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### The Solar System on a square-root scale

Posted: Thu Jul 07, 2016 7:48 pm UTC
Hello,

I was looking on the web for a representation of the Solar System on a square-root scale, that is, representing lengths (some of them, of course) in proportion with the square root of their actual value. To my surprise, I didn't find any.

I find this scale interesting in the case of the Solar System, because it allows to put both the size of objects and some of the distances between them on the same scale in a somewhat meaningful way.

I made such a picture (generated an svg). EDIT: Here it is. Click for a description and higher-quality versions.

I was surprised to not find this because the idea seems simple enough to me. I wonder whether physicists would find such a representation tolerable... What do you think ?

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 12:04 am UTC
Are you thinking of something like a logarithmic-scale model? That seems to fulfill the kind of thing you're asking for, and might explain why you didn't find anything if looking for square-root scale. Or is there something about the logarithmic scale that makes you not want to use it?

Log-scale, out to some nearby stars:

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 6:58 am UTC
DaBigCheez wrote:Or is there something about the logarithmic scale that makes you not want to use it?

Well, the picture you give shows distances to the Sun nicely but the size of objects looks arbitrary. (If not, what is their scale?) Also the position of the Sun is arbitrary because there is no 0 on the axis.

On the other hand, a square-root scale allows to compare both sizes and distances (including sizes to distances).

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 10:57 am UTC
I still don't get the Square Root thing. It's highly unit-magnitude sensitive, for a start (root of 93 million miles is 9500-odd, whilst root of 1AU is 1AU), and the size of the planets, square-rooted by some arbitrary unit-length to make one planet fit (and not dominate) the scale would badly mismatch for the others.

Perhaps a diagram of your own?

The log-scale is self-consistent despite the unit (though the base of the log alters things in a trivial way, c.f. natural log amd base 10) with the same sublength along the line being the same simple multiple, as proportion. And the sun's position is zero (or perhaps infinitesimally close to, which is close enough!). The size of the objects might be arbitrary, but they look reasonably well atuned to logorithmic scaling, too (as radius, cross-section area or volume), possibly such that they'd be correctly-sized (by that metric) if overlaid on the zero-point as per the Sun, just shifting their centres to their orbital distances, but I'd have to check pixel-lengths to be sure. Well, except for the neighbouring stars section. But the planets (and Moon) are sensibly related by non-linearsize, definitely.

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 11:41 am UTC
Yes, separate values are highly dependent on the chosen unit; however ratios are not. sqrt(A/B) = sqrt(A)/sqrt(B) in any unit, which allows for meaningful comparisons of distances and sizes. The same goes for units: 1 AU ≈ 1.4960×1011 m so on the drawing it would appear that 1 AU ≈ sqrt(1.4960×1011) m.

Link to the picture I made given in PM as I still have <5 posts.

On the log-scale diagram the Sun is drawn at abscissa 0.1 AU which I wouldn't call infinitesimal. As for the size of objects, well for instance we can see that Saturn's rings are larger than the Sun... And I don't see a Moon, only Venus (I'm not saying that this diagram is particularly bad, only not perfect.)

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 12:25 pm UTC
You got in there whilst I was editing my reply with your PMed link (you sent me an https, my browser complained that it wasn't secure, and so there was some caveat emptor to whether I should publish the http version, for you instead...).

I was primarily confused by the "it allows to put both the size of objects and select the distances between them on the same scale in a somewhat meaningful way". Earth radius is 6371km, 4.259x105AU. Earth orbit is (slightly less than) 150m km or 1AU.

By either measure, depicted Earth radius (linear) would be 23,500-ish times smaller than its position from origin, a tad over 150 times the rooted pixelature. It's not so bad as I thought, but pluto up to 49AU, or 7x root-AU), that's 1000+px width image for single-pixel (give or take antialiasing) Earth, and then you don't really see Pluto (needs blurring across Neptune's position, anyway), if you want to include it.

Bigger root-bases might help. Cube root? Maybe 100 pixels of width to Plutonian orbit, per Earthly pixel, if I've not messed up the sums. That might work better, under your basic non-log plan. Or base e, not that that's any more meaningful, in tjat context, I'm fairly sure.

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 4:27 pm UTC
The planets were not drawn to scale. Primarily because drawing them to scale, even in a log-scale system, would render them invisible.

Space is big.

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 5:18 pm UTC
@Soupspoon, yes, the square root is only a good compromise to show the planets and their biggest satellites, and it leaves Pluto far away.

On the other hand, the cubic root can reasonably go to the heliosheath (~4.63 AU) and show Pluto etc., but it makes satellites too big (they overlap). It is possible to hide satellites, but I find it a bit unsatisfactory to hide some objects just because there's something bigger next to them...

(Sorry, I had forgotten about that HTTPS thing. You could just overstep it; it's no worse than plain HTTP.)

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 5:40 pm UTC
DaBigCheez wrote:The planets were not drawn to scale. Primarily because drawing them to scale, even in a log-scale system, would render them invisible.

I disagree. The planets were not drawn to scale because the log scale, by definition, does not allow it in a straightforward way like the square-root scale does. By "straightforward way" I mean putting the center of the object on the origin (0) of an axis, then drawing it with a scaled radius. This is not possible with the log scale because "0" is infiniteley far left on the axis. That's also why it allows to display objects by their distance to the Sun, but the Sun has to be placed arbitrarily or be infinitely big or something.

PS: I'll update the first post with the link.

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 6:52 pm UTC
Here is a cubic-root version, showing objects bigger than Pluto included, up to the termination shock.

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 7:08 pm UTC
The sun isn't at a distance of 0 from the center of the solar system though, we can get some non arbitrary tiny number if we are worried about that.

### Re: The Solar System on a square-root scale

Posted: Fri Jul 08, 2016 9:17 pm UTC
doogly wrote:The sun isn't at a distance of 0 from the center of the solar system though, we can get some non arbitrary tiny number if we are worried about that.
True, the barycentre of the solar system isn't always within the sun, but it's not a big deal to 'pretend' that the position of each planet is perhaps the mean of the planet's distance from the Sun/barycentre, and the Sun at (as close as you can get to) zero on the scale, certainly within a solar radius.

### Re: The Solar System on a square-root scale

Posted: Sat Jul 09, 2016 12:56 am UTC
yeah but if homie is upset up log(0) then you can just not, and it isn't an arbitrary thing

basically, logarithms are the best

### Re: The Solar System on a square-root scale

Posted: Sat Jul 09, 2016 12:02 pm UTC
More importantly, for the sun to show up on the diagram, you just need its surface to be visible and log(Rsol) is definitely a nice sensible number to plot.

### Re: The Solar System on a square-root scale

Posted: Sat Jul 09, 2016 12:47 pm UTC
It's funny how I was wondering about the possible shortcomings of the little-used square-root scale and we end up discussing mostly the fine and well-established log scale instead...

My main cause for dissatisfaction with the log scale is the inability (I think) to draw objects (not only the Sun) at scale. Objects can be drawn at logarithmic coordinates, but what does it mean to draw a planet at scale with a log scale?

(EDIT: Sorry, weird thinking and phrasing. Of course there is a natural way to draw a planet at scale with the log scale, which is to draw it between the marks representing its closest and farthest points from the Sun on the log scale. That makes the planet too small to see on a diagram (like a linear scale), and also it make the planet's representation depend on its distance from the Sun, so you couldn't compare the represented size of two planets. Which you can with their square-root representation because it is based on the radius only.)

### Re: The Solar System on a square-root scale

Posted: Sat Jul 09, 2016 5:08 pm UTC
nahoj wrote:It's funny how I was wondering about the possible shortcomings of the little-used square-root scale and we end up discussing mostly the fine and well-established log scale instead...

Well, inevitably, yes. When there's an established, well-worn, and effective tool for the job, any alternative is going to be judged by its merits in comparison, not in a vacuum. You also started the thread without even mentioning log scale, so DaBigCheez and others thought perhaps you were unfamiliar with it; it'd be a pretty obvious alternative if you weren't.

nahoj wrote:My main cause for dissatisfaction with the log scale is the inability (I think) to draw objects (not only the Sun) at scale. Objects can be drawn at logarithmic coordinates, but what does it mean to draw a planet at scale with a log scale?

(EDIT: Sorry, weird thinking and phrasing. Of course there is a natural way to draw a planet at scale with the log scale, which is to draw it between the marks representing its closest and farthest points from the Sun on the log scale. That makes the planet too small to see on a diagram (like a linear scale), and also it make the planet's representation depend on its distance from the Sun, so you couldn't compare the represented size of two planets. Which you can with their square-root representation because it is based on the radius only.)

You could do the same if you're just taking the root of the distance from the sun for each point (creating little guitar-pick planets, I guess). Using the root of the radius isn't really "natural" in the sense that it comes out of the scale as marked, unless you're thinking of the scale as working in two dimensions. You could do something similar with a log scale by just creating a separate log scale for radii scaled to demonstrate whatever you want it to demonstrate.

### Re: The Solar System on a square-root scale

Posted: Sat Jul 09, 2016 5:49 pm UTC
Copper Bezel wrote:Well, inevitably, yes. [...]

Right.

Copper Bezel wrote:You could do the same if you're just taking the root of the distance from the sun for each point (creating little guitar-pick planets, I guess). Using the root of the radius isn't really "natural" in the sense that it comes out of the scale as marked, unless you're thinking of the scale as working in two dimensions.

Indeed I'm thinking of the scale as working in 2 dimensions. In fact, the way I generated the picture, each object has their own scale (with the origin at their center) used to represent their sizes. (Then for orbitee-orbiter distances, the orbitee's scale is used.) But since planets are aligned, their axis (for their orbiters) are too.

Copper Bezel wrote:You could do something similar with a log scale by just creating a separate log scale for radii scaled to demonstrate whatever you want it to demonstrate.

Yeah, I guess so. I'm not entirely convinced it would make a lot of sense because of the origin issue (even with the center of the system / orbitee dichotomy trick), but why not.

### Re: The Solar System on a square-root scale

Posted: Mon Jul 11, 2016 11:39 pm UTC
This is only vaguely on topic, but I'm excited to have an excuse to share the link. My high school was near the sun end of a linear scale model of the solar system. The sun is 15m in diameter and pluto is about 2.5cm in diameter and 65km away. Clearly linear scales are the best.