longwave radiation:

(10)

Note that other energy fluxes relate the to surface longwave emissions flux, , via a factor . This factor is present because the conservation of energy equation directly involves , yet is defined so that .

## Temperature

### Non-Equilibrium Temperature Formula

Going one step further, we can solve for the global mean surface temperature (GMST), :

(11)

### Multiplicative Non-Equilibrium Temperature Formula

One way of organizing the above equation is to break it into distinct factors as follows:

(12)

(13)

where

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

The factors , , , , , and indicate the multiplicative effect on temperature of, respectively, the Greenhouse effect, surface temperature variations, surface emissivity, albedo, non-solar heating and non-thermal-energy storage, and the thermal-energy storage rate.

The factor simplifies to in equilibrium (when the amount of stored thermal energy is not changing), or simplifies to if the non-solar heating rate and non-thermal-energy storage rate are negligible (as scientists typically believe to be true for Earth). As I’ll show in a subsequent section, the factor depends directly on the TOA energy imbalance, which can be measured by satellites.

The temperature represents the equilibrium temperatures of a solar-heated-only uniform-temperature blackbody version of the planet.

The non-multiplicative and multiplicative forms of the temperature formula each have their advantages and disadvantages. Either formula produces the same results. The non-multiplicative version is in some ways simpler or more straightforward. However, the multiplicative formulation can be useful in that it expresses the effect of each climate variable in consistent way; so, the multiplicative formulation makes it easy to compare the relative sizes of the the impacts of each climate variable.

Note that:

• , and
• Temperature increases as GHE, emissivity or non-solar heating increase. Temperature also increases if solar-heating increases, increasing .
• and
• Temperature decreases as TVEB or albedo increase.
• if and if
• if stored thermal energy is being released as is associated with global cooling; and if thermal energy is being stored as is associated with global warming.
• The factor represents the discrepancy between the actual temperature and the equilibrium temperature that the system is moving towards. The system continually shifts towards , unless other factors shift the target (i.e., equilibrium) temperature.

### Equilibrium Surface Temperature Formula

When a planet is in thermal equilibrium, the net thermal-energy storage term, is zero.1An argument could be made for instead defining equilibrium as occurring when the the net overall energy-storage rate, , is zero. I’ve chosen to use zero net thermal-energy storage as the reference for equilibrium, because this choice simplifies explanations of why temperature always moves towards the “equilibrium temperature.” I will denote the mean surface temperature when the planet is in equilibrium by GMST or . Setting to zero leads to the factor simplifying to become the factor , defined as follows:

Thus, the above expression for global mean surface temperature becomes:

(22)

Alternatively, this can be expressed without the decomposition into multiplicative factors:

(23)

At any given point in time, it will not necessarily be the case that equals . However, it will inevitably be the case that, as changes, will “chase” it, i.e., will shift towards the current value of . This is clear because:

• When the temperature is lower than the equilibrium temperature (), this corresponds to a situation in which the stored internal energy of the climate system is increasing (), as occurs when temperature rises. Given that temperatures are rising, is tending towards .
• When the temperature is higher than the equilibrium temperature (), this corresponds to a situation in which the stored internal energy of the climate system is decreasing (), as occurs when temperature is falling. Given that temperatures are falling, is tending towards .

Thus, although GMST might not be the current temperature of a planet, it represents the temperature that the planetary surface is heading towards.

### TOA Energy Imbalance and the Temperature Formula

One of the quantities that satellites orbiting Earth measure is the TOA radiation flux Net Energy Imbalance, . This is equal to the net absorbed solar irradiance minus the outgoing longwave radiation:

(24)

If we go back to the original energy conservation equation, and expand only the term for , and combine it with this equation for , we find:

(25)

This allows us to rewrite the factor , which appeared above, as:

(26)

The factor conveniently depends only on the quantities which are measurable by satellites, and not on the individual terms, , , and , which might be more difficult to measure.

## Discussion

### Status of these Formulas

The above formulas for mean non-equilibrium and equilibrium global temperature are direct consequences of fundamental principles of physics:

• The non-equilibrium result is simply a transformed version of an equation for conservation of energy.
• All mathematical transformations were exact, involving no approximations.
• The only assumption made about the planet is that materials on the planetary surface have a defined temperature (at any given position on their surface and at any given moment).2Technically, the condition is that the surfaces of materials are in Local Thermal Equilibrium. This does not preclude temperature changing fairly rapidly in time, or varying over relatively short distances. Under natural conditions at the Earth’s surface, this condition is known to be reliably satisfied. This condition being satisfied means that it is valid to use the Sefan-Boltzmann law (with emissivity included to account for real materials) to specify the power of thermal radiation emitted by surfaces. This law, and the law of conservation of energy, are the only principles of physics required in order to derive these results.

Thus, the formulas for equilibrium and non-equilibrium global mean surface temperature are scientifically rigorously and mathematically exact.

### What affects Mean Surface Temperature

The derived formula indicates that the following quantities directly affect the equilibrium Global Mean Surface Temperature (GMST):

1. Incoming solar irradiance,
2. Greenhouse effect (i.e., reduction of thermal radiation emissions to space in comparison to thermal radiation emissions from the surface),
3. Albedo,
4. Emissivity,
5. Temperature variations,
6. Non-solar heating,
7. Non-thermal-energy storage rate,

These items are listed in approximate order of decreasing importance with respect to Earth’s baseline temperature, as indicated by the magnitude of , where is the temperature multiplication factor associated with each quantity. Mainstream science estimates the effects of non-solar heating and non-thermal-energy storage on Earth as being far smaller than the effects of the first 5 factors.

These are the ONLY factors that directly effect the equilibrium temperature.

This result does not preclude other factors affecting planetary temperature indirectly. However, any phenomenon that affects planetary temperature does so by affecting one of the six quantities listed above. For example:

• cloud coverage affects both albedo (reflection of shortwave radiation) and the Greenhouse effect (net absorption of longwave radiation);
• the extent of ice and snow coverage in high latitudes affects albedo;
• when deciduous forests drop their leaves in the fall, this produces a small change in emissivity which is measurable by satellites;
• the magnitude of the Greenhouse effect is affected by the concentrations of Greenhouse gasses (including water vapor), cloud coverage, and the atmospheric temperature profile (i.e., the lapse rate);
• ocean currents affect the distribution of temperatures on the Earth’s surface, affecting temperature variations, and also affect humidity levels (and possibly local lapse rates), and hence there may be impacts on the Greenhouse effect.

Consequently, each of these phenomena (as well as others) can potentially affect the global mean surface temperature.

### Other Reference Temperatures

The temperature formulas above are expressed in terms of and , the equilibrium and non-equilibrium solar-heating-only uniform-temperature blackbody temperatures. In some cases, it may be useful to consider other reference temperatures. For example, we can define:

(27)

(28)

(29)

(30)

These temperatures are:

• — The equilibrium temperature of a solar-heating-only reflective uniform-temperature planet which radiates as a blackbody.
• — The equilibrium temperature of a solar-heating-only reflective uniform-temperature planet which radiates as a grey body (i.e., emissivity less than 1).
• — The equilibrium temperature of a solar-heating-only reflective variable-temperature planet which radiates as a blackbody.
• — The equilibrium temperature of a solar-heating-only reflective variable-temperature planet which radiates as a grey body (i.e., emissivity less than 1).

One could also define reference temperatures with subscripts beginning with instead of , which include the factor so that they account for both solar and non-solar heating.

### Temperature “Discrepancy” due to the Greenhouse Effect

In the case of Earth, it is often said that the Greenhouse effect resolves a “33℃ discrepancy” in Earth’s temperature.

Talk of a 33℃ discrepancy arises from the observation that 33℃. If we were to consider only the GHE (as it is now defined technically), we would look at 35℃. So, the warming that the GHE accounts for on Earth is actually 35℃. Though, most people reference the “33℃” figure because the narrative to arrive at that figure is simpler.

Other heating mechanisms are sometimes proposed as a hypothetical alternative to the Greenhouse effect. However, such alternative heating mechanisms would not and could not address the issue. Such mechanisms would alter the term . However, the “33℃ discrepancy” specifically relates to the the measured value of the factor being about 1.14 rather than 1.

Changing the value of the term would would alter the planetary temperature. But, it would in no way address the empirical fact that the term is not 1. The “33℃” specifically relates to the measured value of the GHE factor (plus, implicitly, the factor). That observation cannot be altered by postulating a different value for the non-solar heating factor . A 35℃ “discrepancy” will always exist in the energy conservation equation when the GHE is omitted from calculations, regardless of the presence or absence of any non-solar heat sources.

### Working with Surface Effective Radiative Emission Temperature

When working with some datasets, it may be convenient to work with the surface “effective radiative emission temperature” (or, more concisely, “effective temperature”) rather than the actual surface temperature. The local surface effectve temperature is given by:

(31)

The surface effective temperature may be useful to work with because it can be determined by knowing the surface longwave emissions, , even if the emissivity of the surface is unknown.

The global mean surface effective temperature, is related to the global mean surface temperature, , by:

(32)

(33)

where

(34)

(35)

(See Averages and Correlations for an explanation of the “weighted average” notation used in defining .)

The global value is related to the local surface effective temperature by the equation:

(36)

where

(37)

The conservation of energy equation can be expressed in terms of surface effective temperature as follows:

(38)

Or, in terms of multiplicative factors, this is:

(39)

where

(40)

Combining this result with the relationship between and , we can express as:

(41)

The above equation is likely to be convenient to use, because it allows one to work with either or , given that the quantity in parenthesis .

These equations are sufficient to allow one to analyze planetary temperature in terms of surface effective temperature, , and then subsequently relate this to the actual surface temperature, , once information on emissivty, , becomes available.

### More on the TOA Energy Imbalance

We can rearrange the equation for the TOA net flux imbalance to be an equation for the energy storage rate:

(42)

From this, we see that the TOA energy imbalance sets a lower bound on how much energy is being added to the store of energy in the climate system, i.e., how much warming is happening.

If a significant non-solar heating term, , was present, this would indicate that more energy is being retained beyond that suggested by , leading to more heating of matter in the climate system. Postulating a heating source that is not accounted for would not explain the observed TOA energy imbalance; it would add to it, creating a larger actual energy imbalance and implying more heating than would be expected from the TOA energy imbalance alone.

If both the TOA imbalance and the energy storage rate are known, this allows one to calculate the size of the additional heating term :

(43)

When applying this equation to estimate , any uncertainty in the values of and translates into uncertainty in the value of .

### Localized Equations

A localized version of the conservation of energy equation requires that one account for lateral heat transport, , as follows:

(44)

The quantity represents the energy flux arriving to (positive) or leaving from (negative) a given location, as a result of lateral heat transport (generally due to heat advection via air or ocean currents). Note that .

For the global equations presented in this analysis, a valid localized version of the equation can typically be obtained by replacing the global variable with the local variable , eliminating any mention of the temperature-variation emissivity boost factor, , and replacing with .

• 1
An argument could be made for instead defining equilibrium as occurring when the the net overall energy-storage rate, , is zero. I’ve chosen to use zero net thermal-energy storage as the reference for equilibrium, because this choice simplifies explanations of why temperature always moves towards the “equilibrium temperature.”
• 2
Technically, the condition is that the surfaces of materials are in Local Thermal Equilibrium. This does not preclude temperature changing fairly rapidly in time, or varying over relatively short distances.