Following the Energy Trail This page will show you how you can work out the mathematics of a planet's productivity

Let's see how the sun's energy flows in the Biosphere.

Warning: I am making a lot of assumptions and doing a lot of rounding in the math here. My numbers may be off by several thousand percent, but I think the process is all right.

Try to understand the basic idea, and don't use my math numbers on any Important Exams unless you have checked them out!

Dr Viau says, "Gerald Nordley and Mark Wistey were kind enough to help with this, but I made the mistakes all by myself!"

The Solar Constant

How much of the sun's energy gets to the surface of the earth?

The earth gets only 2 billionths of the sun's energy, but that is still a lot. However, you can see on the chart that life (through photosynthesis) uses only .023% of the energy that reaches the surface of the earth.

34% of the sun's energy is reflected back into space by snow and clouds. This reflective quality of a planet is called its albedo.

42% of the energy goes to warm the land and water. The warmth of the earth is constantly being radiated into space, and the sun's energy replenishes this warmth.

The water cycle -- evaporation and precipitation -- uses 23% of the solar energy.

Winds and ocean currents use 1%.

The amount of energy that gets to the earth's surface at the equator at noon is called

The Solar Constant.

Let's think about how many of the KiloCalories that we use to measure food fall onto an area that is one meter square.

(A meter is pretty close to a yard in length, so a square yard and a square meter are roughly the same size.)

At the Equator:

We find how many KiloCalories falls onto one square meter each day

This turns out to be about 19.7 kilocalories per minute on each square meter at the equator above the atmosphere.

A kilocalorie (kcal) is a food Calorie, the kind of Calorie that we count when we are on diets.

19.7 kilocalories per meter squared per minute multiplied by 60 minutes = 1,180 kilocalories per hour

1,180 kilocalories per hour multiplied by 24 hours = 28,320 kilocalories per square meter per 24 hour day (about).
(We will adjust for the night in the next step!)

We correct for the rotation of the earth

However, because the sunlight is slanted in the morning and the evening, and because of night, we need to divide this 28,320 kilocalories per square meter per 24 hour day by 3.

28,320 kilocalories per square meter per 24 hour day divided by 3 = 9,440 kilocalories per square meter per 24 hour day.

We correct for the presence of the atmosphere

However, all this has been going on at the very top of our atmosphere. Only about 70% of that energy gets down to sea level.

70% of 9440 kilocalories = 6,600 kilocalories

About 4/9 of the solar energy that actually falls on a plant is energy that the plant can use. (Some of the radiation does not help with photosynthesis.)

Let's figure out how much energy is actually useful.

At the equator, a square meter densely covered with plants is receiving useful radiation of about 4/9 of 6,600 kilocalories per square meter per 24 hour day.

6,600 * 4/9 =2933 kilocalories per day

We will round this up to 3000 kilocalories per day for the sale of simplicity.

Most of this energy is used up by the plant just being a plant: it has to use energy to do its life processes, an activity which is called respiration. Under ideal conditions  plants might be able to turn up to about 10% of those 3000 kilocalories into biomass, which is food that the animals could eat and also stalks and thorns and roots that may not be digestible, except by detritovores such as bacteria and fungi.

Let's take 10% of the 3000 calories, remembering that plant tissue may not always be  produced at the maximum rate.

3000 kilocalories per day divided by 10 = 300 kilocalories per square meter per day.

So there are 300 KiloCalories in the new plant tissue that was added that day. This is called the Net Primary Productivity per day.

Over a year, how many calories of primary productivity are produced in our square?

300 Kilocalories * 365 days = 109500 Kilocalories per square meter per year.

Well, there are clouds and rainstorms that would bring that number down. Dr Viau found this table which will be very helpful to all world builders!

 Ecosystem Type Net Primary Productivity (Kilocalories / square meter / year) Approximate Kilocalories per square meter per day Rainfall per year in inches Tropical Rain Forest 9000 25 More than 60 Estuary (the place where a river meets the sea) 9000 25 Swamps and Marshes 9000 25 Savanna (grass, scattered trees,               little or no winter snow) 3000 8 Deciduous Temperate Forest 6000 16 30-60 Boreal Forest (Evergreen Coniferous Forest) 3500 10 12-33 Temperate Grassland (cold winters) 2000 6 10-30 Polar Tundra 600 2 Less than 10 Desert < 200 1 Less than 10

This table is from http://www.geog.ouc.bc.ca/physgeog/contents/9l.html an online Geology course created by Dr, Michael Pidwirny at Okanagan College, British Columbia, Canada.
(Dr Viau's additons are in red)

At 60 Degrees of Latitude we calculate 4/9 times 3,300 which equals

Kilocalories per square meter at 60 Degrees = 1500 kilocalories per day

These figures are about maximum possible production during a day which has 12 hours of daylight and12 hours of darkness.

At 60 Degrees Latitude:

As we travel away from the equator, the curvature of the earth causes the solar energy to be spread out over a larger area.

At 60 degrees North the amount of energy received is about half that at the equator. There will be more discussion of this further down on the page.

There are spaces between leaves: the energy falling there is not used.

There are rocks and bare patches on the ground.

Water and nutrients affect growth -- abundant solar energy is not enough.

Plants grow when it is warm. Brilliant sunlight on a frosty day is not as effective as brilliant sunlight at the equator! Animals must eat all year.

Probably the plant yield will usually be lower than the maximum possible.

These figures are for the energy budget of the earth -sun system: Check the AU Equivalent in the Star Tables for your world.

Formulas:

Kcal yield for year = (Calories per day per square meter) multiplied by (number of days in growing season).
 Latitude Maximum Kcal for plants per square meter per 24 hour day Plants use 4/9 of the available light Biome Growing Season in days per  year Maximum possible light received per square meter per year 10% turned into plant tissue per square meter per year 10% turned into animal tissue by grazing animals per square meter per year Predator eats animal,  10% becomes predator tissue 0 Equator 6000 (average) 2666 Rain Forest, desert 365 days 973090 97309 K/Cal 97* 5 = 500 K/cal 30 Degrees 5200 (average) 2311 Deciduous Forest 120-250   days 843515 60 Degrees 4800(midsummer) 3000 (average) 731 (midwinter) Grasslands 120-200   days 70,000 kcal 70 Degrees 4100(midsummer) 0 (midwinter) Coniferous Forest 90-120   days 80 Degrees 3300(midsummer) 0 (midwinter) Tundra 60-100   days 42,000 - 90 Degrees At and below a possible  120 days, earth photo synthesis is very difficult. This is true in all zones where plants cannot get enough light.

Special Cases

The earth's axis is tilted at 23 degrees to the plane of its rotation around the sun. In the high latitudes near the poles, winters are dark, and summers have long days. We have heard of "the midnight sun", which refers to the period when the sun does not set at the poles.  When we think about the high latitudes, both north and south, we must remember that, although light is available in abundance, temperatures remain low. Life processes are chemical processes, which work more quickly as temperatures rise.

Go on to

The Caloric Content of Foods

Some Animal Weights and Caloric Requirements on Earth