<p>The &quot;orbits&quot; in this case have no closed-form in terms of circular or elliptic functions.  The locus of the body will tend to fill space in an annular region around the central point.</p>
<p>This image of a <span class="math-container">$F=-1/r$</span> orbit was produced by &quot;<a href="http://tp.lc.ehu.eus/jma/mechanics/central.jar" rel="nofollow noreferrer">central force workbench</a>&quot; by Juan M. Aguirregabiria.</p>
<p>The potential is now of the form <span class="math-container">$V= \ln|r|$</span>. The only formula that is unchanged is Kepler's second law: Equal areas are swept out in equal time, as this is true for all central forces.</p>
<p><a href="https://i.sstatic.net/6G3Uk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6G3Uk.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-10-11T21:37:11.063Z
2021-10-11T21:37:11.063Z
James K
<p>Imagine that Flatland habitats go into space. All stars and planets are the circles instead of spheres and gravity law is <span class="math-container">$F=Gm_1m_2/R$</span> instead of <span class="math-container">$F=Gm_1m_2/R^2$</span>.</p>
<p>How the other formulas will be changed in this case?</p>

<solar-system><gravity><space><newtonian-gravity>
How will be changed Hill's radius and Kepler's formulas in case of 2D space instead of 3D?
Robotex
# How will be changed Hill's radius and Kepler's formulas in case of 2D space instead of 3D?

(Asked by: username_0 on 2021-10-11)

Imagine that Flatland habitats go into space. All stars and planets are the circles instead of spheres and gravity law is $F=Gm\_1m\_2/R$ instead of $F=Gm\_1m\_2/R^2$.

How the other formulas will be changed in this case?

**Tags:** solar-system, gravity, space, newtonian-gravity


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## 1 answers

### Answer by username_1 on 2021-10-11. Score: 3

The "orbits" in this case have no closed-form in terms of circular or elliptic functions. The locus of the body will tend to fill space in an annular region around the central point.

This image of a $F=-1/r$ orbit was produced by "[central force workbench](http://tp.lc.ehu.eus/jma/mechanics/central.jar)" by Juan M. Aguirregabiria.

The potential is now of the form $V= \ln|r|$. The only formula that is unchanged is Kepler's second law: Equal areas are swept out in equal time, as this is true for all central forces.

[![enter image description here](https://i.sstatic.net/6G3Uk.png)](https://i.sstatic.net/6G3Uk.png)


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