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Rediscovering Kepler’s Third Law

Stephen Wolfram

Author

Stephen Wolfram

Title

Rediscovering Kepler’s Third Law

Description

Our Solar System

Rediscovering Kepler’s Third Law

Our Solar System

Find the orbit period and semimajor axis for all the planets in our Solar System:

In[]:=

EntityValue

planets

PLANETS

,{"OrbitPeriod","SemimajorAxis"}

Out[]=



87.96926

days

0.38709893

,

224.70080

days

0.72333199

,

365.25636

days

1.00000011

,{

1.8808476

1.52366231

},{

11.862615

5.20336301

},{

29.447498

9.53707032

},{

84.016846

19.19126393

},{

164.79132

30.06896348

}

Make a log-log plot:

In[]:=

ListLogLogPlot[%]

Out[]=

The fact that it’s close to a straight line implies a power law relationship.

Find a formula for the semimajor axis as a function of the period:

In[]:=

FindFormula[%%,p]

Out[]=

0.0135544

0.7

QuantityMagnitude[p,Days]

There’s an exponent of 0.7, close to the 2/3 of Kepler’s third law.

Exoplanets

Pick 10 random exoplanets:

In[]:=

RandomEntity["Exoplanet",10]

Out[]=



Kepler 603 d

Kepler 52 c

Kepler 1207 b

Kepler 436 c

HATS-36 b

Kepler 1031 b

EPIC 211945201 b

Kepler 869 b

Kepler 1283 b

Kepler 1392 b



Only a few have known semimajor axes:

In[]:=

EntityValue[%,{"OrbitPeriod","SemimajorAxis"}]

Out[]=



6.22138751

days

,Missing[NotAvailable],

16.39622470

days

,Missing[NotAvailable],

13.69174268

days

,Missing[NotAvailable],

16.80864363

days

,Missing[NotAvailable],{

100.2731

0.0523

},{

29.4493727

,Missing[NotAvailable]},

19.50556

days

0.1493

,

40.45646644

days

,Missing[NotAvailable],

12.95496499

days

,Missing[NotAvailable],

15.15141028

days

,Missing[NotAvailable]

FInd results for 100 random exoplanets:

In[]:=

data=EntityValue[RandomEntity["Exoplanet",100],{"OrbitPeriod","SemimajorAxis"}];

Plot semimajor axis against orbit period:

In[]:=

ListLogLogPlot[data]

Out[]=

Find a fit to the data, removing all cases in the data that contain missing values:

In[]:=

FindFormula[DeleteMissing[data,1,1],p]

Out[]=

0.672582

QuantityMagnitude[p,JulianYears]

The exponent is extremely close to 2/3, validating Kepler’s third law.

Cite this as: Stephen Wolfram, "Rediscovering Kepler’s Third Law" from the Notebook Archive (2019), https://notebookarchive.org/2019-08-98zx44w

Wolfram Foundation

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