Upgrade your planet: add rings!

This is the first on two or three posts on adding rings to Furaha, or any other fictional planet. The reason to add any is simple: rings add drama! Ron Miller, an excellent artist specialising in astronomical illustration, provided fascinating examples of how dramatic rings can look by transplanting Saturn's rings to Earth. He has shown and discussed the results a few years ago on this io9 page (the comments are also interesting).
 

Click to enlarge; copyright Ron Miller (used with permission)

Here is one of his examples, showing a view from Guatemala. Impressive, isn't it? Somehow I only started wondering whether rings would look just as good on Furaha a short while ago. I certainly studied rings before, in the context of the planet Ilo, as can be seen from this post and this one.

Click to enlarge; copyright Gert van Dijk

The animation above shows the planet Ilo and was taken from the main Furaha site (under the 'astronomy/seasons' tab. It shows how the shape of the areas lit by the sun changes over a year; the curved horizontal band is the ring shadow.

Adding rings to a view as expertly as Mr Miller did is not a simple matter of drawing some curves in the sky. Once you start thinking about the problem, you will realise that their aspect depends on how large they are, but also on where you are on the planet, on the direction your 'camera' is pointing in, and finally on the 'lens' you are using, by which I mean the visual angle. And as if that is not enough, the lighting of the rings differs for every hour and day of the year. Some places on the surface will be in the shadow of the rings, and at other times the planet will cast a huge shadow on the otherwise sunlit rings.    

So I started exploring these matters. I am not certain I will actually add rings to Furaha, but it is fun to explore. In this post I will start with basic ring astronomy (mind you, the only astronomy I can deal with is of a very basic variety). Planetary rings consist of many pieces of ice or rock, from boulders to dust particles, all circling a planet in its equatorial plane, and only there. There can be no rings directly overhead on the poles. In Earth's solar system, rings usually do not form one continuous band from the inner to the outer radius, but are divided into several distinct ringlets with gaps in between. These are apparently the effect of moons that sweep clean parts of the rings through their own gravitational effects. While there must be strict Newtonian rules underlying where these ringlets and gaps are placed, the large variety of ring shapes in our own solar system suggests that there is wiggle room here, allowing some creativity on the wordbuilder's side. Just compare the rings of the gas planets, for which this Wikipedia article is a good start.

click to enlarge; copyright:  http://astronomy.nju.edu.cn

The image above can be found at this website; it compares the various ring systems and stresses another important feature: rings are always situated inside a planet's Roche limit, or 'tide limit'. The word 'tide' here has nothing to do with seas, but concerns 'tidal forces' acting on a piece of ice or rock coming close to a large mass (a planet or sun). The gravitational forces act differently on the parts of an object closest and farthest away from the planet. These tidal forces will break up an object if it is within a certain distance of the planet, and that distance is the Roche limit. The breaking of the object, say an asteroid, goes on until the remnants are small enough to stay whole. If you start this process with a large enough object, the resulting pieces may collide with one another, breaking off more pieces, adding to the fun. The Wikipedia page on Roche's limit shows very clearly what happens to a heavenly body when it crosses the limit, and also contains a formula to calculate how far out from a planet the limit lies (this is the formula for 'fluid' objects).

distance= 2.44 * radiusP * (densityPlanet /densityAsteroid)^1/3

The distance is in km, and is governed by the planet's radius (radiusP, in km) and on a fraction, involving the density of the planet divided by that of an asteroid coming near the planet. If you wish to express the Roche limit in units of the planetary radius, you can just leave RadiusP out. If the asteroid has the same density as the planet, the fraction is one, and then the formula simply reads: 'The Roche limit is at 2.44 planetary radii'. But asteroids are probably less dense than a terrestrial planet, making the fraction greater than 1, so the Roche limit will be further out. I found estimates for various types of asteroids ranging from 2 to 5 gram per cubic centimeter, and the density of Furaha is 5.9 gram per cubic cm (the Furaha planetary system was kindly worked out by Martyn Fogg). So we can take the Roche limit for Furaha up to 3.5 radii.

Click to enlarge; copyright Gert van Dijk

Here is a ring of which the outer margin is exactly 3.5 planetary radii away from the centre of the planet. The planet is shown at the point in its orbit where the northern pole points directly towards the sun (the summer solstice). The glowing orange arrow show the direction of sunlight, and the thin lines show the plane of the orbit (and the direction perpedicular to it). The rings are 50% transparanet. You can see the shadow of the planet on the rings, and you can just also see the shadow of the rings on the planet.

That ring seems overly large, so I will set for a safer maximum value for the outer radius of the ring at 2.5 radii out from the centre of Furaha. But what should the value for the inner radius be? That is less clear; the outer parts of the atmosphere would form an effective limit, but I do not think I want the rings to come that close: bits and pieces might rain down continuously, and if these are large enough they will mess up the biosphere. So let's start with a fairly narrow ring with an inner radius of 1.8 to 2.0 radii.

click to enlarge; copyright Gert van Dijk

So here we are; that looks better. I also adjusted the transparency of the rings: this one is 67% transparent. How much light should the the rings let through? If the density of rocks in the rings is high and the rings are thick, not much light will come through. The reason this matters is because the rings casts shadows on the daylight side of the planet. They do so on the Northern or Southern hemisphere depending on the time of year.

click to enlarge; copyright Gert van Dijk

Here is a simple scheme explaining that. Furaha's axis is tilted by 18.3 degrees, and so are the rings. When it is summer in the Northern hemisphere (at the left), the North pole is tilted towards the sun, and the shadow of the rings falls on the Southern hemisphere. When it is winter in the Northern hemisphere (at the right) the ring shadow darkens the Northern hemisphere. I do not want these shadows to be big and black, because that will have large and unpredictable effects on the climate, the weather and the life forms (well, I certainly cannot predict them, and that is the actual limiting factor). So I prefer narrow transparent rings: I still get drama without too many unknown problems. I will assume that a small asteroid, breaking up and filling a large area, would result in largely transparent rings. In fact, it is the other way around: the rings are about two thirds transparent, because the asteroid forming the rings contained exactly the right amount of matter to make it so.

The final matter is the composition of the rings; they can theoretically be made of ice or of rocks. I doubt ice would last long this close to the sun, so the rings are made of rock. Earth's moon consists of  very dark rock, and yet its appears bright enough in the night sky to have inspired generations of poets; that seems a suitable unit of measurement for drama. Let's therefore assume that the putative Furaha ring system is made of dark rock.

So far we have the main features of the ring system in place, based on science, as it should be, but with a fair amount of handwavium. I am not certain about whether I should give up on Furaha's two tiny moons. Ideally, I should calculate appropriate gaps in the rings, but admit that at present I do not have the knowledge to do so. Any readers who can do so are friendly invited to comment on the matter!

Click to enlarge; copyright Gert van Dijk

This image shows Furaha with some fancy rings, for even more drama. The inner one is nearly completely transparent.  Another step in deciding whether or not I should add rings to Furaha may be to calculate the effects of this particular ring system on the lighting of the planetary surface. But, having done so once for Ilo, I already have a fairly good idea of what the effects for Furaha will be.

click to enlarge; copyright Gert van Dijk

This is another image of the same fancy rings, now from a lower point of view that lets you see the Southern hemisphere with the ring shadows. These shadows do not look too bad. Now that we have reasonable options to design rings, it is time to take the next step: what do the rings look like from any point on the surface of the planet, for any point in time of the year? That will require another post...                          

The Cycle of Cyann 3: the last of the rings...

I did mention that I thought some small errors had crept into the way Bourgeon and Lacroix had depicted the shape of the terminator (that's the edge between the lit and dark halves of a planet). How do I know?

Well, the terminator is supposed to be a great circle on a globe; it just looks distorted on a flat map, but is a circle in 3D. But if we take that flat map and drape it around a globe and have a look, the terminator ought to show up as a perfect circle again. I did just that with an image taken from the book 'La clé des confins', and here it is. The flat image is shown alongside the sphere it is draped around.

Click to enlarge

The thin latitude and longitude lines are drawn by the program I used to depict the globes (Matlab); the vaguer lines were in the original image. You can see the two equators line up nicely, so the map isn't too inaccurate. However, the terminator at the pole does not form a smooth circle, but has a distinct 'peak' there. Anyone who has tried to draw flat maps for the purpose of having them look good on a globe knows that polar areas are very difficult to get right. The map by Bourgeon and Lacroix shows some 'pinching' there.

To be fair, I also did the same trick with my own projection. Here we go:

Click to enlarge

It does work better, but I am not making this point to gloat. The opposite, in fact: I am astonished how accurate this was all done by Bourgeon and Lacroix. Remember that the album 'Six saisons sur Ilo' appeared in 1996 or 1997. I first worked out the mathematics of the terminator shape in 1985, and in 1996 this was the best printer output I could obtain:

Not exactly impressive, is it? So, no doubt, to produce good drawings, Bourgeon had to redraw the image by hand, and I guess a small error crept in. I am curious to hear whether this explanation is correct, though, so if anyone knows how to reach the authors, please let me know.

I also promised you animations of the ring and its shadows. They're done, and as I cannot post them here, I decided to incorporate them in the Furaha website. Simply go to the site, choose the planet icon and then 'some more examples'. If that sound too complicated, go there directly by clicking here (but you will lose the context).

Cyann and Ilo's rings

The last time I introduced 'Le Cycle de Cyann', focusing on the wildlife. On browsing through 'La clé des confins' my attention was drawn to some astronomical explanations. In the second album of the series Cyann travels over the planet Ilo (there should really be a dot in the letter O, but I cannot reproduce that here). That planet is of interest as it has a very long and narrow continent winding almost like a snake over the globe. This provides an opportunity for Cyann to travel through a vast array of landscapes along with their accompanying biotopes, much to the enjoyment of the reader.

But there is another thing that struck me, and that is that Ilo has rings. I cannot say whether it is at all probable that an Earth-like planet has rings, but their presence certainly has some interesting consequences. From seeing photographs of Saturn, everyone knows that the rings cast a shadow on the planet. But that shadow will not always fall in the same area, provided the planet is tilted to a degree.

In midwinter the North pole is tilted maximally towards the sun, and so will the rings; hence they cast a large shadow on the southern hemisphere. But at the spring and autumn equinoxes the plane of the rings present to the sun edge on, so the rings cast a shadow on the planet no wider than the rings themselves. Here is one picture of Cyann, waiting for her lover on the equator, on the day of the equinox:

Click to enlarge

She is facing the sun, and the rings behind her are lit, forming a luminescent vertical stripe in the sky. Looking in the other direction the rings form a dark stripe. Looks like an excellent setting for a romantic encounter.

The consequences of all this tilting are fairly complex. Think of how the ring shadows complicate winter: the shadow of the rings falls on the hemisphere that is tilted away from the sun, where it is winter anyway. But the shadows make the winter days even darker, but in complex ways: it is theoretically possible for the sun to come up, disappear behind the rings, come back again in midday, disappear behind the rings again, and finally to come back only to set on the horizon. All such matters are brilliantly handled in the book. Here are two pictures from 'Le clé des confins' showing how it works. One is an equinox picture, and the other is a solstice picture. If you need more explanations on why the sadows change shape, please go to the planet page on the Furaha site.

Click to enlarge

These caught my attention as they resemble the images on the astronomy page of Furaha quite closely, in particular the way the terminator is drawn. The 2x1 ratio of the maps is a giveaway that there is some serious thought here. So I wondered whether I could copy all these ring effects, and sat down to some good old-fashioned Matlab programming. Here is how that started:

Click to enlarge

You will see rays of the sun -the blue lines- starting from the edges of the rings. They are of course all parallel, and strike the planet (many don't of course, and these are not shown). The relative sizes of the planet and rings were taken from the book, and I estimated axial tilt to be 21 degrees. Once the x,y,z-coordinates of the rays falling on the planet are known, which only takes high school mathematics, these can be converted to latitudes and longitudes, et voilà. Project these over a rough map of the planet, redrawn from the book, and there we are:

Click to enlarge

Nice, isn't it? It is quite close to the drawing in the book, but small changes in ring diameter and axial tilt have large consequences. I also found that ths shape of the terminator in the solstice picture in the book is not exactly right. I can only assume it is some copying error, because the rest is much too good to be guesswork. Do not underestimate the amount of thought that has gone into this. I know from experience that working out the form of the terminator can take quite a bit of effort. Bourgeon and Lacroix not only did that too, but also worked out correct ring shadows, as well as thinking up cultures, animals, etc. It's almost enough to make you desperate...

But I won't be! Instead, I think I will produce some animations of the shadows as they change in the course of a year. Later.