The Future Hub of Earth and Space: The Space Elevator

Casual Talk

BoJone believes that the significance of science does not lie in endless calculations, but in using limited scientific theories to explain as many natural and life phenomena as possible. For this reason, scientists pursue harmonious, concise, and elegant scientific theories. Science is about finding ways to turn the unknown into the known and developing further on that basis.

With the development of media technology, we have more and more channels to access information. Whenever we see a piece of scientific news on the Internet or in a newspaper, we are almost always excited by it. However, as the saying goes, "the layman looks at the excitement, while the expert looks at the essence." For friends who truly love science, they might be more interested in the origin of the news content. That is to say, we hope to further understand how the conclusions were reached—even if only at a very superficial level.

The Space Elevator

The following content is sourced from the Internet:

In 1895, a Russian scientist first proposed the concept of a space elevator. After more than a hundred years of research, this dream has yet to be realized. A space elevator is essentially a long cable—one end fixed on Earth and the other fixed to a counterweight (such as a large satellite) in geostationary orbit. Under the interaction of gravity and centripetal acceleration, the cable is kept taut. The space elevator will move up and down along the cable using solar or laser energy.

Since the end of the last century, NASA and several high-tech companies have begun intensive research into the feasibility of building a "space elevator" from Earth to a height of 100,000 kilometers in space. However, due to the extreme difficulty of the carbon nanotube cable technology and the high costs involved, the "space elevator" remains at the stage of a beautiful blueprint.

Recently, both the United States and Japan have announced plans to build space elevators tens of thousands or even over a hundred thousand kilometers long. In fact, NASA has indeed placed great importance on the space elevator project in recent years, listing it as the first project in its "Centennial Challenges" program. Michael Laine founded the LiftPort Group in 2003, investing heavily in research for a space elevator from Earth to space.

Analysis

Some friends might wonder: why do space agencies insist on building elevators tens of thousands of kilometers long right away, rather than starting with hundreds or thousands of kilometers? Let us analyze the length of the space elevator. To minimize energy consumption, we hope to utilize gravity and inertial centrifugal force to maintain balance. This requires the height of the elevator to meet certain conditions. To simplify the calculation, let us imagine the elevator as a uniform wire, as shown in the figure, with a linear density \lambda. Since the elevator is stationary relative to the Earth, every point on the wire is in circular motion with an angular velocity equal to the Earth’s rotation (currently considering only Earth’s gravity). First, we calculate the Earth’s gravitational pull on this elevator. Let the mass of the Earth be M and the total mass of the elevator be m.

For a point on the elevator at a distance r from the center of the Earth with mass dm, the gravitational force exerted by the Earth is dF_G = \frac{GMdm}{r^2} = \frac{GM\lambda dr}{r^2}. Thus, the total gravitational force is: F_G = \int_R^{R+h} \frac{GM\lambda dr}{r^2} = \frac{GM\lambda h}{R(R+h)} = \frac{GMm}{R(R+h)}

Next, we analyze the inertial centrifugal force. For a point on the elevator at a distance r from the center of the Earth with mass dm, the inertial centrifugal force is dF_C = dm \cdot r\omega^2 = \lambda \omega^2 r dr. Thus, the total centrifugal force is: F_C = \int_R^{R+h} \lambda \omega^2 r dr = \frac{\lambda h \omega^2}{2}(2R+h) = \frac{m\omega^2}{2}(2R+h)

When the two forces are equal: \begin{aligned} \frac{GMm}{R(R+h)} &= \frac{m\omega^2}{2}(2R+h) \\ h &= \sqrt{\frac{2GM}{\omega^2 R} + (R/2)^2} - \frac{3R}{2} \\ h &= \sqrt{\frac{v_2^2}{\omega^2} + \frac{R^2}{4}} - \frac{3R}{2} \end{aligned}

Where v_2 is the second cosmic velocity (escape velocity). For Earth, h is approximately 140,000 kilometers.

Cost

From the above calculation, it can be seen that h is independent of the mass of the elevator. So why is there still a desire to use lighter materials? In fact, a 100,000-kilometer elevator must be transported by rockets to a distance of 100,000 kilometers, so mass must be considered. Essentially, transportation costs are proportional to both mass and distance (this statement is not strictly rigorous; strictly speaking, the greater the mass and the longer the distance, the higher the cost). Furthermore, a lower mass means the mass can be more easily ignored, leading to greater stability.

Thoughtful friends might propose a scheme like this: place a point mass m' at the top of the elevator, and use uniform material for the part below. Indeed, this is a good idea as it would shorten the length of the elevator; however, the total mass of the elevator would increase. In this case, the equilibrium equation is: \frac{GM\lambda h}{R(R+h)} + \frac{GMm'}{(R+h)^2} = \frac{\lambda h\omega^2}{2}(2R+h) + m'(R+h)\omega^2

We generally find that the total mass \lambda h + m' will be larger. Nevertheless, this idea is not without merit. Because "transportation costs are proportional to both mass and distance," when it comes to actual construction, we will find an optimal solution where perhaps the total mass increases, but the cost decreases.

"Space Railway"

In some more long-term visions, once the space elevator on Earth is built, it will be integrated with a lunar elevator. People will be able to reach the Moon conveniently after a few transfers. An article on CNET News described a scene in a future space elevator with a touch of humor: "The elevator door opens, and an elderly elevator operator politely asks the space passengers, ’Which floor in space, please?’"

Could this be the legendary "Space Railway"?