Question 62 - Problems - Chapter 1

Problem:

The table below shows the masses and the diameters of the different planets in our solar system.

import numpy as np
import pandas as pd

PlanetName = ['Mercury','Venus', 'Earth', 'Mars', 'Jupiter', 'Saturnus', 'Uranus', 'Neptunus', 'Pluto']
DistanceSun = [57.9e6, 108.0e6, 150.0e6, 228.0e6, 778.0e6, 1430.0e6, 2870.0e6, 4500.0e6, 5890.0e6]
OrbitalPeriod = [0.241, 0.615, 1, 1.88, 11.9, 29.5, 84, 165, 248]
Mass = [3.3e23, 4.87e24, 5.98e24, 6.42e23, 1.9e27, 5.67e26, 8.7e25, 1.03e26, 1.5e22]
Radius = [2439, 6052, 6378, 3393, 71399, 60000, 25400, 24300, 1500]
Gravity = [0.38, 0.91, 1, 0.38, 2.53, 1.07, 0.92, 1.19, 0.045]
RotationPeriod = [58.6, 243, 0.997, 1.026, 0.41, 0.43, 0.65, 0.77, 6.39]
PlanetDataSet = list(zip(PlanetName,DistanceSun,OrbitalPeriod,Mass,Radius,Gravity,RotationPeriod))
df = pd.DataFrame(data = PlanetDataSet, columns=['PlanetName','DistanceSun','OrbitalPeriod','Mass','Radius','Gravity','RotationPeriod'])
df.set_index('PlanetName', inplace=True)
df

	DistanceSun	OrbitalPeriod	Mass	Radius	Gravity	RotationPeriod
PlanetName
Mercury	5.790000e+07	0.241	3.300000e+23	2439	0.380	58.600
Venus	1.080000e+08	0.615	4.870000e+24	6052	0.910	243.000
Earth	1.500000e+08	1.000	5.980000e+24	6378	1.000	0.997
Mars	2.280000e+08	1.880	6.420000e+23	3393	0.380	1.026
Jupiter	7.780000e+08	11.900	1.900000e+27	71399	2.530	0.410
Saturnus	1.430000e+09	29.500	5.670000e+26	60000	1.070	0.430
Uranus	2.870000e+09	84.000	8.700000e+25	25400	0.920	0.650
Neptunus	4.500000e+09	165.000	1.030000e+26	24300	1.190	0.770
Pluto	5.890000e+09	248.000	1.500000e+22	1500	0.045	6.390

Question:

a) Calculate the average density of each plannet and make a list of the planets in order of decreasing densities.
b) Is there a corelation between the density of the planet and the distance to the sun?

Solution:

To calcculate density Rho, we use the formula:

$$ \textrm{Density}\;\varrho =\;\frac{\textrm{Weight}\;\left(\textrm{in}\;\textrm{kg}\right)}{\textrm{Volume}\;\left(\textrm{in}\;m^3 \right)} $$

Therfore, we need the volume of each planet

$$ \textrm{Volume}\;v\;=\frac{4}{3}*\pi \;r^3 $$

# Add two columns
df['Volume'] = 0
df['Density'] = 0

#iterate the dataset and calculate volume and density 
for index, row in df.iterrows():
    Volume = (4/3) * np.pi * (row["Radius"]*1000)**3
    df.loc[index,['Volume']] = Volume            # add values to the table
    Density = (row['Mass']*1e3) / (Volume * 1e6) # in g/cm3
    df.loc[index,['Density']]  = Density
    print(f'For planet {index} the density is {Density:.2g} g/cm3')
    
df.sort_values(by=['Density'], ascending = False)

For planet Mercury the density is 5.4 g/cm3
For planet Venus the density is 5.2 g/cm3
For planet Earth the density is 5.5 g/cm3
For planet Mars the density is 3.9 g/cm3
For planet Jupiter the density is 1.2 g/cm3
For planet Saturnus the density is 0.63 g/cm3
For planet Uranus the density is 1.3 g/cm3
For planet Neptunus the density is 1.7 g/cm3
For planet Pluto the density is 1.1 g/cm3

	DistanceSun	OrbitalPeriod	Mass	Radius	Gravity	RotationPeriod	Volume	Density
PlanetName
Earth	1.500000e+08	1.000	5.980000e+24	6378	1.000	0.997	1.086781e+21	5.502487
Mercury	5.790000e+07	0.241	3.300000e+23	2439	0.380	58.600	6.077487e+19	5.429876
Venus	1.080000e+08	0.615	4.870000e+24	6052	0.910	243.000	9.285074e+20	5.244977
Mars	2.280000e+08	1.880	6.420000e+23	3393	0.380	1.026	1.636214e+20	3.923691
Neptunus	4.500000e+09	165.000	1.030000e+26	24300	1.190	0.770	6.010456e+22	1.713680
Uranus	2.870000e+09	84.000	8.700000e+25	25400	0.920	0.650	6.864197e+22	1.267446
Jupiter	7.780000e+08	11.900	1.900000e+27	71399	2.530	0.410	1.524632e+24	1.246202
Pluto	5.890000e+09	248.000	1.500000e+22	1500	0.045	6.390	1.413717e+19	1.061033
Saturnus	1.430000e+09	29.500	5.670000e+26	60000	1.070	0.430	9.047787e+23	0.626673

b) Density is decresing if planets are further from the sun but not in a linear way so this can be a coincidence.