<p>Copernicus estimated the ratio of each planet. Kepler either used these numbers or recalculated for himself.</p>
<p>For the inner planets (Mercury and Venus), he used the max elongation angles, which is the max angle between the planet and the Sun as observed from Earth. Note this forms a right triangle, so <span class="math-container">$d= sin(\theta)$</span>, where <span class="math-container">$d$</span> is the radius of the inner planet in AU (Earth's orbital radius), and <span class="math-container">$\theta$</span> is the max elongation angle.</p>
<p>For the rest of the planets, he measured the time between quadrature and opposition. Knowing the sidereal period of earth and the planet, he estimated the angle swept in that time as <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span> respectively. Then <span class="math-container">$d = 1/ cos(\alpha-\beta)$</span>. From <a href="https://wisp.physics.wisc.edu/astro104/lecture4/lec4i.html#:%7E:text=Copernicus%20used%20simple%20geometry%20to,of%20an%20inferior%20planet%20orbit." rel="noreferrer">University of Wisconsin astronomy notes</a> here is a figure for clarity:</p>
<p><a href="https://i.sstatic.net/lQXyd3r9.jpg" rel="noreferrer"><img src="https://i.sstatic.net/lQXyd3r9.jpg" alt="enter image description here" /></a></p>
<p>Copernicus did a pretty good job at estimating planet radii (same source as above):</p>
<p><a href="https://i.sstatic.net/QsuZaIYn.jpg" rel="noreferrer"><img src="https://i.sstatic.net/QsuZaIYn.jpg" alt="enter image description here" /></a></p>
<p>To me, the more interesting part of Kepler's observations is when he realized ellipses better fit the orbital paths of the planets than the circles used by Copernicus.</p>

CC BY-SA 4.0
2024-10-16T05:48:45.163Z
Thanks. This helps me understand better. I tried looking up how the radii of planets’ orbits were calculated, but most search results just kept giving me Kepler’s law and not how Kepler (or Copernicus) calculated them.
Priscilla Thomas
CC BY-SA 4.0
2024-10-16T15:03:40.363Z
Note that Kepler only foumd the distances in units of  Earth's distance from the Sun, which wasn't known.  Fifty years later Cassini found the distances in units of length.
stretch
CC BY-SA 4.0
2024-10-16T16:32:43.073Z
@stretch, exactly, and also that’s why the “Astronomical Unit” (AU) is so important. I don’t know the full history, but we didn’t invent AU’s so “we could get” a relative sense of distances, and didn’t name it with such a _grandiose_ name for egotistical reasons. It’s because for a long time we had really precise measurements of the solar system distances but only _relative to each other_. The size of the orbit of the earth itself was then a convenient yardstick for solar system measurements. When we nailed the size of that orbit in “earthly units”, everything else could be pinned down in turn
Euro Micelli
CC BY-SA 4.0
2024-10-15T22:55:17.710Z
2024-10-15T22:55:17.710Z
Connor Garcia
<p>I don’t understand how Johannes Kepler got the data to formulate his third law. How was the radius of each planet’s orbit determined?</p>

CC BY-SA 4.0
2024-10-16T02:14:25.190Z
Related: https://astronomy.stackexchange.com/questions/8849/how-did-kepler-guess-his-third-law-from-data
Nilay Ghosh
CC BY-SA 4.0
2024-10-16T02:16:00.917Z
And same discussion in HSM.SE: https://hsm.stackexchange.com/questions/1948/how-did-kepler-guess-his-third-law-from-data
Nilay Ghosh
<kepler>
How did Johannes Kepler figure out the radius of each planet’s orbit?
Priscilla Thomas
# How did Johannes Kepler figure out the radius of each planet’s orbit?

(Asked by: username_0 on 2024-10-15)

I don’t understand how Johannes Kepler got the data to formulate his third law. How was the radius of each planet’s orbit determined?

**Tags:** kepler

> **Comments:**
> **username_1**: Related: https://astronomy.stackexchange.com/questions/8849/how-did-kepler-guess-his-third-law-from-data  
> **username_1**: And same discussion in HSM.SE: https://hsm.stackexchange.com/questions/1948/how-did-kepler-guess-his-third-law-from-data  

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## 1 answers

### Answer by username_2 on 2024-10-15. Score: 29

Copernicus estimated the ratio of each planet. Kepler either used these numbers or recalculated for himself.

For the inner planets (Mercury and Venus), he used the max elongation angles, which is the max angle between the planet and the Sun as observed from Earth. Note this forms a right triangle, so $d= sin(\theta)$, where $d$ is the radius of the inner planet in AU (Earth's orbital radius), and $\theta$ is the max elongation angle.

For the rest of the planets, he measured the time between quadrature and opposition. Knowing the sidereal period of earth and the planet, he estimated the angle swept in that time as $\alpha$ and $\beta$ respectively. Then $d = 1/ cos(\alpha-\beta)$. From [University of Wisconsin astronomy notes](https://wisp.physics.wisc.edu/astro104/lecture4/lec4i.html#:%7E:text=Copernicus%20used%20simple%20geometry%20to,of%20an%20inferior%20planet%20orbit.) here is a figure for clarity:

[![enter image description here](https://i.sstatic.net/lQXyd3r9.jpg)](https://i.sstatic.net/lQXyd3r9.jpg)

Copernicus did a pretty good job at estimating planet radii (same source as above):

[![enter image description here](https://i.sstatic.net/QsuZaIYn.jpg)](https://i.sstatic.net/QsuZaIYn.jpg)

To me, the more interesting part of Kepler's observations is when he realized ellipses better fit the orbital paths of the planets than the circles used by Copernicus.

> **Comments:**
> **username_0**: Thanks. This helps me understand better. I tried looking up how the radii of planets’ orbits were calculated, but most search results just kept giving me Kepler’s law and not how Kepler (or Copernicus) calculated them.  
> **username_3**: Note that Kepler only foumd the distances in units of  Earth's distance from the Sun, which wasn't known.  Fifty years later Cassini found the distances in units of length.  
> **username_4**: @username_3, exactly, and also that’s why the “Astronomical Unit” (AU) is so important. I don’t know the full history, but we didn’t invent AU’s so “we could get” a relative sense of distances, and didn’t name it with such a _grandiose_ name for egotistical reasons. It’s because for a long time we had really precise measurements of the solar system distances but only _relative to each other_. The size of the orbit of the earth itself was then a convenient yardstick for solar system measurements. When we nailed the size of that orbit in “earthly units”, everything else could be pinned down in turn  

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