• This presentation shows how does a planet’s orbital period relates to its distance from the Sun.
• This is related to Kepler’s Third Law: The squares of the orbital periods of the planets are directly proportional to the cubes of the semi-major axes of their orbits (T^2 ∝ R^3).
• We’ll verify this with a log–log linear regression and visualize the Solar System with ggplot2 and a 3D Plotly plot.

• Loads required R libraries and displays the data as a table.
• Creates the Solar System dataset with each planet’s distance(AU), orbital period(days), mass(Earth Masses), and radius(km).

  Solar System Planets

  name	distance_AU	orbital_period_days	radius_km	mass_earths
  Mercury	0.39	88	2440	0.055
  Venus	0.72	225	6052	0.815
  Earth	1.00	365	6371	1.000
  Mars	1.52	687	3390	0.107
  Jupiter	5.20	4333	69911	317.800
  Saturn	9.58	10759	58232	95.200
  Uranus	19.18	30687	25362	14.500
  Neptune	30.07	60190	24622	17.100

• Transforms the data using logarithms and fits a linear regression model log(T) vs log(R).
• Plots the relationship with a regression line and planet labels to visualize Kepler’s Law.

• Plots residuals versus fitted values from the regression model to check model fit and assumptions.
• Ensures there’s no strong pattern, confirming a good linear relationship in log–log space.

• Creates an interactive 3D scatter plot of planet mass, radius, and distance from the Sun using Plotly.
• Lets you rotate and explore planetary relationships in three dimensions.

• Extracts the slope estimate, standard error, p-value, and confidence interval from the regression.
• Tests whether the slope is approximately 1.5, confirming consistency with Kepler’s Law.

## Estimated slope: 1.5000

## SE(beta1): 0.0015

## t-stat (H0: beta1=1.5): 0.021

## p-value: 0.9837

## 95% CI: [1.4964, 1.5037]

If the 95% CI contains 1.5 and the p-value is large, the data are consistent with Kepler’s law.

• This slide displays the core R code used to reproduce the analysis of Kepler’s Third Law. It shows how the Solar System data are created, transformed into logarithmic form, and modeled using simple linear regression. -The final lines use ggplot2 to visualize the relationship between log(T) (orbital period) and log(R) (AU), producing the log–log plot seen earlier.

# Build data
planets <- tibble::tibble(
  name = c("Mercury","Venus","Earth","Mars","Jupiter","Saturn","Uranus","Neptune"),
  distance_AU = c(0.39, 0.72, 1.00, 1.52, 5.20, 9.58, 19.18, 30.07),
  orbital_period_days = c(88, 225, 365, 687, 4333, 10759, 30687, 60190)
)

# Transform + fit model
df <- planets %>% mutate(logT = log10(orbital_period_days),
                         logR = log10(distance_AU))
m <- lm(logT ~ logR, data = df)

# Visualize
ggplot(df, aes(logR, logT)) +
  geom_point() +
  geom_smooth(method = "lm")

• The log–log regression slope is ~1.5, which matches Kepler’s Third Law.
• Diagnostics show the linear model is a good approximation on the log scale.
• 3D Plotly view adds intuition about mass and radius vs distance.