Saturday, 13 December 2014

Astrophysics Corner, Part 15 – Changing the Earth's Rotation Axis Tilt:

Astrophysics Corner, Part 15 – Changing a Planet’s Rotation Axis Tilt: the Effect on the Sun’s Altitude and the Length of the Day at Different Altitudes
In an earlier blog, I noted that solar panels on a planet are affected by both the latitude of the location that the panels are placed and the tilt of the planet’s axis of rotation, relative to its orbital plane.
  • The latitude of the location of the panels is obviously important. Essentially, the solar flux that is experienced on the surface varies with the latitude of the site as the sine of the sun’s elevation above the ground. The farther from the planet’s equator, the smaller that angle will be.
  • That will be further compounded by the tilt of the planet’s axis, relative to the plane of its orbit around the star. The greater the tilt, the more the elevation of its sun will vary throughout the year, so the solar panels output will be greater or lesser as the season’s progress.
Earlier, we looked at how the latitude of the location of the observer (person or solar panel) influenced the altitude of the sun and the length of the day. In this blog, we will be a bit more speculative and discuss how the tilt of the rotation axis affects the length of the day, the altitude of the sun (or the planet’ s star in the case of planets other than the Earth) and the distribution of solar energy on that planet.

Case 1 - 23.5 Degree Tilt (Standard Earth tilt)
Below is the standard picture, the Earth with a 23.5 degree tilt. These are the relative positions of the Earth and Sun in the Northern hemispher mid-summer. It is pretty clear from the picture that the North Pole will be bathed in sunlight for an entire rotation (day) at this time of the year, and that northern areas in general will have longer days than southern areas. It is also clear that the sun’s rays will be perpendicular to the ground at mid-northern latitudes, which means the sun will be directly overhead then. Thus, longer, hotter days.

If we reverse the picture above, the northern hemisphere will now be in relative darkness, as it is in mid-winter. The sun’s rays will hit at a very oblique angle in the north, thus providing relatively little energy per square meter. The situation will, of course be the opposite in the south, which will be experiencing the long, warm days of summer.
As for the spring and fall equinoxes, we have to imagine the sun being out of the picture (where the reader’s head is now) looking down on the Earth. It is obvious then, that the sun will warm the north and south equally, and there will be no difference in the length of the day.  
The next two graphs show the altitude of the sun at mid-day, and the length of the day, at various latitudes throughout the year.

The final two graphs show the effective power of the sun at various latitudes, throughout the year and aggregated over the year. As you can see, the sun’s power drops off almost linearly with latitude.

Case 2 - 0 Degree Tilt (Straight Up and Down Planet)
Now, let’s imagine tilting the Earth’s axis, or alternatively, being on a different planet with 0 tilt.

In this case, it is clear that the strength of the sun’s rays will not very throughout the year, at a given latitude, nor will the length of the day vary. It will always rise at the same time, climb to the same height in the sky at noon, then sink to the horizon at the same time and place.

If we multiply the sine of the angle of the sun by the number of hours of daylight, to get a measure of how much effective sunlight falls at different latitudes, we get the graphs below. As you can see, the effective solar insolation at various latitudes never changes. Furthermore, it is linear with latitude.

There will be no seasons - hot places will always be hot and cold places will always be cold. Kind of monotonous. Assuming the planet had an atmosphere and oceans like the Earth’s, one suspects that there would have to be steady, strong winds and currents redistributing the heat from the equator to the poles year round. It might be quite a wild place to live.

Case 3 - 90 Degree Tilt (Tipped Over Planet)
The other extreme is the 90 degree tilt, such that the equator and the line through the poles have effectively swapped places.
Now, it is obvious that the length of the day will vary drastically throughout the year. At summertime at the North Pole, the sun will be up all day, directly overhead. In fact, the North Pole will have 24 hour days half the year, and 0 hour days the other half of the year. The situation will be the reverse at the winter solstice. The equator will have 12 hour days year-round, but even relatively “tropical” latitudes a few degrees above the equator will experience very large differences in the length of the day throughout the year.

As before, when we multiply the sine of the angle of the sun by the number of hours of daylight, to get a measure of how much effective sunlight falls at different latitudes, we get the graphs below. In this case, the solar insolation at the North Pole is actually much higher than at the equator in mid-summer.

Interestingly, although seasons would vary tremendously at any particular latitude during the course of the year, on an annual basis, the energy of the sun is distributed quite evenly throughout the planet. The form of the function approximates a quadratic quite well, with the maximum solar energy being at the mid latitudes, dropping off towards the poles and the equator. So, the weather would probably be quite strange, with great temperature shifts from season to season, but not very much rebalancing of heat between low latitudes and high latitudes.

Heat Balances by Tilt
Perhaps the most interesting result, is the variation in the planet’s heat balance between the equator and the poles, when different axis tilts are applied. I ran the simulation for a number of other planet tilts, and the results are in the graph below.
It appears that the equator gets about the same amount of solar energy over the course of a year, regardless of the planet’s tilt. But the amount of solar energy gained by higher latitudes is strongly dependent on the planet’s tilt.
  • The stronger the tilt, the more solar energy falls on the poles. This means that the planet has a more even heat balance over the course of a year.
  • The stronger the tilt, the more solar energy the planet appears to capture over the course of a year, via the combined effect of length of day and altitude of sun at the various altitudes. This seems a bit counter-intuitive, but that’s the result that the simulations give.

So, if humanity ever wants to build a planet, scout for alternative planets, or just change the tilt of the Earth, we might want to keep this in mind.
Actually, it is thought that the presence of the moon maintains the Earth’s axis tilt at a near constant level. If the moon weren’t so large and so close, the Earth’s axial tilt would vary over geologic time, something like how a spinning top can wobble. But the moon is slowly drifting away from the Earth, as its orbital momentum is stripped away due to tidal interactions with the Earth. So, in some distant future the Earth might actually go through these large variations in axial tilt. Wouldn’t that be interesting.

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