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insertcleverphrasehere
2014-Jun-22, 09:40 AM
I am wondering if anyone has ever seen a log scale diagram of the solar system. By this I mean the orbits of the planets at distances in a log base 10 scale from the sun in a top down view.
I suspect it has not been done or at least it hasn't been uploaded to the internet at any time as i have looked pretty thoroughly.
I'd like to make one but am having a bit of trouble translating the elliptical orbits to log scale. If anyone has seen such a diagram or could assist in making one, please let me know.
the purpose is to produce a diagram in which the entirety of the solar system orbits can be seen easily but is still scientifically accurate.
thanks for any help.

Jeff Root
2014-Jun-22, 11:11 AM
I've never drawn such a diagram but I'm fairly sure I've thought
about it in years past, for the same purpose.

Why are you concerned about elliptical orbits? I expect that the
only orbit which will not be drawn as a perfect circle will be that
of Pluto.

If you *can* show the elliptical shape of orbits, I don't know
whether this is perfectly accurate, but it should be more than
close enough for any diagram that isn't going to be measured
with a micrometer caliper: Just convert the perihelion and
aphelion to their log values, and draw an ellipse with those
values. If you are doing it on a computer, I expect that you
will find the difference between the perihelion distance and
the aphelion distance is only a couple of pixels even for Mars
or Mercury. Maybe less than a pixel!

Something vaguely similar that I did-- This shows part of the
orbits of the Moon and the Earth-Moon barycenter around the
Sun, to scale: http://www.freemars.org/jeff2/MoonOrb1.png

Ask again if you have further questions or want some *real*
answers to the questions you asked.

-- Jeff, in Minneapolis

antoniseb
2014-Jun-22, 12:44 PM
I am wondering if anyone has ever seen a log scale diagram of the solar system. ... am having a bit of trouble translating the elliptical orbits to log scale. ...
the purpose is to produce a diagram in which the entirety of the solar system orbits can be seen easily but is still scientifically accurate.

You can easily just make a one-dimensional diagram showing the aphelion and perihelion values for each object. I'm not sure how much extra value you'd get by making a two dimension system of near-circles, as it doesn't give that much more accuracy of representation over a 1-D diagram, since the orbits are not co-planar (i.e. you'd need a 3D diagram to be more accurate.

One other thing I thought I'd mention. The log of the radius of the Earth's orbit in miles is almost 8.0. The log for Neptune is about 9.5. The log for the aphelion of Sedna is less than 11.0. Your diagram if drawn as near circles will mostly be concentrated at the edge.

Hornblower
2014-Jun-22, 02:02 PM
You can easily just make a one-dimensional diagram showing the aphelion and perihelion values for each object. I'm not sure how much extra value you'd get by making a two dimension system of near-circles, as it doesn't give that much more accuracy of representation over a 1-D diagram, since the orbits are not co-planar (i.e. you'd need a 3D diagram to be more accurate.

One other thing I thought I'd mention. The log of the radius of the Earth's orbit in miles is almost 8.0. The log for Neptune is about 9.5. The log for the aphelion of Sedna is less than 11.0. Your diagram if drawn as near circles will mostly be concentrated at the edge.
If we convert to larger units of measure for the planetary orbits to make the numbers smaller, the result will be subtracting the same number from the original logarithm for each planet. With a suitable choice of units we can get them nicely spaced over the range of the diagram.

For a 2D face-on view, I would just plot the orbits as circles and make them eccentric as needed. Even in the linear original they do not differ noticeably from circles at a casual glance, even for Mercury and Pluto. Measuring them with a compass shows the ellipse clearly, but the difference is surprisingly slight at these eccentricities.

insertcleverphrasehere
2014-Jun-22, 05:36 PM
If we convert to larger units of measure for the planetary orbits to make the numbers smaller, the result will be subtracting the same number from the original logarithm for each planet. With a suitable choice of units we can get them nicely spaced over the range of the diagram.

For a 2D face-on view, I would just plot the orbits as circles and make them eccentric as needed. Even in the linear original they do not differ noticeably from circles at a casual glance, even for Mercury and Pluto. Measuring them with a compass shows the ellipse clearly, but the difference is surprisingly slight at these eccentricities.

actually it is the orbit of mercury, and to a lesser extent mars that are very visible eccentric, this is because toward the base of the log plot (closer to the sun) the eccentricities actually become a little MORE pronounced, not less. attached is an excel chart showing MIN, AVE and MAX distances from the sun for each planet and plotted to a log scale. it is rough but shows that the eccentricities are easily visible. 19664

StupendousMan
2014-Jun-24, 03:12 PM
Bird's eye view of the inner solar system from above, with ordinary scales on the axes.

19670

Bird's eye view of the outer solar system from above, with ordinary scales on the axes.

19669

Bird's eye view of the entire solar system from above, with one choice of logarithmic scaling of the orbital sizes (there are many different possible ways to choose the scale). With this scaling, the orbital shapes are no long ellipses with the proper eccentricity. I don't think it's possible to scale the distances logarithmically and keep the eccentricities accurate. In any case, I don't think that using a logarithmic scale helps in any way.

19671

Hornblower
2014-Jun-25, 06:53 PM
While it is true that the ratio of perihelion to aphelion is not to scale in the log plot, it still is clearly visible.

The log plot is helpful in displaying the overall pattern in a single view when the ratio of the largest to the smallest value is so large that the small stuff becomes vanishingly small when the scale is set so the largest will fit on the chart. There is no sacrifice of scientific accuracy as long as we recognize and understand the mathematical basis for the display.