PlanetData
FindFormula

Astronomy
Physics

Rediscovering Kepler's Third Law

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

In[1]:=

EntityValue

planets

PLANETS

,{"OrbitPeriod","SemimajorAxis"}

Out[1]=



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[2]:=

ListLogLogPlot[%]

Out[2]=

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[3]:=

FindFormula[%%,p]

Out[3]=

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
(5)

Pick 10 random exoplanets:

In[1]:=

RandomEntity["Exoplanet",10]

Out[1]=



Kepler 474 b

Kepler 142 c

Kepler 1099 b

Kepler 181 b

K2-75 b

HAT-P-25 b

Kepler 272 b

Kepler 675 b

Kepler 1060 b

HD 1605 c



Only a few have known semimajor axes:

In[2]:=

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

Out[2]=

{

135.949203

,Missing[NotAvailable]},{

114.3591

0.057

},{

52.0785080

,Missing[NotAvailable]},{

75.36053

0.04

},

7.8196

days

0.081

,{

87.72811

0.0466

},{

71.36132

0.038

},{

56.1369359

,Missing[NotAvailable]},

46.91004488

days

,Missing[NotAvailable],{

5.784

3.52

}

FInd results for 100 random exoplanets:

In[3]:=

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

Plot semimajor axis against orbit period:

In[4]:=

ListLogLogPlot[data]

Out[4]=

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

In[5]:=

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

Out[5]=

1.22107

0.6

QuantityMagnitude[p,JulianYears]

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

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Contributed by: Stephen Wolfram

Wolfram Language Example Repository

Rediscovering Kepler's Third Law

Exoplanets
(5)

See Also

Publisher Information

Rediscovering Kepler's Third Law

Exoplanets (5)

See Also

Related Symbols

Publisher Information

Exoplanets
(5)