A Holistic Assessment of Electromagnetic Acceleration for Use in Lunar Space Launch

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Author: Aryan Prasad
Peer Reviewer: Hyunjin Christina Lee
Professional Reviewer: Josh Bilak

Abstract

Rockets powered by chemical reactions are the only mechanism currently available for accessing space. The rocket equation dictates that the vast majority of a rocket’s mass must be fuel instead of payload, making them inherently inefficient machines. A better mechanism that is based on the ground could rapidly commercialize space and open new frontiers for humanity. A railgun launcher is one such solution. A capsule would be loaded between two parallel rails, forming a complete circuit through which current would flow. The current, combined with an external magnetic field, would produce a force that accelerates the cargo outward. This study assesses the feasibility of developing such a mechanism from both a physical and engineering perspective. The first major section develops the theory of the launcher, using fundamental equations to find the forces and acceleration within the mechanism. It then analyzes the energy and power requirements before concluding with a cost estimate based on raw materials. Following the theoretical section, numbers were entered into the equations based on previous calculations and reasonable estimates to find actual values. While the total energy required is not mammoth, discharging all of it within the short launch window would require the use of a large-scale capacitor bank. The cost estimate amounted to 500 million dollars. The total cost would be much higher due to the logistics of megaprojects and other factors that were not considered, but the launcher’s long-term viability would serve as a strong incentive for its development. This study concludes by discussing its limitations and options for future researchers to conduct deeper analyses.

A Holistic Assessment of Electromagnetic Acceleration for Use in Lunar Space Launch

This study aims to assess the use of a launcher powered by electromagnetic acceleration for sending payload from the Lunar surface to Lunar orbit. The focus is on analyzing how that mechanism would function, what would be required to construct it, and whether it can serve as a potential alternative to chemical rocketry. The broad scope of this study necessitates conducting a high-level analysis. As such, the goal is not to find a definitive answer, but rather to explore this possibility and find avenues for future more-detailed studies.

Conventional rockets rely on the thrust generated by chemical reactions of propellants. Their motion is governed by the rocket equation, which dictates an exponential relationship between the velocity required to reach a given orbital height and the fuel requirement as a fraction of the total mass of the rocket. To achieve Low Earth Orbit, a rocket’s mass needs to be approximately 80-90% fuel. After accounting for the rocket’s structure, the remaining mass of the actual payload – satellites, scientific instruments, humans – can often comprise as little as 2% of the total mass (Pettit 2012). Adding payload mass requires more fuel. More fuel adds more mass, creating a positive feedback loop that is difficult to break. In addition to this physical inefficiency, rockets have no permanent infrastructure. They must be rebuilt and refueled for each launch attempt. Combined, these factors make rocketry expensive and limit our access to space. 

One proposal for a cheaper, more efficient mechanism is to replace chemical propulsion with electromagnetic acceleration. This mechanism would take the form of a railgun, which consists of two parallel rails surrounded in a magnetic field. The payload would be loaded in between the rails, forming a complete circuit. The flow of current and the magnetic field would generate a Lorentz force on the capsule, accelerating it up and outwards to orbital velocities. The launcher would remain on the ground and the infrastructure would only need to be installed once. The only input would be electrical energy to power the launcher and generate the magnetic field and routine maintenance. 

The analysis will begin with developing the equations relevant to analyzing the launcher and its components starting from fundamental physical principles. Following that, numerical values will be generated from those equations and assessed. To make a determination on the mechanism’s feasibility, I will make an assessment from multiple angles: the cost estimate based on raw materials, values for the voltage and magnetic field, and other major factors that could hinder the operation or development of the electromagnetic launcher. Some key numerical values will be compared to those found in existing technology.

Due to the difficulty of analyzing such a complex system, many intricacies, such as whether the cargo could survive the acceleration experienced in the railgun, will be left to future study. Some of the key findings in this exploratory study can serve to provide different pathways for future researchers, which is discussed near the conclusion. The primary reason for considering the launch mechanism for use on the Moon instead of Earth is because the Moon poses fewer natural barriers to operation. Its lack of an atmosphere prevents air resistance from affecting the launch and it has a much weaker gravitational field than Earth. Staging the mechanism on the Moon increases the favorability of it being a feasible alternative to conventional rocketry and simplifies the analysis. Furthermore, as humanity seeks to build long-term outposts on the Moon, a mechanism to launch satellites and other cargo to Lunar orbit will be of vital importance. This analysis lays the rudimentary groundwork for that future.

Theoretical Analysis

This study is fundamentally theoretical. The equations that drive the analysis include physical laws and derivations based on those laws. Unless otherwise noted, all fundamental equations come from (Knight 2017). In the next section, numerical values will be found from the equations described here.

Gravity and Orbital Velocity

Finding the required velocity for the cargo to achieve orbit is necessary to consider the forces within the launcher. The first step involves Newton’s Law of Universal Gravitation, which gives the force due to gravity from two objects. In this case, those bodies are the Moon and the cargo.

(Equation 1)

where Fg is the force due to gravity, G is the Universal Gravitational Constant, M and m are the masses of the Moon and cargo, respectively, and r is the separation between their centers of mass. When the cargo is orbiting the Moon in a circular path, the force of gravity on it is related to its centripetal acceleration by Newton’s Second Law.

(Equation 2)

Centripetal acceleration is defined in (3).

(Equation 3)

where v is the tangential velocity of the orbiting body. Equating (2) and (3) yields the equation:

(Equation 4)

Equating (1) and (4) results in (5), which can be solved algebraically for the orbital velocity of the cargo.

(Equation 5)

Equation (6) gives an expression for the velocity of the cargo in terms of the Moon’s mass and its orbital radius.

(Equation 6)

Projectile Analysis

Once the orbital velocity for the cargo is found, an exit velocity from the railgun tube can be calculated. The exit velocity is necessary to analyze the forces needed in the launcher. Once the cargo leaves the acceleration tube, the only force acting upon it is gravity, causing a radially-inward acceleration. As the cargo ascends to orbital heights, the gravitational force decreases as per (1), causing a diminished acceleration compared to that at surface level. For this analysis, I chose to use the Lunar Reconnaissance Orbiter – a NASA satellite that surveys the Lunar surface – as a test cargo. It is a typical satellite carrying scientific instruments and it’s likely that potential human civilizations would seek to launch similar satellites for other uses. The LRO orbits 50,000 meters above the Lunar surface (Dunbar 2011). It is only a small fraction of the Moon’s 1.74*106 meter radius (Knight 2017). 

Per the inverse square relationship in (1), the gravitational force experienced at the satellite at orbital height would be approximately 94% the magnitude of the force at the surface. The acceleration experienced, which is force divided by mass, would decrease by the same factor. The decrease in acceleration is small enough to justify treating the gravitational acceleration the cargo experiences at the Lunar surface as a constant throughout its ascent, which simplifies the analysis by allowing use of the kinematic equations for constant acceleration. Furthermore, this approximation would result in an overestimate of the necessary exit velocity, which would provide a worst-case scenario for the strength of the forces that the launch mechanism must produce.

Prasad, Figure #1: Diagram of the Launcher
Figure 1: Diagram of the Launcher

Figure 1 depicts the launch mechanism, which is a straight tube of length d, inclined at angle , and has a height h. The projectile also exits the tube with a velocity vector inclined at that same angle and can be separated into horizontal and vertical components.

(Equation 7)
(Equation 8)

Since there is no horizontal acceleration after the capsule leaves the launcher, the final velocity equals the initial velocity. In this case, the horizontal velocity upon exiting the tube needs to be the required orbital velocity.

(Equation 9)

Equation (10) is a constant acceleration kinematic equation for the vertical motion of the cargo.

(Equation 10)

where vy is the final vertical velocity. In this case, it is zero, as that allows the cargo to transition from a parabolic trajectory to a stable circular orbit around the Moon. This orbital insertion would require the use of thrusters or other propellants, which is discussed further in a later section. gM is the acceleration due to gravity on the Moon, and hP is the difference between the orbital height and the height of the top of the launcher. The height of the launcher would necessarily be a small fraction of the total ascent because of the fundamental challenges with building structures that are several kilometers high. Equations (9) and (10) both contain two unknowns: and vE. They can both be solved for vE, resulting in (11) and (12).

(Equation 11)
(Equation 12)

Equations (11) and (12) can be equated, eliminating the exit velocity.

(Equation 13)

Equation (13) can be solved for the angle.

(Equation 14)

can then be substituted into either (11) or (12) to find the exit velocity. With the kinematics of the launch established in these first two subsections, the acceleration and forces within the railgun mechanism needed to reach orbital velocity can be considered.

Force and Acceleration in the Railgun

"Railgun-1" 2005, Figure 2: Force Diagram of
Figure 2: Force Diagram of the Launcher (“Railgun-1” 2005)

As depicted in Figure 2, the railgun consists of two parallel rails. Current flows up through one, across the projectile (known as the “armature”), and down the other rail. The current, in conjunction with an external magnetic field, produces a Lorentz Force on the projectile, described by (15).

(Equation 15)

where FL is the Lorentz Force, I is the current flowing through the circuit, L is the length of the armature, and B is the strength of the magnetic field. By design, the projectile is perpendicular to both rails, while the magnetic field vector points upwards out of the rube. Thus, the orthogonal force vector (cross product) of length and magnetic field points outward, parallel to the rails. With a known direction, (15) can be rewritten in terms of magnitude only.

(Equation 16)

Ohm’s Law relates current to voltage and resistance (V and R respectively)

(Equation 17)

Equation (17) can be substituted into (16), yielding:

(Equation 18)

Resistance is given by (19).

(Equation 19)

where <#RHO> is the resistivity of the wires, x is the distance the projectile has travelled when in the rails, L is the separation between the rails, and A is the cross-sectional area of the wires. The sum in the numerator represents the length of the entire circuit. As the projectile moves forward within the rail, the length of the circuit increases by twice that distance, as the current must travel up and down the rails. The numerator term also includes the separation between the rails at both ends. Equation (19) can be substituted into (18), which, with some minimal simplification, yields (20).

(Equation 20)

The advantage of substituting resistance into (16) is that the Lorentz Force can now be thought of in terms of the distance of the rails and the cross sectional of the wires. To examine the acceleration that will take place in the railgun, the forces acting on the cargo parallel to the rails must be summed.

Prasad, Figure #3: Forces Acting on Cargo
Figure 3: Forces Acting on Cargo

Figure 3 depicts those forces: the Lorentz Force points outward and a component of gravity and friction between the capsule and rails points inwards. The tube is exposed to the vacuum of space, so there is no air resistance. If the reference frame used is inclined at angle <#THETA>, the component of gravity acting against the cargo is:

(Equation 21)

The force of friction is equal to the product of the coefficient of friction between the surfaces and the normal force.

(Equation 22)

The only forces acting in the vertical direction are the vertical component of gravity and the normal force. Since, within this reference frame, this is no vertical acceleration, these forces must be equal.

(Equation 23)

Substituting (23) into (22) yields (24).

(Equation 24)

Equation (25) sums the forces.

(Equation 25)

Equations (20), (21), and (24) can be substituted into (26).

(Equation 26)

Newton’s Second Law states that the acceleration an object experiences is equal to the net force acting upon that object divided by its mass. Applying that law to (23) yields an expression for acceleration.

(Equation 27)

As stated previously, x changes as the projectile moves in the railgun, meaning that acceleration is not constant throughout the launch. The other terms – such as the magnetic field experienced, width of the armature, and resistivity of the wires – are fundamental to the structure of the mechanism and can be ensured to be constant during normal operation. Acceleration is the second derivative of position with respect to time, so (27) can be rewritten as a differential equation.

(Equation 28)

Equation (28) is a complicated second-order differential equation. Attempting to solve for a function of x with respect to time using the computer algebra system Mathematica yielded (29).

(Equation 29)

where c1 and c2 are constants of integration and z is an intermediate variable created solely for the purpose of the integration. c2 is the initial displacement, which is known to be zero. c1 must have the same units as the other terms within the radical, such as VLBA/<#RHO>m (the logarithm of any physical quantity is dimensionless). The units of those physical quantities simplify to meters squared per seconds squared. Once a square root is taken, they become the units of velocity. Thus, c1 can reasonably be assumed as the initial velocity of the capsule: zero. Thus, both constants of integration are discounted from the final equation.

The integral in (30) is, in all likelihood, impossible to evaluate analytically. However, the function of position with respect to time that results from its solution can still be graphed with software such as Desmos. To do so, the value of each constant in the equation needs to be known. This is problematic for the variables V, B, and A. Unlike gM and , they have not been calculated previously in the analysis. Unlike <#RHO> , m, and L, there is no immediate precedent for their values (L would approximately be the width of the capsule in use).

In Desmos, I will collate these three variables into a single term, b, which can be varied as needed. Taking the derivative of (30) with respect to time numerically within Desmos will enable me to find the values of time and distance that result in the needed exit velocity found in the preceding subsection. Dividing the change in velocity by time will give a figure for average acceleration. I will attempt this calculation with several different values of b to determine how the required distance, time, and average acceleration are affected. Ultimately, the value of b that I settle on will be based on balancing the time the capsule spends in the launcher, the total length of the launcher, the acceleration, and the magnitudes of the three variables encapsulated within b.

One peculiarity when graphing (29) for any values of the constants is that the graph is shifted to the left by three. That results in the velocity being approximately zero at -3 seconds, whereas it should be zero at zero seconds. Subtracting three from the upper limit of integration solves this issue without altering the nature of the function, resulting in (30).

(Equation 30)

Once b is known, I will find values for A and V based on precedent from current railgun technology or reasonable estimates. The magnetic field needed is more complex, so its value will be left to be calculated once the others are known. With the dynamics of the launch established, the focus of the analysis can shift to the specifics of the circuit and magnetic field required to power the launch.

Capacitors and Power

The amount of electrical energy required to generate the Lorentz force for the launch would be mammoth. The current flowing through the circuit can be calculated by substituting V and R – calculated with (19) using values found in the previous the previous subsection – into (17). A capacitor would be required to discharge the stored energy to the circuit in the short duration of the launch.

Prasad, Figure 4: Diagram of a Parallel-Plate Capacitor
(Figure 4: Diagram of a Parallel-Plate Capacitor)

A capacitor stores charge on conducting plates that are separated by an insulating material, known as a dielectric. The simplest type of capacitor is a parallel-plate design, shown in Figure 4, but their forms vary immensely to suit different uses. When large amounts of energy are involved, as is the case with this electromagnetic launcher, smaller capacitors are often joined together in a single circuit (known as a “bank”) to act as one large capacitor. The capacitance of individual capacitors in a parallel circuit sum algebraically, so linking them together is a powerful way to increase the total quantity of charge stored (Knight 2017). Due to the diversity of capacitor types and the equal diversity of equations used to analyze different types of capacitors, an analysis based on the specifics of any given system of capacitors would be impractical for this study and miss the broader picture. Instead, using the law of conservation of energy, the total amount of energy that any capacitor system would need to store for the launch can be found. That quantity of energy would be compared to that stored in existing capacitor systems in intensive-use environments to determine whether it would be feasible with contemporary technology.

(Equation 31)

Equation (31) is a statement of conservation of energy for the launcher-capsule-Moon system, with the left side of the equation representing the instant before the launch begins and the right side representing the moment the capsule exits the launch tube. UE is the electric energy stored in the capacitor and UG1 is the potential energy held by the capsule-Moon system when it rests at the Moon’s surface. During the launch, the initial electric and gravitational potential energy is converted into a higher gravitational potential, UG2, and kinetic energy of the capsule, K.

Of course, some of the energy is dissipated due to friction and other nonconservative forces – such as counter-electromotive force, discussed in a later section – acting within the launcher. Finding an expression for the energy lost would require an integration of those forces over the length of the launcher, which would be tedious at best. Instead, (32) gives the energy lost during the launch as a fraction of the initial energy of the system. 

(Equation 32)

The specific fraction would be determined based on the proportion of energy lost in similar systems or estimated as a reasonable, but high, value to ensure that this analysis considers the worst-case scenarios. Equation (32) can be substituted into (31).

(Equation 33)

Equation (33) can be solved for UE and the standard equations for gravitational potential energy – as derived from Newton’s Universal Law of Gravitation – and kinetic energy can be substituted in as follows.

(Equation 34)

The gravitational potential energy of the system as the capsule exits the launcher would be equal to the total distance between the capsule and the center of the Moon: the Moon’s radius plus the height of the launcher. The launcher’s height can be found by taking the product of the total length of the launcher as found in the previous subsection and the sine off <#THETA> found in the projectile analysis subsection.

In addition to comparing the total energy found in (34) to other capacitor systems, it can be used to find the power output required to charge the capacitors before launch. Given the scale of the launcher, a large energy source will be needed to charge the capacitors. Solar energy is the best solution, given that the Moon receives continuous sunlight for two-week intervals, and, unlike Earth, has no obstructing atmosphere. The greatest advantage of solar is that no materials would need to be extracted or sent to the Moon to generate electricity. The power that the solar array would need to produce would be the total energy stored in the capacitors divided by the time taken to charge them.

(Equation 35)

The time to charge is variable and depends on how frequently the system needs to be ready to launch new cargo. Once a time is settled upon, the power required can be compared to the output of solar panels under optimal conditions to determine how many panels are needed to charge the system in the given time.

Generating the Magnetic Field

To generate the Lorentz force that accelerates the cargo, the entire launcher must be encapsulated in an external magnetic field that is directed upwards. Furthermore, the magnetic field must be roughly constant throughout the length of the launcher. A constant current through a wire will generate a constant magnetic field with a direction given by the right-hand rule.

“Coil-1” n.d., Figure 5: Magnetic Field Produced by a Coil of Current-Carrying Wire
(Figure 5: Magnetic Field Produced by a Coil of Current-Carrying Wire (“Coil-1” n.d.))

Figure 5 shows how a current running through a coil of wire produces a magnetic field directed along the lateral axis of the coil. The magnetic field of approximate strength B is only felt in the interior of the loop. To cover the entire launcher, a series of these coils would need to be lined up together underneath the launcher with the orientation shown in Figure 5.  

The Biot-Savart Law describes the magnitude of a magnetic field generated by a current in a specific configuration of wires. Equation (36) is a simplified version of that Law that gives the magnetic field experienced along the central axis of symmetry of a coil of wire.

(Equation 36)

where <#MU>0 is the permeability of free space, RC is the radius of the loop, I is the current through the loop, z is the distance between the center of the coil and the point where B is felt, and N is the number of loops in the coil. The coil’s radius and number of turns in the coil could be found based on other electromagnet systems or reasonable assumptions. The distance z would be based on the height of the coil and the size of the launcher, which can be reasonably approximated. That leaves the current as the major unknown, and, as such, (36) can be rearranged to solve for current.

(Equation 37)

The voltage required can be found from (17) using the current from (37) and the resistance calculated analogously to (19). The material of the wires will be the same, so the resistivity won’t change. The length will be the product of the circumference of the coil and the number of loops: 2<#PI>RCN. The cross sectional-area can take upon the same value as that for the wires of the railgun.

The power required for a single coil can be found by the current and the voltage. The total energy released can then be determined by multiplying the power by the time taken to launch.

(Equation 38)

The energy needed for the entire coil system would be the result in (38) multiplied by the total number of coils, which would just be the total length of the launcher divided by the diameter of a single coil. In an analogous fashion to the last subsection, the energy requirement can be compared to that of existing capacitor systems to assess feasibility and cost and determine the number of solar panels needed to charge the magnetic field’s capacitor system. It is worth noting that while an individual capacitor does not create a constant current, a capacitor bank can be designed such that the total current produced is constant even while the currents from individual capacitors fluctuate (Knight 2017).

The materials costs of the wires that generate the magnetic field can be found in addition to the costs of the capacitors and solar panels. The cost would be based on the volume of the coils. The volume is multiplied by the density of the relevant material and then the cost per kilogram – found based on common prices for the materials – to get an estimate for the total raw material cost. That relationship is described by the skeletal equation below.

(Equation 39)

The volume of a single coil, given by (40), would be the product of the circumference of one loop, the cross-sectional area of the wire, and the number of loops in the coil. Naturally, the total volume will be the volume of a single coil multiplied by the number of coils.

(Equation 40)

where Rw is the radius of the wire itself. As stated above, it can be the same as the area for the railgun wire for simplicity. With that set of cost estimates complete, the final major remaining cost are the main wires within the launch mechanism.

Final Cost Assessment

The wire that comprises the bulk of the railgun circuit is essentially two large cylinders. The total volume is twice the volume of a single wire, which would be the cross-sectional area multiplied by the length of the railgun. Applying (39) using the material of the wire gives its cost. Adding all the costs found in the previous few subsections provides a rough estimate for the cost of the electromagnetic launch mechanism based solely on the costs of the key raw materials. This estimate is only partial and excludes many of the major costs for the entire project, which are discussed later. This estimate can be divided by the mass of the payload used to get a cost per kilogram for the system, which can be compared to competitively-priced chemical rocketry. With the collected information about the launcher’s physical properties and cost, I can judge whether it would be viable to construct and would be a feasible alternative to chemical rocketry.

Numerical Results

Here, the equations and procedures developed in the previous section are used to generate numerical values that will lead to the assessment of viability in the next section. Other than universal constants, all numerical values are summarized in table form and their origin is explained, whether it be from an equation or an estimate based on precedent in current technology or a reasonable assumption. The subheadings from the previous section are reused to keep the data organized. All values are placed in the most relevant section they appeared in first. Since there are no measured values, significant figure rules are disregarded, and all decimal values are rounded to the hundredths place unless more is needed for a specific value.

Gravity and Orbital Velocity

(Table 1: Calculations for Gravity and Orbital Velocity)

The orbital radius used for (6) is the distance between the centers of mass of the two gravitationally bound bodies. That distance is the sum of the radius of the Moon and the orbital height of the Lunar Reconnaissance Orbiter.

Projectile Analysis

(Table 2: Calculations for Projectile Analysis)

The change in the projectile’s height from the moment it leaves the launch mechanism to the moment it enters orbit is approximately its orbital height above the Lunar surface: 50,000 meters. Since the launcher is inclined at a gentle angle, the launcher would not be very tall in comparison to the orbital height even if it were several kilometers long. As such, the launcher’s height would not have a significant impact on the exit velocity and all following numerical results. If the height of the launcher were taken into consideration, the exit velocity needed would be lower. Ignoring its height provides another worst-case scenario for the launcher.

Force and Acceleration in the Railgun

(Table 3: Important Values for Railgun Acceleration)

Copper’s durability and high conductivity make it a natural choice for the wiring. The mass of the payload consists of the mass of the LRO satellite and the capsule that it needs to be contained in; the former has a mass of around 1900 kilograms while the latter – a modern capsule that is heavily used – has a mass around 4200 kilograms. The armature length is based upon the width of the capsule, which must fit between the rails. The coefficient of friction was taken from a study of the friction in a model railgun by Zhu and Li. They found that – due to a variety of factors involved in a railgun launch, such as heating – the coefficient of friction between the armature and the rails can vary drastically, reaching a maximum of approximately .9 towards the end of the launch. This study uses that maximum value to find an upper-bound on the capabilities of the launcher. If it can handle that large coefficient of friction applied over the entire launch, it will handle the actual lower average coefficient of friction.

Prasad, Figure 6: Graph of Equation (30) in Desmos for a selected value of VBA
(Figure 6: Graph of Equation (30) in Desmos for a selected value of VBA)

All the constants in (30) are now known with the exceptions of V, B, and A. It was graphed in Desmos to find the lengths and times where the instantaneous rate of change equaled the exit velocity found above. That was done for several different values of VBA and the results are collated in Table 4.

(Table 4: Values from the Solved Differential Equation)

The term b encapsulates multiple physical aspects of the launcher that can be varied to fit the needs of the mechanism. Interesting to note is that for values of b less than 7, the capsule never obtains the necessary launch velocity. Instead, the acceleration decreases in magnitude and eventually becomes negative when the theoretical length of the launcher is in the hundreds of thousands of meters. From a physical perspective, this would indicate that the Lorentz force – which is intrinsically tied to V and B – becomes weaker than the forces of friction and gravity. Conversely, for large values of b, the launcher becomes too powerful, as demonstrated by the monstrously-large accelerations and comparatively tiny lengths for b=20 and b=30. Any larger b and the launch time becomes negative, indicating some physical abnormality that should be avoided.

At the low-end of the acceptable range of b values, the function is extremely sensitive to changes in b. The required distance decreases by 85% after increasing b from 8 to 10 and the time taken to launch decreases by a similar amount. For larger b, the drops in length and time are comparatively much smaller in both absolute and relative terms. The decision on which value of b to adopt depends on the balance we want to achieve between a short launcher length – minimizing material costs and structural engineering challenges – and a reasonable time and modest acceleration. Ultimately, the decision made here is subjective and numerous arguments could be made for or against any particular value of b.  I decided to continue this analysis with b=10. The launcher is almost two kilometers in length and several hundred meters tall, which would be a monumental obstacle to its construction. Still, it is much shorter than the length for  b=8, which would be akin to hoisting a particle collider into the sky. The launch takes over a full second and the resultant acceleration is much more favorable than that for higher values of b.

(Table 5: Structural Values of Launcher)

The radius of the wires in the railgun was taken to be .1 meters, and the cross-sectional area follows from the formula for the area of a circle. This is scaled up from the .02-meter radius that Zhu and Li used for their model railgun. This larger mechanism ought to have larger wires but making them much larger would pose a difficult engineering challenge and require significantly more material. 

A typical military railgun that accelerates a projectile to a speed of several kilometers per second within one second has a voltage of over 1000 volts (Starr and Youngquist 2021). On the lower-end, one railgun in a study at the Naval Postgraduate School study was powered by a capacitor system of 330 volts (Hartke 1997), while another study at the same school modeled a railgun with at least 10,000 volts (Warnock 2003). Given this variation in values, a mid-range voltage of 2000 was chosen, although further study would be needed to determine the optimal voltage given the scale of this mechanism. The resulting magnetic field strength is .159 teslas. One benefit of having a larger voltage is that it minimizes the strength of the magnetic field, as a large magnetic field is much harder to generate than a large voltage.

Capacitors and Power

(Table 6: Capacitor and Power Requirements for Launch)

The launcher’s height is comparable to some of the tallest structures built on Earth, and its splayed design would make it a considerable civil engineering challenge even considering the Moon’s lower gravitational field. It is a mere .8% of the capsule’s orbital height, validating the approximation used for the projectile analysis that the total ascent of the capsule is approximately equal to its orbital height.

As with the voltage values for a railgun, the amount of energy dissipated during the launch varies considerably. Bauer (1995) cites 53% as the proportion of electrical energy lost during the launch, with potential decreases to only 34% with fundamental design improvements. Sung (2008) provides a range of 50-90% of initial energy lost, with larger industrial-grade railgun mechanisms having a greater efficiency than smaller ones. As such, considering the efforts that would be made to minimize energy losses in a launcher for the Moon, I choose .5 for the proportion of energy lost. This gives an energy requirement of half a billion joules.

That energy requirement is an order of magnitude higher than that of the largest existing capacitor banks. Capacitors used for navy railguns can store and discharge approximately 2*107 to 3*107 joules of energy (Hundertmark and Liebfried 2017). Capacitor banks also find extensive use in areas of scientific research, especially for generating extreme magnetic fields. The Wuhan National High Magnetic Field Center uses a capacitor bank storing 2.56*107 joules of energy to generate a magnetic field of up to 90 teslas (Han et al. 2017). The largest capacitor bank in the world is used at the Dresden High Magnetic Field Laboratory, holding an impressive 5*107 joules (Wosnitza et al. 2006). However, the amount of energy that the launcher’s capacitor system will need to store is 10.8 times that of the one in Dresden.

The capacitor bank at the Dresden Lab is comprised of 500 individual high-energy capacitors collated into 19 separate modules that together can store up to 50 megajoules of electrical energy (Wosnitza et al. 2006). This modular system allows for the bank to be flexible and could allow for the energy stored to be scaled up by a factor of ten. There is no fundamental physical limitation to that. The cost of the bank is estimated at $12,076,500 (“Prime Minister Milbradt” 2006). It’s unclear if that is just the material cost or inclusive of research and development and other less-quantifiable expenses. However, for the estimate in this analysis, that value will be multiplied by 10.8 to correspond with the increase in energy: $130,545,182.10. 12 hours (43,200 seconds) was arbitrarily chosen as the amount of time to recharge the system in preparation for a launch. That would allow the mechanism to launch two batches of payload each day, a far brisker pace than any schedule for rockets. It would enable a Lunar civilization to harness the full potential of this launch mechanism’s reusability. The most powerful solar panel on the commercial market has a power output of 435 watts under optimal conditions (Aggarwal 2021). The price of a single solar panel depends on the wattage, but given a high-end price of $4.43 per watt, the total price of the array was found. The actual cost of a Lunar solar array would vary, and some major factors are considered in the next section.

Generating the Magnetic Field

(Table 7: Magnetic Field Values for One Coil)

Instead of taking the values for the radius of the coil and number of loops from existing electromagnet systems, I used appropriate values scaled in comparison to the launcher. Engineers would have a lot of flexibility with implementing the system, so they need not be constrained by existing values. The distance along the axis is taken to be reasonably large considering the width of the armature. With the energy required and volume of a single coil known, the necessary values for the entire electromagnet system can be found.

(Table 8: Values for the Entire Magnetic Field System)

he energy needed for the electromagnet systems is twice as large as the one for the main railgun, and, as such, the ensuing solar panel requirements are doubled. The energy required here is 20.35 times greater than the energy stored in the capacitor bank at the Dresden Laboratory. Assuming that the cost scales linearly with energy stored – as in the previous section – the cost for the capacitor to generate the magnetic field would be $245,782,722.30. One tesla is an extraordinarily strong magnetic field. As such, it makes sense that the magnetic field for the launch has a high energy requirement, especially considering that the entire launcher must be encompassed in the field. Note that any values omitted from the table – such as the time to recharge and the cost per solar panel – were reused from before as there was no need to change them.

The price per kilogram of copper was found from an online marketplace. The market forces of supply and demand will make it fluctuate over time, meaning that this price is not definitive. The bulk-buying of materials required for the project would likely decrease the overall price, reiterating the fact that the cost estimates in the analysis are simply meant to be a rough gauge of the order of magnitude of expense for the project.

Final Cost Assessment

(Table 9: Material Cost Estimate)

The railgun wire runs up and down the entire length of the railgun, so its volume is twice the product of its cross-sectional area and the length of the launcher. As with the wiring for the magnetic field, the wiring of the launcher is made of copper. The same material cost procedure applied in the previous subsection was used again to get the cost of the railgun wire. The limitations of the total cost estimate will be discussed in the next section, where it will be compared to other aerospace projects and used to assess feasibility. Finally, the cost per kilogram of payload will be compared to that for rockets to see if this alternative space launch mechanism is more economical.

Discussion

The implications of the numerical results from the previous section are considered to assess whether this study found any fundamental barriers to building an electromagnetic launcher. The limitations of this analysis and several options for future study based on the findings of this analysis are discussed.

Assessment of Feasibility

None of the results of this study prohibit the construction and operation of an electromagnetic launcher from a physical standpoint. The key hurdles are all engineering problems and logistical challenges that can likely be solved through careful research and planning. The least troublesome part of this mechanism from that perspective is generating the electricity needed to power it. Only 84 standard-wattage solar panels would be needed to charge the mechanism’s capacitors over 12 hours. A brisker launch schedule would require more solar panels, but it would never become a prohibitive amount. Large solar farms on Earth have power outputs of megawatts, whereas this array would only need to produce a few tens of thousands of watts. This presents a clear advantage for electromagnetic launch over rockets, as the latter need specialized fuels to be synthesized and transported. Electricity is much easier to generate, even on the Moon. Solar panels used on the Moon would need to be specialized to survive the challenges of the Lunar surface, such as dust particles and tiny meteorite impacts. However, this is a comparatively small challenge that would not add a significant cost. Solar panels power the International Space Station, so their viability beyond Earth has already been demonstrated.

While energy generation is not a significant problem, energy storage is. The required capacitor banks would be much larger than any existing ones. The modularity system used at the Dresden Magnetic Field Laboratory may be the key to achieving such a high capacitance: keep adding more capacitors until the energy storage requirement is met. Analyzing the rate of discharge for that massive system will be crucial to ensuring that it successfully powers the launcher. Transmitting that energy from the bank to the launcher and magnetic field wires will be another major challenge. This study’s broad scope means that the specifics of the capacitor system must be left to other studies. Determining if the necessary capacitance could be achieved by simply adding to the Dresden system without major overhauls should be a key focus of such studies. Still, the existence of high-capacitance systems that can power existing railguns and generate extreme magnetic fields is a promising sign that the energy storage needed for the launcher is feasible with contemporary technology.

Aside from the capacitor bank, the largest engineering challenge of this project would likely be the structure of the mechanism. Its length is comparable to that of particle colliders and its height would make it a formidable skyscraper. Unlike a skyscraper, however, its mass would not be distributed over a central point, ensuring that many structural supports would be needed to keep it standing. The Moon’s smaller gravitational field would make it easier to construct than on Earth, although transporting that mammoth amount of material from Earth would pose a different challenge.

The next aspect of the feasibility check is the partial cost estimate: approximately 500 million dollars. It’s crucial to discuss what is considered in this estimate. It includes the costs of some of the main materials that were easily-quantifiable: the solar panels, capacitor banks, and wires for the launcher and magnetic field. Perhaps the largest material cost excluded from this estimate is all the material required for the structure, such as the rails and supports. Given the mechanism’s scale, the costs of these materials could easily dwarf the initial estimate. Furthermore, the materials used for the structure of the railgun would have to be strong and durable to survive the extreme conditions of the launch. Specifically, with half or more of the initial electrical energy being converted into heat throughout the launch, these materials would have to be built to resist melting and wear from frequent use. Other materials not included are additional wiring, batteries for longer-term energy storage, and controllers and sensors for the launch. Compared to the actual structure and the initial estimate, those costs would likely be minimal.

No non-material costs were considered due to the difficulty in quantifying them. These include manufacturing, shipping, and installation of materials, labor expenses, and the research and development needed before the construction of this system could begin. Considering that novelty of the launcher and the fact that it would be built on the Moon, these costs would likely dwarf the material costs. Taking these factors into consideration would likely give a cost estimate in the tens of billions of dollars. The complexity of administering a megaproject would likely inflate that cost further. Still, this partial cost estimate provides a basis for comparison with other megaprojects. For example, NASA’s super-heavy Space Launch System, which is designed to take astronauts back to the Moon and on to Mars, has cost nearly 20 billion dollars to develop and it is still far from being operational (Berger 2020). This is not a direct comparison, as the SLS is a complex system designed for human spaceflight while the mechanism in this study launches cargo on the Moon. However, the SLS is not fundamentally new technology and it faces the same problems as other rocket systems. Aerospace projects are simply expensive, so the cost of developing this launcher would not be abnormally-high.

Even if the cost of development is greater than that of chemical rockets, this launch mechanism outcompetes its competitors by being entirely reusable. Once the infrastructure is installed, it does not need to be altered or added on to. The only inputs would be energy, from the Sun, and regular maintenance. In contrast, a new rocket needs to be built and fueled with chemicals for each launch, a tedious and expensive process. The current leader in economical space launch is SpaceX with its Falcon 9 rocket. Its cost per kilogram to launch to low Earth orbit is around $2700 (“Falcon 9” 2020). This may seem significantly cheaper than the cost per kilogram of over a quarter of a million dollars for this launcher, as calculated in the previous section. However, the capsule that the study used was only loaded with the LRO, which does not approach its maximum capacity. In reality, the capsule would be filled before launch. Furthermore, the cost estimate for the launcher uses the development price, whereas the calculation for SpaceX uses the cost for a single launch only. To remedy this disparity, the following equation can be used to find the number of launches (n) that it would take for the launch mechanism to have the same per kilogram cost as the Falcon 9.

(Equation 41)

Solving for n results in (42).

(Equation 42)

With all the appropriate values substituted, n evaluates to 98 launches (rounded up). Assuming a brisk two launches per day schedule, it would take only about a month and a half for this mechanism to eclipse the current best method for sending payload to space. All future launches would only make the cost per launch cheaper and solidify the electromagnetic launcher as the more economical method of accessing space.

Thus, I conclude that, on a high-level, this electromagnetic launch mechanism is feasible. This analysis found no fundamental physical barriers. Further study and deeper analysis are needed to uncover potential physical and engineering challenges. Although not immediately cheaper than chemical rockets, it has the potential to be far cheaper over the long-term due to the permanence of the infrastructure.

Options for Future Study

The goal of this study was to explore the logistics and challenges of building an electromagnetic launcher. Many decisions were made about which paths to take and what values should be classified as reasonable. This leaves many opportunities for future study. Of particular concern and curiosity are the capacitors. For this mechanism to be built, extensive research needs to be conducted into high energy capacitors, particularly regarding efficient use of physical space. As the Apollo program did with computers, the development of a railgun launcher may galvanize the creation of a new class of powerful capacitors that have widespread applications beyond the original field of use.

The forces and acceleration within the railgun launcher were analyzed at a very high-level. A more detailed analysis relying on computational modelling software would be necessary to ensure that the launcher and the capsule would behave as expected throughout the entire duration of the launch. An immense quantity of energy is transferred and dissipated within a short burst of time. The effect of those transformations would have to be assessed to ensure that both the launcher and the cargo can withstand such shocks. The launcher would have to be designed with materials that could withstand the stresses and heat released during the launch. The delicate cargo would have to be insulated from the heat and high accelerations.

One factor ignored throughout this study is the counter-electromotive force that would be generated by the movement of the capsule in the launch tube. The launcher is a closed circuit with some defined area. The length of the circuit increases as the capsule advances in the tube, increasing the enclosed area. The magnetic field is constant over the length of the launcher, so the magnetic flux of the circuit loop – which can be thought of as a measure of the number of magnetic field lines passing through the area enclosed by the circuit – will increase. Faraday’s Law states that a change in the magnetic flux of a current loop will induce an electromotive force in the loop. However, Lenz’s Law states that the current produced from this emf will flow in a direction that induces a magnetic field in the opposite direction as the initial changing magnetic field.

In this case, the changing magnetic field comes from the acceleration of the cargo outwards. The induced current from this changing flux will create a magnetic field that points down rather than up, reducing the magnitude of the Lorentz force that accelerates the cargo. Furthermore, the magnitude of the induced current is proportional to the rate of change of magnetic flux with respect to time. As the capsule moves faster and faster, the counter-electromotive force will increase and counteract its motion even more. This negative feedback loop could prevent the cargo from ever reaching the exit velocity needed to achieve orbit. This force was excluded from the analysis because it is mathematically-complex, but it would need to be considered when developing the actual mechanism. A projectile in a military railgun can overcome this force and accelerate to immense velocities, so its existence does not preclude the development of this launch mechanism. However, extensive research needs to be conducted to analyze how the counter-electromotive force behaves when the entire system is scaled up immensely.

This entire study has focused on the Moon, primarily because it lacks an atmosphere that would exert resistance on the capsule as it moves through the launch tube or ascends to orbital heights. Of course, Earth has a significant atmosphere and its effects on the projectile’s acceleration and ascension would need to be assessed to deploy the system on Earth. Considering the high demand for cargo heading to Earth orbit, the system could be even more successful on Earth than on the Moon.

Finally, when analyzing the projectile aspects of the cargo, it was assumed that the capsule would simply transition to circular orbit when it had reached the top of its trajectory. This is a flawed assumption, because the capsule cannot spontaneously transition from a parabolic trajectory to a circular one. Instead, the capsule would need to be propelled into a circular orbit. The propulsion would most likely come from chemical rockets. As the cargo is already at orbital heights, the fuel would be minimal compared to the amount needed if it was launched from the surface, but it would still need to be factored in to prevent the cargo from continuing on the parabolic trajectory and crashing back to the surface.

Conclusion

This study has determined that an electromagnetic launch mechanism has the potential to serve as a viable alternative to chemical rocketry for Lunar space launch. The main constraints and avenues for future study come from the energy storage in capacitor banks and the actual structure of the mechanism. Its scale would make it a remarkable engineering challenge. The costs would likely be in a similar range as other aerospace megaprojects. The mechanism’s long-term utility provides a great incentive to conduct further research into the development of this mechanism. If ultimately viable, it could lead to a mass commercialization of space. If not, the research spurred on by this goal would find plenty of applications elsewhere. Those details are left to more focused studies. This project combines the frontier of science and technology and the rewards could be literally astronomical.

Acknowledgements

I would like to thank Mr. Bilak, my physics teacher and project mentor. In addition to being a great teacher, he guided me throughout the course of this project, from the genesis of the research question to reviewing the final version of the paper for errors. His enthusiasm for physics is contagious. Additionally, I must extend my gratitude to Jacob Terkel, whose assistance was instrumental in solving the second-order differential equation. Without him, I would have remained stuck in the land of constants. Finally, I would like to thank Leone Huang for helping me work out the gradient descent methods used in an earlier version of this work.