<p>Kepler's third law is <em>not</em> that <span class="math-container">$a^3/T^2$</span> is a constant. It is that
<span class="math-container">$$ \frac{a^3}{T^2} \propto (M_\odot + M_{\rm planet}) $$</span>
and so the left hand side depends on which planet you are looking at and should be larger for Jupiter than it is for the Mars for example.</p>
<p>Does that account for your discrepancy? Almost. It should change the 4th significant figure, because for Jupiter, <span class="math-container">$(M_\odot + M_{\rm planet}) = (1 + 0.000954)M_\odot$</span>, whilst for Mars it is <span class="math-container">$(1 + 0.0000003)M_\odot$</span> - this gives a ratio of 1.00095 between the RH sides of Kepler's third law for Jupiter vs Mars, compared with the discrepancy you have found that has a ratio of 1.0012.</p>
<p>Further discrepancies are of course caused by the fact that the Sun/planet system is not an isolated 2-body system. The other planets will perturb each others' orbits.</p>

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2021-11-29T14:38:22.750Z
2021-11-29T14:38:22.750Z
ProfRob
<p><a href="https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law" rel="nofollow noreferrer">The Wikipedia article</a> about Kepler's third law includes a nice table about the ratio between <span class="math-container">$a^3$</span> and <span class="math-container">$T^2$</span>. However, for Mars, Jupiter, Saturn, Uranus, and Neptune, the ratio <span class="math-container">$\frac{a^3}{T^2}$</span> in <span class="math-container">$10^{-6} \text{AU}^3 \text{ day}^{-2}$</span> deviates from the supposed constant <span class="math-container">$7.495564252$</span>. For instance, Mars' ratio is <span class="math-container">$7.4950842239$</span> when applying the formula, which is off by <span class="math-container">$0.0004800281$</span>, while Jupiter's ratio is <span class="math-container">$7.50430207179$</span>. Why is this happening? I don't think that it's because of orbital eccentricity because Mercury contradicts this hypothesis, nor is it gravitational perturbations between the body and the Sun because of Mars' case. I'm suspecting it to be inaccurate orbital measurements, but I would like to know for sure.</p>

<orbit><planet><mathematics><keplers-laws>
Why is there a deviation between the ratio of $a^3$ and $T^2$ for the outer planets?
WarpPrime
# Why is there a deviation between the ratio of $a^3$ and $T^2$ for the outer planets?

(Asked by: username_0 on 2021-11-29)

[The Wikipedia article](https://en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion#Third_law) about Kepler's third law includes a nice table about the ratio between $a^3$ and $T^2$. However, for Mars, Jupiter, Saturn, Uranus, and Neptune, the ratio $\frac{a^3}{T^2}$ in $10^{-6} \text{AU}^3 \text{ day}^{-2}$ deviates from the supposed constant $7.495564252$. For instance, Mars' ratio is $7.4950842239$ when applying the formula, which is off by $0.0004800281$, while Jupiter's ratio is $7.50430207179$. Why is this happening? I don't think that it's because of orbital eccentricity because Mercury contradicts this hypothesis, nor is it gravitational perturbations between the body and the Sun because of Mars' case. I'm suspecting it to be inaccurate orbital measurements, but I would like to know for sure.

**Tags:** orbit, planet, mathematics, keplers-laws

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## 1 answers

### Answer by username_1 on 2021-11-29. Score: 9 | Accepted answer

Kepler's third law is *not* that $a^3/T^2$ is a constant. It is that
$$ \frac{a^3}{T^2} \propto (M\_\odot + M\_{\rm planet}) $$
and so the left hand side depends on which planet you are looking at and should be larger for Jupiter than it is for the Mars for example.

Does that account for your discrepancy? Almost. It should change the 4th significant figure, because for Jupiter, $(M\_\odot + M\_{\rm planet}) = (1 + 0.000954)M\_\odot$, whilst for Mars it is $(1 + 0.0000003)M\_\odot$ - this gives a ratio of 1.00095 between the RH sides of Kepler's third law for Jupiter vs Mars, compared with the discrepancy you have found that has a ratio of 1.0012.

Further discrepancies are of course caused by the fact that the Sun/planet system is not an isolated 2-body system. The other planets will perturb each others' orbits.


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