An orbital ring connected to the Earth by space elevators would reduce the cost of going to space to an amount comparable to an airplane ticket. This would cause a boom in the space tourism industry and eventually millions and even billions of people and tons of cargo will be moving from the Earth’s surface to space annually, and vise versa. This would necessitate an expansion in our space-based infrastructure to include space-based solar panels, a lunar mass driver, the routine mining of asteroids, and especially enormous space habitats (for all those billions of people to live in) such as the Standford Torus, the Bernal Sphere, or the O’Neil Cylinder. Orbital rings also allow you to build artificial planets and Dyson spheres, which would allow us to completely colonize the solar system. They would also allow us to build a Birch planet, a single planet with a surface area which exceeds the total surface area of all the planets in the Milky Way galaxy.

# Megastructures: Shkadov Thrusters (VIDEO)

# The Diversity of Exoplanets in the Galaxy

In this lesson, we’ll give a brief catalog of the various different classes of planets in the universe. We'll discuss Pulsar planets, hot Jupiters, Super Earths, ice and water worlds, archipelago worlds, diamond worlds, and rogue planets. Most of the planets we’ll be discussing were discovered using the Kepler Space Telescope and the transit method. We once believed that the formation of planets was rare and that there probably weren’t many planets beyond our solar system. We couldn’t have been more wrong.

# Drake's Equation and Searching for Life in the Milky Way

In this lesson, we’ll discuss the prospect of life in the Milky Way galaxy beyond the Earth. We'll begin by discussing the speculations made in a paper written by Carl Sagan about the possibility of life in Jupiter's atmosphere. From there, we shall derive a formula which describes the habitable zone of a star. Using this formula and data obtained by the Kepler Space Telescope, we can estimate the total number of "Earth-like" planets in the Milky Way. From there, we discuss the fraction of those planets on which simple and intelligent life evolve; then we'll discuss the fraction of those planets on which advanced communicating civilizations evolve and what fraction of those civilizations are communicating right now.

# Proof of the Theorem: \(\lim_{ϴ→0}\frac{sinϴ}{ϴ}=1\)

In this lesson, we’ll prove that \(\lim_{ϴ→0}\frac{sinϴ}{ϴ}=1\). We'll prove this result by using the squeeze theorem and basic geometry, algebra, and trigonometry. In a future lesson, we'll learn why this result is important: the reason being because knowledge that \(\lim_{ϴ→0}\frac{sinϴ}{ϴ}=1\) is required to find the derivatives of the sin and cosine functions. But we'll save that for a future lesson.

# Proof of Green's Theorem

For a vector field \(\vec{F}(x,y)\) defined at each point \((x,y)\) within the region \(R\) and along the continuous, smooth, closed, piece-wise curve \(c\) such that \(R\) is the region enclosed by \(c\), we shall derive a formula (known as Green’s Theorem) which will allow us to calculate the line integral of \(\vec{F}(x,y)\) over the curve \(c\).

# Gravitational Force Exerted by a Rod

# Gravitational Force Exerted by a Sphere

To find the gravitational force exerted by a sphere of mass \(M\) on a particle of mass \(m\) outside of that sphere, we must first subdivide that sphere into many very skinny shells and find the gravitational force exerted by anyone of those shells on \(m\). We'll see, however, that finding the gravitational force exerted by such a shell is in of itself a somewhat tedious exercise. In the end, we'll see that the gravitational force exerted by a sphere of mass \(M\) on a particle of mass \(m\) outside of the sphere (where \(D\) is the center-to-center separation distance between the sphere and the particle) is completely identical to the gravitational force exerted by a particle of mass \(M\) on the other particle of mass \(m\) such that \(D\) is there separation distance.

# Introduction to Double Integrals

In previous lessons, we learned that by taking the integral of some function \(f(x)\) we can find the area underneath that curve by summing the areas of infinitely many, infinitesimally skinny rectangles. In this lesson, we'll use the concept of a double integral to find the volume underneath any smooth and continuous surface \(f(x,y)\) by summing the volumes of infinitely many, infinitesimally skinny columns.

# Colonizing and Terraforming Venus

The first serious proposal in scientific literature on terraforming other worlds in the universe was about terraforming Venus. The planetary scientist Carl Sagan imagined seeding the Venusian skies with photosynthetic microbes capable of converting Venus's \(C0_2\)-rich atmosphere into oxygen. Other proposals involve assembling a vast system of orbital mirrors capable of blocking the Sun's light and cooling Venus until this hot and hellish world became very frigid and rained \(C0_2\) from its atmosphere. The solleta would also be capable of simulating an Earth day/night cycle. To create oceans and an active hydrosphere on Venus, we could hurl scores of icy asteroids from the Kuiper belt to Venus and, upon impacting the Venusian atmosphere, would rapidly disintegrate releasing enormous quantities of water vapor into the atmosphere which subsequently condense to form the first seas on Venus. Or perhaps Saturn's moon Enceladus—containing a colossal subsurface ocean dwarfing that of the Earth's—could be sacrificed towards the end of creating the first seas on Venus. But even if humans never terraform this hellish world, they could still live their—partially at least—by deploying thousands of blimps into the Venusian skies capable of supporting a long-term, human presence of perhaps over a million people. Venusian sky cities. But eventually, after many millennia of terraforming Venus, a rich ecosystem of life—including us—could live on Venus's surface.