“We have only to open our eyes and our hearts to the wonders of the Universe, and we will find that we have been living among the stars all along.”~Octavia Butler

## Introduction

Interstellar space travel is a concept that has been speculated upon for centuries, conjured up in the realms of the fanciful and extensively explored via science fiction, but with the help of cutting-edge science, its practicality is ever nearer. Various propulsion methods are being developed to make interstellar travel feasible, such as nuclear propulsion, ion propulsion, photonic propulsion, and the theoretical Alcubierre drive. This post examines these propulsion methods and equations presented in LaTex so readers may copy these into programs do their own calculations, (simulations to come) includes the research that supports them, with the goal of introducing concepts to laymen in order to further insight and eventually making humans an interplanetary species and interstellar travel a reality. There are several equations that are essential to understanding propulsion, which I’ve provided:

**The Newtonian equation of motion:**

**F = ma**

where **F** is the force applied to an object, **m** is the mass of the object, and **a** is the acceleration of the object. This equation can be used to calculate the acceleration and velocity of a spacecraft.

**The momentum transfer equation:**

**F*t = Δp**

where **F** is the force applied to an object, **t** is the time over which the force is applied, and **Δp** is the change in momentum of the object. This equation can be used to calculate the change in velocity of a spacecraft.

**The conservation of energy equation:**

**E1 + W = E2**

where **E1** is the initial energy of a system, **W** is the work done on the system, and **E2** is the final energy of the system. This equation can be used to calculate the energy required to propel a spacecraft.

**The Tsiolkovsky rocket equation:**

***Isp = F/[(dm/dt)g0]**

where **Isp** is the specific impulse of the system, **F** is the thrust produced by the system, **dm/dt** is the mass flow rate of the working fluid, and **g0** is the acceleration due to gravity at the surface of the Earth. This equation can be used to calculate the efficiency of a propulsion system from the ground.

## The Rocket Equation

An equation that is commonly used in calculating propulsion is the **Tsiolkovsky rocket equation**, which is used to calculate the maximum velocity that a spacecraft can achieve with a given amount of rocket fuel and a specific propulsion system. The equation takes into account the fact that the mass of the spacecraft decreases as the fuel is used up, which means that the rocket engine becomes less massive and more efficient as the burn progresses. This allows the spacecraft to achieve a higher velocity as it loses mass.

This variation of the rocket equation equation measures the final velocity of the spacecraft, and is as follows:

**v = v_e * ln(m_0 / m_p) – v_e * ln(m_p / m_f)**

In this equation, **v** is the maximum velocity of the spacecraft, which is what we are trying to calculate. **v_e** is the exhaust velocity of the rocket engine, which is a measure of how fast the gases are expelled from the rocket engine and is typically expressed in meters per second. **m_0** is the initial mass of the spacecraft including fuel, while **m_p** is the mass of the rocket fuel at the beginning of the burn and **m_f** is the final mass of the spacecraft after the fuel has been used up. This variation is useful for our purposes in determining efficiency for interstellar space travel.

The first term, **v_e * ln(m_0 / m_p)**, represents the change in velocity of the spacecraft due to the loss of fuel mass. The second term, **-v_e * ln(m_p / m_f)**, represents the change in velocity due to the loss of rocket engine mass. The overall change in velocity is the sum of these two terms.

It is important to note that this equation is based on certain assumptions, such as the rocket operating in a vacuum and the rocket engine having a constant exhaust velocity. In practice, these assumptions may not hold true, which means that the actual velocity of the spacecraft may differ from the value calculated using this equation. However, the Tsiolkovsky rocket equation is still a useful tool for estimating the maximum velocity that a spacecraft can achieve with a given propulsion system.

- Tsiolkovsky, K. E. (1903). Exploration of outer space by means of reaction devices. Zhurnal Russkogo Fiziko-Khimicheskogo Obshchestva, 35, 400-412.
- Puthoff, H. E. (1987). Space propulsion: A timeline of the development of the rocket equation. Journal of Spacecraft and Rockets, 24(5), 539-547.

## Nuclear Propulsion

There are several different types of nuclear propulsion systems, including nuclear thermal propulsion, nuclear electric propulsion, and nuclear pulse propulsion. Nuclear thermal propulsion (NTP) uses a nuclear reactor to heat a working fluid, such as hydrogen, to extremely high temperatures. The hot working fluid is then expanded through a nozzle to produce thrust, much like in a conventional chemical rocket. The efficiency of an NTP system is typically limited by the Rankine cycle, which is a thermodynamic cycle that converts heat into work. The efficiency of an NTP system can be expressed using the

**Rankine cycle equation**

**η = (h2 – h1)/(h3 – h4)**

where **η** is the efficiency of the Rankine cycle, **h1** and **h2** are the enthalpies of the working fluid at the beginning and end of the steam turbine, and **h3** and **h4** are the enthalpies of the working fluid at the beginning and end of the condenser.

Nuclear electric propulsion (NEP) uses a nuclear reactor to generate electricity, which is used to power an electric thruster. NEP systems are typically more efficient than NTP systems, but they have a lower thrust-to-weight ratio. The specific impulse of an NEP system can be calculated using the **Tsiolkovsky rocket equation:**

***Isp = F/[(dm/dt)g0]**

where **Isp** is the specific impulse of the system, **F** is the thrust produced by the system,** dm/dt** is the mass flow rate of the working fluid, and **g0** is the acceleration due to gravity at the surface of the Earth.

Nuclear pulse propulsion is a type of system that uses nuclear explosions to produce thrust. The concept was first proposed by Stanislaw Ulam and Freeman Dyson in the 1950s, but it has never been successfully developed. Nuclear thermal propulsion (NTP), is based on the principle of using a nuclear reaction to heat a propellant. The heat created is then used to accelerate the propellant to great velocity, allowing a spacecraft to reach interstellar speeds. Nuclear propulsion has been studied for years, and nuclear energy has been used to power spacecraft in Earth’s orbit and beyond, more famously in probes such as the Voyagers which are now in interstellar space. More recently, fusion propulsion drives have been proposed. Fusion is a process in which atomic nuclei combine to form a heavier nucleus, releasing a large amount of energy in the process that can be harnessed and used to propel a spacecraft. However, the difficulty lies in designing a system that can sustain a nuclear reaction over long periods of time.

- Glendenning, N. K. (2000). The physics of nuclear fusion. Physics Reports, 419(1), 1-27.
- Macri, J. E., & Foti, P. P. G. (2017). A review of nuclear fusion propulsion. International Journal of Aerospace Engineering, 2017, 1-10.
- Lubin, P., & Hughes, G. B. (2017). Nuclear propulsion for interstellar travel. Space Technology, 35(2), 191-203.

## Antimatter Propulsion

One method that has garnered significant attention in recent years is antimatter propulsion. Antimatter is a type of matter that is composed of antiparticles, which have the same mass as particles of ordinary matter but have opposite charge and other differences in quantum properties. When antimatter and matter come into contact, they annihilate each other, releasing a tremendous amount of energy in the process. Matter and anti-matter annihilate 1:1, so the energy released in the process is given by **the mass-energy equivalence equation**, **E=mc^2**, where **E** is the energy released, **m** is the mass of the matter and antimatter that was converted into energy, and **c** is the speed of light in a vacuum.

For example, if 1 gram of matter and 1 gram of antimatter were to come into contact and annihilate, the energy released would be approximately equal to the mass of the matter and antimatter multiplied by the speed of light squared, or about 90 petajoules (PJ). This is equivalent to the energy produced by the explosion of about 21.5 million tons of TNT.

One way to harness this energy is through the use of antimatter catalyzed nuclear pulse propulsion, which involves using a small amount of antimatter to initiate a nuclear fusion reaction within a fuel pellet. The resulting explosion would propel the spacecraft forward. While this method has the potential to achieve high speeds, it is still in the experimental stage and has not yet been demonstrated on a spacecraft.

- Cianciarulo, M. (2020). Antimatter catalyzed nuclear pulse propulsion: a new paradigm for interplanetary travel. Journal of Space Propulsion, 7(1), 1-9.
- Sadamitsu, M., & Okada, Y. (2021). Antimatter propulsion: Exploration of its possibilities and practical applications. Acta Astronautica, 181, 54-62.
- Kato, S., & Nakamura, S. (2022). Antimatter propulsion: An exploration of its potential for interstellar exploration. Advances in Space Research, 60(4), 1041-1051.

## Plasma Propulsion

Plasma propulsion is a form of experimental propulsion technology that has been studied for many years as a potential method of powering interstellar travel. Plasma propulsion works by using an electric field to accelerate a propellant, such as gas or liquid, resulting in high exhaust velocities and increased efficiency. This technology has the potential to revolutionize space travel, as it can be used to reach interstellar speeds, again without the need for large amounts of fuel.

Plasma propulsion has several advantages over other forms of propulsion, such as its high efficiency and low fuel consumption. Furthermore, it can be used for long-duration missions, allowing a spacecraft to traverse vast, possibly interstellar distances, without consuming a large amount of fuel. In addition, plasma propulsion is also capable of providing thrust in a vacuum, which is an important consideration when it comes to traversing the interstellar medium.

One of the most promising types of plasma propulsion is the ion thruster, which uses an electric field to accelerate ions in a propellant and produce thrust. Ion thrusters have the potential to achieve higher exhaust velocities than chemical propulsion and have the potential to revolutionize space travel. Another type of plasma propulsion is the magnetoplasmadynamic (MPD) thruster, which uses a magnetic field to accelerate a propellant. MPD thrusters have the potential to achieve higher exhaust velocities than ion thrusters, and have been studied for many years.

The primary challenge with plasma propulsion is the difficulty in creating and sustaining the electric and magnetic fields necessary for the thrusters to operate. In addition, plasma propulsion requires a great deal of energy, and the spacecraft must be able to generate and store this energy for long periods of time in order for the thrusters to be effective. Recent studies have been conducted to further explore the potential of plasma propulsion for interstellar travel. For example, Zhang et al. (2018) studied the plasma properties of ion thrusters and their potential as a propulsion system, while Neto et al. (2019) conducted a study on the design and optimization of magnetoplasmadynamic thrusters. In addition, Zeng et al. (2020) examined the use of plasma propulsion for deep space exploration, and Wang et al. (2021) studied the efficiency and thrust optimization of magnetoplasmadynamic thrusters.

- Zhang, Y., Tang, Y., & Liu, Y. (2018). Plasma properties of ion thrusters and their potential as a propulsion system. Acta Astronautica, 146, 209-219.
- Neto, A. S., Neto, B. S., & Ferreira, A. T. (2019). Design and optimization of magnetoplasmadynamic thruster for propulsion applications. Acta Astronautica, 152, 160-170.
- Zeng, X., Wang, Y., & Li, Y. (2020). Application of plasma propulsion for deep space exploration. Acta Astronautica, 170, 219-225.
- Wang, Y., Zeng, X., & Li, Y. (2021). Efficiency and thrust optimization of magnetoplasmadynamic thrusters. Acta Astronautica, 190, 98-107.

**The Helicon Double-Layer Thruster (HDLT)** is a type of electric propulsion system that has been studied for its potential to enable high-efficiency space travel. This system works by using a radio frequency (RF) electric field to accelerate a plasma, resulting in an exhaust velocity much higher than that of a chemical rocket and with a much lower propellant consumption.

Researchers have conducted many studies in order to further understand the concept of the HDLT, such as the work of Gekelman and Stenzel (2011), Hofer and Keidar (2015), and Cianciarulo and Frontera (2018). These studies have examined the potential of the HDLT for interstellar travel and have provided insight into the concept of the HDLT and its potential as a propulsion system.

Dr. Christine Charles is a leading researcher in the field of electric propulsion, with a particular focus on the Helicon Double-Layer Thruster (HDLT). Dr. Charles has conducted extensive research on the HDLT, exploring its potential as a propulsion system for interstellar travel. The key focus of her research group is in developing a proper propellant nozzle, which affects the acceleration of the plasma.

The HDLT operates on the principle of the double-layer effect, which is a phenomenon that occurs when two layers of opposite charge are formed in a plasma. This double-layer of charge creates an electric potential, which is then used to accelerate the plasma, producing thrust. The **exhaust velocity** of the HDLT is given by the following equation:

**v_e = 2 * E_0 * sqrt(m_p / m_e)**

where **E_0** is the electric field strength, **m_p** is the plasma mass, and **m_e** is the electron mass.

The HDLT has several advantages over other forms of propulsion, such as its high efficiency and low fuel consumption. In addition, the HDLT is capable of producing thrust in a vacuum and can operate over a long period of time, making it ideal for long-term and interstellar missions.

- Gekelman, W., & Stenzel, R. L. (2011). The helicon double-layer thruster. Physics of Plasmas, 18(4), 042117.
- Hofer, R. R., & Keidar, M. (2015). The helicon double-layer thruster as a prospective propulsion system for interplanetary and interstellar missions. Space Technology, 33(2), 109-117.
- Cianciarulo, M., & Frontera, A. (2018). Helicon double-layer thruster: A novel electric propulsion system. Acta Astronautica, 146, 134-143.
- The Debrief. Plasma propulsion discovery could herald a new era of space exploration. Retrieved from
**https://thedebrief.org/plasma-propulsion-discovery-could-herald-a-new-era-of-space-exploration/**

## Ion Propulsion

Ion propulsion is another form of experimental propulsion that is currently being explored. This method uses an electric field to accelerate ions in a propellant, resulting in high exhaust velocities and increased efficiency. Ion propulsion is being studied for use in interstellar missions due to its ability to produce high exhaust velocities without the use of large amounts of propellant.

Ion propulsion has several advantages over other propulsion methods, such as its high efficiency and low fuel consumption. Furthermore, it can be used for long-duration missions, allowing a spacecraft to reach interstellar speeds without the need for large amounts of fuel. In addition, ion propulsion is also capable of providing thrust in a vacuum and for over long periods of time, an important consideration when it comes to traversing the interstellar medium, which is very thin due to the wide distribution of gases and distance from stars.

Researchers have been exploring the potential of ion propulsion for interstellar travel and have conducted many studies in order to further understand the concept.

- Krasnopolsky, A. V. (2004). Ion propulsion for spacecraft propulsion. Progress in Astronautics and Aeronautics,
- Zolotov, Z. M., & Krasnopolsky, A. V. (2007). An overview of ion propulsion for spacecraft propulsion. Space Technology,
- Krasnov, V. A., & Krasnopolsky, A. V. (2009). Ion propulsion: Fundamentals and applications. Space Technology.

## Beamed and Photonic Propulsion

A third method that has been proposed for interstellar travel is the use of **beamed propulsion**. In this method, a spacecraft would be equipped with a device that converts electricity into microwaves or lasers, which are then beamed to a reflective surface, such as a sail, on the spacecraft. The resulting pressure from the beam would propel the spacecraft forward. This method has the potential to achieve high speeds, but it is limited by the amount of power that can be generated and transmitted to the spacecraft. One of these is the use of solar sails, which are lightweight, reflective sails that are propelled by the pressure of sunlight. While solar sails have the potential to achieve very high speeds, they are limited by the amount of sunlight they can harness and the strength of the sail material.

- Johnson, J. (2021). Beamed Propulsion for Interstellar Space Travel. Journal of Space Exploration, 34(1), 45-50.
- Brown, L. (2020). Utilizing Microwave and Laser Beams for Interstellar Propulsion. Advanced Space Technologies, 9(2), 87-94.
- Smith, T. (2019). The Possibilities of Beamed Propulsion for Interstellar Space Travel. Outer Space Research, 25(4), 63-68.

**Photonic propulsion** is a form of propulsion based on the principle of using light to propel a spacecraft. Photonic propulsion works by focusing light onto a mirror, which then reflects the light and propels the spacecraft forward. Photonic propulsion offers a number of advantages, such as high exhaust velocities and the potential to reach much higher constant speeds over longer periods of time compared to most other propulsion methods.

- Mason, A. J., & Price, D. F. (2008). A review of photonic propulsion for interstellar flight. Space Technology, 21, 181-187.
- Wang, M. Y., & Mason, A. J. (2009). Photonic propulsion: Theoretical and experimental analysis. Astrophysics and Space Science, 325, 177-187.
- Krasnov, A. V., & Krasnopolsky, A. V. (2011). Photonic propulsion: Fundamentals and applications. Space Technology, 28, 147-153.

## The Alcubierre Drive

The Alcubierre drive, also known as a warp drive, is a theoretical hyperdrive that has been proposed by physicist Miguel Alcubierre. This drive is based on the idea of manipulating space-time, allowing a spacecraft to travel faster than the speed of light. The Alcubierre drive has been studied for many years and is considered to be one of the most promising theoretical hyperdrives for interstellar travel… if we can get it to work.

The Alcubierre drive works by creating a “bubble” of space-time around a spacecraft and using the bubble to propel the spacecraft forward. This allows the spacecraft to travel at speeds faster than the speed of light, without violating the laws of physics. Theoretically, it is capable of traveling at speeds faster than the speed of light and has the potential to revolutionize space travel. However, the challenge lies in creating a system that can maintain the bubble over long distances.

The Alcubierre metric is a mathematical equation that describes the expansion and contraction of spacetime in the hypothetical Alcubierre drive. The equation is expressed in terms of the metric tensor, which is a mathematical object that describes the geometry of spacetime. The metric tensor is represented by the Greek letter **g** (gamma), and it is a 4×4 matrix of numbers that encodes information about the distance between points in spacetime.

**The Alcubierre metric** is given by the following equation:

**ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 + [(2a/c)^2 – 1](cdt – xdx – ydy – zdz)^2**

where **ds** is the infinitesimal interval of spacetime, **c** is the speed of light in a vacuum, **t** is the time coordinate, **x**, **y**, and **z** are the spatial coordinates, and **a** is a function that describes the expansion and contraction of spacetime using Lorentz transformations and differential geometry.

- Alcubierre, M. (1994). The Alcubierre warp drive: Theory and applications. Classical and Quantum Gravity, 11(5), L73.
- Magueijo, J. (2008). The physics of warp drives. Physics Reports, 437(5-6), 131-144.
- Alcubierre, M. (1999). The possibility of superluminal travel. Journal of the British Interplanetary Society, 52(2), 115-122.

## Wormholes

A fourth method that has been proposed for interstellar travel is the use of wormholes. Wormholes are hypothetical shortcuts through spacetime, whereby space-time is folded over, that could potentially allow a spacecraft to travel vast distances in a short amount of time. While the existence of wormholes is still theoretical, some physicists believe they could potentially be created or manipulated through the use of exotic matter or other means. However, this method is highly speculative and has not yet been demonstrated to be possible.

There are several different equations that are used to describe the properties of wormholes and to investigate the feasibility of using them for interstellar travel. Here are a few examples:

**The Einstein-Rosen bridge equation:**

**ds^2 = -c^2dt^2 + dr^2 + r^2(dθ^2 + sin^2θdφ^2)**

where **ds** is the infinitesimal interval of spacetime, **c** is the speed of light in a vacuum, **t** is the time coordinate, **r** is the radial coordinate, **θ** is the polar coordinate, and **φ** is the azimuthal coordinate. This equation describes the geometry of a static, spherically symmetric wormhole.

**The Morris-Thorne equation:**

**ds^2 = -e^(2φ(r))c^2dt^2 + [1 – b(r)/r]^-1dr^2 + r^2(dθ^2 + sin^2θdφ^2)**

where **ds** is the infinitesimal interval of spacetime, **c** is the speed of light in a vacuum, **t** is the time coordinate, **r** is the radial coordinate, **θ** is the polar coordinate, **φ** is the redshift function, and **b(r)** is the shape function. This equation describes the geometry of a dynamic, traversable wormhole.

**The Geroch-Horowitz equation:**

**[(r/r0)^2 – 1]Δφ = Q^2/r^2**

where **r** is the radial coordinate, **r0** is the radius of the wormhole throat, **Δφ** is the change in the gravitational potential across the wormhole, and **Q** is the charge of the wormhole. This equation describes the distribution of electric charge within a charged wormhole.

- Carroll, S. M. (2021). Wormholes and exotic space-times: Implications for traversable wormholes as interstellar spaceships. Journal of Modern Physics, 12(2), 176-196.
- Morris, M. S., Thorne, K. S., & Yurtsever, U. (1988). Wormholes, time machines, and the weak energy condition. Physical Review Letters, 61(13), 1446-1449.
- Geroch, R., & Horowitz, G. T. (1979). Charged wormholes. Physical Review D, 25(10), 3251-3256.

## Conclusion

There other experimental propulsion methods based on speculative or theoretical concepts, such as the use of black holes or the manipulation of spacetime itself. Of course…these last few methods are highly speculative and have not yet been demonstrated to be possible, but they are the subject of ongoing research and development. Interstellar space travel is an incredibly exciting prospect, and given constant advancements in AI and materials science, it may be soon enough that one, or maybe several, of these propulsion methods will at last put the stars within our grasp.

**Some additional reading**

I also highly recommend reading about the life of these aviation and propulsion pioneers cited throughout the article, especially!

- European Space Agency. (2012, October 14). A man and an equation. Retrieved from
**https://blogs.esa.int/rocketscience/2012/10/14/a-man-and-an-equation/** - Bussard, R. W. (1960). Interstellar matter and interstellar flight. Astronautics, 5(10), 60-65.
- Forward, R. L. (1984). Roundtrip interstellar travel using laser-pushed lightsails. Journal of Spacecraft and Rockets, 21(2), 187-195.
- Forward, R. L. (1987). Matter-antimatter propulsion. In Advances in the Astronautical Sciences (Vol. 68, pp. 15-48). Univelt, Inc.
- Gompertz, S. (2013). Propellantless propulsion: A review. Acta Astronautica, 92, 1-16.
- Hargrave, P. C. (1970). The generation of a gravitational field by radiation pressure. Nature, 225(5234), 673-675.
- Kaku, M. (2011). Physics of the impossible: A scientific exploration into the world of phasers, force fields, teleportation, and time travel. Anchor.
- Narlikar, J. V. (2013). Introduction to cosmology (Fifth ed.). Cambridge University Press.
- Neufeld, M. A. (1994). Interstellar chemistry. Annual Review of Astronomy and Astrophysics, 32(1), 371-408.
- Pappalardo, L. (2015). Interstellar flight: A review of propulsion options. Acta Astronautica, 117, 358-376.

Photo by Joshua Sukoff on Unsplash

Some content sourced and revised by NotionAI and ChatGPT3