Phil Plait's Bad Astronomy: Misconceptions

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What causes the seasons?

Bad Astronomy: The seasons are caused by the change in the distance of the Earth to the Sun.

Good astronomy: The seasons are mostly due to the axial tilt of the Earth. The change in distance of the Earth to the Sun is a very minor player.

[Note added January 21, 1998: This page had a math error in it when I originally published it. The error is not a huge one, and has only a small impact on the conclusions, so I simply corrected it. At the bottom of the page I include the original incorrect calculation, just to keep me honest.]

How it works: This is one of the most pernicious types of ideas: one that sounds reasonable, and so it propagates easily. Unfortunately, it's wrong. Well, not completely wrong; certainly the Earth's distance from the Sun has something to do with the temperature, but it is a relatively minor effect.

First, a sanity check: The Earth's orbit is an ellipse. The Earth reaches perihelion (the point in its orbit closest to the Sun) in January, and it reaches aphelion (farthest point from the Sun) some six months later. If that were all that governed weather, we'd have summer in January, and Winter in July! This may be true for our Southern Hemisphere friends, but not up in the North. Something else must be going on.

We can check our qualitative conclusion above with some (simple!) math. The math involved in calculating a planet's gross temperature has been known for a long time. Basically, the temperature depends only weakly on distance changes; the temperature goes as the inverse square root of the distance of the planet to the Sun. What does that mean? In other words, if you double the distance of a planet from the Sun, the temperature will drop by the square root of 2, or about 1.4. Doubling the Earth's distance from the Sun will drop the mean temperature by about 80 degrees Celsius (Careful here! You cannot use Celsius units for the actual calculation. You have to use the Kelvin scale, which has the same units as Celsius, but starts at 0. In other words, 0K = -273 C. If you take the square root of the temperature using Celsius you'll get the wrong answer! However, since the units are the same, an 80 degree drop is the same in both scales). Specifically, the Earth's average temperature is about 280 degrees Kelvin (10 Celsius). 280/1.4=200, or a drop of 80 degrees.

At perihelion (nearest point) the Earth/Sun distance is about 147,000,000 km, and at aphelion (farthest point) it's about 152,000,000 km. The change in temperature is then

     square root( 152,000,000 / 147,000,000 ) =1.017

or just less than 2 percent. This turns out to be only 5 degrees Celsius, which is quite a bit less than the temperature change we see between winter and summer. Obviously, something else must be going on.

The largest contributor to the change in seasons is the tilt, or inclination, of the Earth's spin axis with respect to its orbital plane (the ecliptic). The usual explanation is as follows: take a flashlight and a piece of paper. Shine the light straight onto the paper, so you see an illuminated circle. All the light from the flashlight is in that circle. Now slowly tilt the paper, so the circle elongates into an ellipse. All the light is still in that ellipse, but the ellipse is spread out over more paper. The density of light drops. In other words, the amount of light per square centimeter drops (the number of square centimeters increases, however, so the total amount of light stays the same-- you expect that, as the light from the flashlight has not changed).

The same is true on the Earth. When the Sun is overhead, the light is falling straight on you, and so more light (and more heat) hit each square centimeter of the ground. When the Sun is low, the light gets more spread out over the surface of the Earth, and less heat (per square centimeter!) can be absorbed. Since the Earth's axis is tilted, the Sun is higher when you are on the part of the Earth where the axis points towards the Sun, and lower on the part of the Earth where the axis points away from the Sun.

For the Northern Hemisphere, the axis points most toward the Sun in June (specifically, around June 21), and away from the Sun on December 21. This corresponds to the Winter and Summer Solstices, or the midpoints of summer and winter. For the Southern Hemisphere, this is reversed.

There is more, too. In the summer, the Sun is higher, and therefore the days are longer. This gives the Sun more time to heat the Earth, so it gets hotter. In the winter, the sun is lower, and the days are short, giving the Sun less time to heat the Earth. This is a secondary effect.

The distance of the Earth to the Sun is a smaller effect yet, but it does exist! So the Southern Hemisphere gets slightly hotter summers and slightly colder winters than the North. But only by a couple of degrees, and only on average. Your mileage may vary!

A good page about seasons can be found at The MSNBC website. They have a nice diagram (though a bit crowded) there as well.

Another one is a discussion of season misconceptions (and he takes to task the MSNBC site I mention above!).

January 21, 1998:
Okay, so I made a small mistake on the original page. I'll quote the original passage, here, and add some notes on the math as well for those of you interested in the details.

We can check our qualitative conclusion above with some (simple!) math. The math involved in calculating a planet's gross temperature has been known for a long time. Basically, the temperature depends only weakly on distance changes; the temperature goes as the distance to the one-fourth power (the square root of the square root!). In other words, if you double the distance of a planet from the Sun, the temperature will drop by 2^(1/4) or 1.18. Doubling the Earth's distance from the Sun will only drop the mean temperature by about 50 degrees Celsius (the Earth's average temperature is about 310 degrees Kelvin or 10 Celsius. 310/1.18=260, a 50 degree drop. The Kelvin scale is absolute, which means it starts at 0, which is why I used it for the calculation).

At perihelion (nearest point) the Earth/Sun distance is about 146,000,000 km, and at aphelion (farthest point) it's about 152,000,000 km. The change in temperature is then

        ( 152,000,000 / 146,000,000 ) ^ 1/4=1.0085

or only 0.85 percent! This turns out to be only 2 degrees Celsius, which is quite a bit less than the temperature change we see between winter and summer! Obviously, something else must be going on.

My mistake was that I put in an additional factor of a square root in there, making the change in temperature a bit too small. I also used 146 million kilometers for the perihelion distance, and 147 million is actually a bit better. The temperature change from winter to summer is about 5 degrees, not 2 as I stated originally. Where I live in Washington, DC, the temperature in summer hits 35 Celsius easily, and commonly drops to 0 Celsius in the winter. 35 degrees is a lot more than 5!

To calculate the temperature of a planet, you basically need to assume that the amount of heat the planet gets from the Sun is balanced by the amount of heat radiated away by the planet. If this were not true, the planet would either heat up (if it didn't radiate the heat away) or it would freeze (if it radiates too much).

Qualitatively: the star gives off heat over its whole surface. That heat expands in a sphere centered on the Sun, and travels to the planet. The planet intersects a small piece of it which is equal to the area of a circle with the same radius as the planet (if I ever get a chance I'll place a diagram here that shows this graphically...). The planet absorbs some of that heat, and, if it rotates quickly, re-radiates it away over its whole surface.

Quantitatively:

sigma * T_planet⁴=
sigma * T_Sun⁴ * 4 * pi * radius_Sun² / (4 * pi * distance²) *
(1-albedo) * pi * radius_planet² / 4 * pi * radius_planet²

where sigma is a constant (not important here, since it cancels out), T is temperature (for the planet or the Sun, each is labeled above), distance is the distance from the planet to the Sun, radius is the radius of the Sun or planet (also labeled), and albedo is a measure of the reflectivity of a planet. An albedo of 1 means the planet is a perfect reflector, like a mirror. An albedo of 0 means the planet absorbs every photon that hits it; it would look black. The Earth has an albedo of 0.39, as it happens.

We can then do a bit of algebra to get:

T_planet=T_Sun * (radius_Sun/2 * distance) ^1/2 * (1-albedo)^1/4

Phew! From here you can see that the temperature of the planet depends on the inverse square root of the distance to the Sun. Note that if you put in the correct numbers for the Earth and Sun (distance=1.5 x 10¹³ centimeters, T_Sun=5780, radius_Sun=7 x 10¹⁰ centimeters and albedo=0.39) you get a temperature of the Earth of about 250 Kelvin. That's about -20 below Celsius, or -10 Fahrenheit! What gives?

Our atmosphere, that's what gives. Our atmosphere helps keep heat in (by absorbing some of the radiation re-radiated by the Earth), so you need a correction factor to our albedo. Without our thin layer of air, the surface temperature of the Earth would rapidly drop, freezing the oceans solid. This is called a "greenhouse effect", and is a very real occurrence. It's when things get out of control that you get a runaway greenhouse effect. Note also that the temperature on the surface of Venus should be about -20 Celsius (distance=1.1 x 10¹³ centimeters, albedo=0.65; although it's closer to the Sun its albedo is higher, so it should have about the same temperature as the Earth), but is actually in excess of 500 Celsius (over 900 Fahrenheit!). Should you worry about runaway greenhouse effect? Take a look at our closest neighbor. You tell me.

My thanks to Bad Readers Darrell Bennett, Eric Carlson and Georg Zemanek for pointing out some of my errors!

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