# Belt of matter at ~14 AU?

In this terrific answer user Hobbes presents a graph of escape velocity as a function of distance from the Sun:

I notice that at about 14 AU the graph goes from decreasing exponentially to decreasing linearly. This looks to me like there must be some significant mass distributed in space more or less evenly starting at about ~14 AU which would account for the increase in speed compared to a continued exponential decline.

My first thought was "dark matter" but I doubt this is the reason for the change in graph profile. So far as I know, the significant large outer-solar-system objects start past Neptune (hence the term TNO), out in the Kupier belt and Oort cloud. What is responsible for the change in graph profile (blue line) at ~14 AU?

• I notice that at about 14 AU the graph goes from decreasing exponentially to decreasing linearly. (1) Does it really? And (2) before asking on the belt of matter, I would ask: Is this accurate?.
– gerrit
Jan 29 '14 at 11:13
• @gerrit: It came from wikipedia! It must be accurate! :) Jan 29 '14 at 15:32

It looks to be merely an approximation in the SVG graph (in vector format) that was prepared for Wikipedia and later used in a converted to a PNG format (raster). If you check the SVG source, the path was defined as:

M 8 -400 C 10 -250 60 -85 140 -68 Q 150 -65 400 -50


Where M is moveto moving the starting point to x = 8, y = -400, C is curveto drawing a Bézier curve from the previously defined starting point with curve control points x1 = 10, y1 = -250 and x2=60, y2=-85 to x=140, y=-68 and then continues with a quadratic Bézier curve Q from x=400, y=-50 to a control point at x1=150, y1=-65.

So it is not linear at all, even though it might appear so (it might even have something to do with browser specific interpretations of SVG format Bézier curves, but I didn't test that). And neither is entirely precise, nor I believe it was meant to be. It's not too hard to find different graphs of Solar system escape velocities required at different distances to the Sun neither, for example:

Heliocentric velocity vs. radial distance from the Sun for a constant thrust (Source: The Dawn Mission, J. Varey, U. of Colorado)

Of course, that's just musing with the impossibile, since IPS on the Dawn spacecraft can't provide constant thrust for so long, but the escape velocity curve is not linear.

Why was the graph for Wikipedia prepared with such a shallow curve beyond 14 AU however remains unclear, as there isn't any explanation for it published on its page. My guess - because it aligned the two curves used nicer and it "looks" better like so. But I personally wouldn't start drawing conclusions that there must be some additional mass in the Solar system because of it. It has little to do with reality. ;)

• Thank you! For some reason, my wife also comments that my conclusions have little to do with reality :) Jan 29 '14 at 15:31