Apr 8

Apr 8 Defying Gravity: Taming the Rocket Equation

Different Types of Rocket Engines | Alex S. Li

“Rocket man burning out his fuse up here alone”

— Rocket Man, Elton John

The rocket equation: the heartbeat of rocketry. This mathematical formula is the governing law of physics for Outer Space launch and travel. The rocket equation is an immensely useful tool to determine the amount of fuel needed for a rocket of a certain mass to reach its Outer Space destination. But, this blueprint also reveals the heavy toll needed to escape Earth’s gravity, leading one astronaut to call it an equation of tyranny.

While rockets have been around for centuries, the technologies behind rocket engines have been constantly evolving. With each scientific advancement, engineers have invented new types of rocket engines that can carry payloads to ever far-reaching corners of the universe, assisting us to better understand our cosmic galaxy.

Rocket engines, the unsung workhorses of our Outer Space explorations, is the topic of this post. I will start by providing a high-level overview of the principles behind rocket propulsion. Then, I will explore the common types of rocket technologies that are currently used and under development for Outer Space. Finally, I will end with a policy perspective on how we can conquer the tyranny of the rocket equation.

Reaching New Heights: Rocket Propulsion

Before getting into current rocket technologies, I want to provide a basic understanding of how a rocket flies. At its core, rocketry is highly dependent on Newton’s three laws of motion. According to Newton’s first law, a rocket will not move until acted upon by another force. A propellant can provide this impetus by creating a forward-moving propulsion force called thrust. Thrust is generated by the propellant in line with Newton’s third law by, depending on the type of rocket engine, some form of reaction that causes mass to be expelled backward through a rocket’s nozzle. How fast this rocket travels will depend on, as described by Newton’s second law, the magnitude of the thrust created.

Through a derivation of Newton’s second law, we can also measure the efficiency of a rocket engine by calculating its specific impulse. Specific impulse is the change in momentum delivered per unit of propellant consumed; this measurement calculates the “time in seconds that one pound of propellant generates one pound of thrust.” The higher the specific impulse, the more efficient the rocket.

But, how do you determine how much propellant is needed to get to a certain spot in Outer Space? Well, the answer can be gleaned from the rocket equation. While several individuals—including British mathematician William Moore, Scottish scientist William Leitch—had independently derived this formula, Russian scientist Konstantin Tsiolkovsky is credited as the first to use it to determine whether rocketry is suitable for Outer Space travel. Fundamentally, the rocket equation relies on the principle of conservation of momentum (a derivation of Newton’s third law): thrust, at any given point, will approximately be equal to the exhaust velocity (the speed at which a propellant mass is expelled) at that point; it is not an exact 1:1 ratio because there are other aerodynamic forces such as lift and drag that come into play.

As a rocket flies, it becomes lighter and lighter as its fuel is being depleted. This continuous decrease in the rocket’s mass also means less fuel is needed to escape Earth’s gravity over time. What makes the rocket equation special is its ability to recursively take this factor into consideration. The formula requires two inputs: delta-v and the exhaust velocity. Once we have a destination in mind, we can solve for the rocket velocity—also known as delta-v—needed to get there. By knowing the type of rocket propellants involved, we can calculate how much exhaust velocity can be generated. With these two factors plugged into the rocket equation, we can determine how much of a rocket’s initial mass must be fuel for that particular rocket to reach that specific spot in Outer Space.

Solving for this relationship, scientists have arrived at several stunning conclusions. First, a rocket’s fuel requirement is extremely large: upwards of 85 to 95 percent of a rocket’s mass must be dedicated to the rocket’s propellants. With a certain percentage of the remaining mass needed for the rocket structure itself, a rocket may only have 4-5 percent of usable mass for payloads. Second, most of the delta-v needed to reach a certain point in Outer Space is expended to escape Earth’s gravitational pull. For instance, while it takes 9.3 to 10 km/s to escape Earth’s gravity, it would only take an additional 10.2 km/s to get from Low Earth Orbit to Mars. Hence, half of the fuel needed for an Earth-Mars trip (the shortest route at 57.6 million km / 35.8 million miles) is spent in the journey’s first 400 km (250 miles). So you can see why the rocket equation is a symbol of the tyrannical grasp gravity has on our ability to leave Earth.

Current Rocket Technologies

Armed with a basic understanding of rocket propulsion principles, in this section, I will provide a general overview of the common rocket engine technologies currently being used and developed.

Solid Propellant Rockets

The OG of rocket engines, solid propellant rockets have been around long before humans ever reached Outer Space. First invented by the Chinese during the 13th century, solid rockets have come a long way since their gunpowder-based beginnings. But, the basic concept of how these rockets generate thrust has remained unchanged. The rocket casing—now typically made of steel—is filled with a solid propellant. An igniter combusts the propellant, forming hot gas as it burns. When the hot gas gets expelled through the rocket’s nozzle, thrust is created. Nowadays, the amount of thrust generated over time can be controlled by how the propellant is packed inside the casing; this science is called grain geometry.

Solid rocket propellant has two primary forms: homogeneous or heterogeneous. Homogeneous fuel usually has a single, double, or triple-base structure and are made of one to three primary ingredients. These ingredients can include the fuel, the oxidizer (required to ignite the fuel in the oxygen-less Outer Space), the plasticizer (which helps to increase energy yield), and the binder (which controls how the fuel is burned over time by creating inert interstitial spacing). Meanwhile, heterogeneous solid propellant has the oxidizer and fuel mixed together as large composite macroscopic particles; oxidizer, plasticizer and other modifiers can be added to this mixture.

Solid rockets have several advantages and disadvantages. Their propellants are easy to store and are relatively safe. Because of their ingredients’ simplicity, solid propellants can be produced at a low cost. However, the rocket’s thrust-generating performance is largely inefficient when compared to other types of propellants. Additionally, while thrust can be modulated through the propellant’s geometry, once ignited, a solid rocket’s propulsion cannot be stopped until the fuel is completely consumed. For these reasons, solid propellant rockets are generally used when a tremendous amount of low-cost uncontrolled thrust is required. Examples of solid rockets include the solid rocket boosters (hence their name) of the Space Shuttle and the SLS rocket.

Liquid Propellant Rockets

A rocket using liquid propellant was first launched on March 16, 1926 by Professor Robert H. Goddard. Named Nell (after his daughter), the rocket reached a height of about 41 feet (12.5 meter) with a flight duration of around 2 seconds. Generally, liquid propellant rockets are made up of storage tanks (for the fuel and the oxidizer), a combustion chamber with attached nozzles (where the fuel is mixed, ignited by the oxidizer, and the byproduct is eventually expelled), and a pumps-and-valves transfer system that can regulate the flow of the fuel and the oxidizer into the combustion chamber.

Liquid propellant rockets generally come in bipropellant form; these rockets use two separate liquid propellants: a liquid fuel (commonly oxygen) and a liquid oxidizer (commonly hydrogen or kerosene). There are several advantages to liquid rocket engines. They are usually more efficient and generate more specific impulse than solid rocket engines. Additionally, since the flow of the liquid propellants can be controlled by intake valves, thrust can be throttled and the engines can be started, stopped, and restarted. However, liquid rockets have their disadvantages with the biggest one being weight. The extra materials needed for the transfer system and the storage tanks can add significantly to these rockets’ mass. Furthermore, liquid propellants are generally more unstable, toxic, and highly reactive.

Hybrid Propellant Rockets

As its name suggests, a hybrid propellant rocket uses different forms of propellants. Generally, solid fuel is used with either a gaseous or liquid oxidizer. Using liquid oxygen as the oxidizer and a gelled form of gasoline as the fuel, Soviet scientists launched the first ever hybrid rocket on August 17, 1933. The hybrid rocket generally contains a pressurized tank for the liquid/gas oxidizer, and a combustion chamber where the solid fuel sits. Whenever thrust is needed, the oxidizer is transferred to the combustion chamber where it reacts with the fuel and the byproduct is released through the nozzle.

Hybrid rocket engines try to combine the best of both worlds. When compared to solid rockets, hybrid rockets have advantages in that their thrust can be throttled and have better specific impulse. When compared to liquid rockets, hybrid rockets are safer and mechanically simpler as they eliminate the need to determine the pumping ratio between the bipropellants. Hybrid rockets also use fewer material than liquid rockets as only one storage tank (for the oxidizer) is necessary. However, hybrid rockets’ weaknesses are that they cost more than a pure solid rocket and unlike a pure liquid rocket, it’s harder to refuel as the hybrid rocket’s solid fuel is pre-placed in the combustion chamber.

Ion Propulsion Rockets

Unlike solid, liquid, or hybrid rockets, which rely on chemical reactions to create propulsion, ion propulsion rockets are a type of chemically-inert/non-reactive rocket engine. While futuristic sounding, ion engines were first theorized by Tsiolkovsky and Professor Goddard and became a reality in 1959 when Dr. Harold Kaufman, a scientist at NASA’s Glenn Research Center, built a broad-beam electron-bombardment working version. While Dr. Kaufman’s model used mercury as the fuel, nowadays, ion thrusters generally use xenon, a colorless, odorless, tasteless and most importantly, chemically inert gas.

Ion rockets generate propulsion by creating plasma and discharging the positively-charged ions. Plasma is a state of matter that consists of a gas of ions (particles with a net electrical charge) and free electrons (negatively-charged particles). Ion propulsion rockets create plasma and positively-charged ions by bombarding its propellant with electrons via a process called, unsurprisingly, electron bombardment. The electrons are typically recollected and use to ionize additional propellant while the ions are accelerated out of the nozzle through either electrostatic (via Coulomb force) or electromagnetic (via Lorentz force) means, thereby generating thrust.

An ion rocket is made up of six main components: a power source, a power processing unit, a propellant management system, a control computer, a propellant tank, and the ion thruster itself. While any power source can be used, ion engines are generally powered by solar cells since sunlight is pretty much boundless and free in the solar system. The power processing unit converts the sun-generated power into energy needed to operate the ion engines. The propellant management system controls the flow of the propellant into the thruster from the propellant tank. The rocket’s control computer monitors and manages this entire process. Examples of working ion thrusters include Deep Space 1 spacecraft’s engine and the NASA’s Evolutionary Xenon Thruster (NEXT), both of which use xenon as the propellant and the latter will be used as a part of NASA’s DART (Double asteroid Redirection Test) mission.

Ion engines’ main advantage over their chemical counterparts is the former’s high specific impulse. Ion thrusters take advantage of the charge-mass ratio principle to accelerate ions, hence a small potential difference could generate high exhaust velocity; in fact, while chemical reactions can only create thrust of 5 km/s, charged particles can create exit velocity of 15km/s to 35km/s. Therefore, less propellant (and thereby mass) would be needed to generate the thrust required.

However, ion thrusters are limited by the “acceleration with patience” principle; currently, these rocket engines can only generate about 0.5 newtons/0.1 pounds of thrust, equivalent to the force a piece of paper exerts on one’s hand. This low acceleration means ion engines cannot generate the amount of thrust required to escape Earth’s gravity. But, once in the near frictionless and zero-gravity realm of Outer Space, ion rockets can be extremely efficient as their momentum build up. Since ion thrusters cannot generate their own power, one other disadvantage is that solar-powered ion rockets will have less and less power to function as they move further away from the sun. In these cases, a different power source, like nuclear, will be needed for the thrusters to continue operation.

Cold Gas Rockets

Cold gas rockets also use inert gas as the propellant. The concept is fairly simple: pressurized gas is released through a de Laval (converging-diverging) nozzle to generate propulsion. The nozzle speeds up the highly pressurized—but low velocity—propellant to an exhaust velocity near that of sound, generating thrust. While the propellant can be stored in solid, liquid, or gaseous form, it is generally kept in gaseous state so conversion is not needed before the propellant is pushed through the nozzle.

The advantages of cold gas rockets are that they are very simplistic, enabling them to be small and not take up much mass. They are also inexpensive, clean, and, when inert gas is used as the propellant, likely the safest of all rocket engines. However, cold gas rockets produce less thrust and are less efficient than chemical rockets. Therefore, they are impractical for launch operations. Yet once in Outer Space, cold gas rockets are great for satellites that just need a tiny amount of thrust to maneuver in the vacuum.

Nuclear Thermal Rockets

Nuclear thermal rockets use heat generated from nuclear fission (when an atom’s nucleus is split into smaller or lighter nuclei) for propulsion purposes. A nuclear thermal rocket’s key component is its small nuclear reactor. Uranium is generally used as the fuel to generate heat which converts the liquid propellant, typically hydrogen, to its gaseous state. Thrust is produced when this gas is then accelerated out of the nozzle. While tests have been performed, to date, nuclear thermal rocket has not made its maiden flight. The closest we came was when NASA and the Atomic Energy Commission established the Nuclear Engine for Rocket Vehicle Application (NERVA) program in 1961. However, the Nixon administration ultimately decided to cancel the program in 1973.

Nuclear thermal rockets can generate higher specific impulse than that of chemical rockets (500-1000 seconds compared to 400-500 seconds). The higher efficiency means that less propellant is needed for long-duration missions, leading to less cost and mass for launch. However, like ion engines, based on the current state of technology, nuclear thermal rockets cannot generate enough thrust to escape the pull of Earth’s gravity. Additionally, nuclear thermal rockets will involve working with highly radioactive material; hence, accidents could have catastrophic effects.

Nuclear Fusion Rockets

Unsurprisingly, a nuclear fusion rocket relies on the power of nuclear fusion (when two atoms collide to form one larger atom) for propulsion. This fusion process generates an exorbitant amount of energy, which if directed properly, can be channeled out of a rocket’s nozzle and generate thrust. Fusion rockets are still theoretical as the process can only occur in environments that are extremely hot (measured in millions of degrees). But, research is underway to create a working fusion rocket with its reactor releasing energy directly into the propellant. With this engine, the energy from the fusion reaction can heat and accelerate the propellant to exit with velocities higher than 30 km/s.

The advantage of nuclear fusion rocket is its high specific impulse (130,000 seconds compared to 400 seconds, so about 325 times more efficient than chemical rockets). Additionally, these rockets can harness the heat of the fusion reaction by converting it back into electricity which can then power other rocket’s functions. Furthermore, when compared to fission rockets, fusion rockets are safer as they generate less radiation. But, the fusion process is still volatile. Additionally, the mass of a fusion reactor can potentially make it cost-prohibitive to launch.

The Hardest Step: Getting Past Earth

While we have a wide selection of rocket technologies that can propel us forward, we are, unfortunately, severely constrained to a subset of these rockets for getting into Outer Space in the first place. Of the rocket engines currently available, only chemical rockets can create enough thrust to escape Earth’s gravity. So in order to successfully venture to the far-reaches of the universe and, perhaps one day, colonize other planets, we must find a way to escape the tyranny of the rocket equation. And for this purpose, I argue that our answer might lie in building off of Earth.

For any current Outer Space exploration, Earth’s gravity takes a heavy toll. For example, about 60% of the delta-v needed to go from Earth’s surface to the Moon’s surface is expended in reaching Low Earth Orbit. For a trip to Mars, this represents about half of the fuel needed for the entire trip. So, if we can find a way to bypass Earth’s gravity, we not only would have a broader set of rockets to choose from but can also go farther and carry more useful mass.

To take advantage of this energy saving, in the long run, we should be focused on establishing production facilities off of Earth. This will not be an overnight task; in order for this plan to work, we have to first master the technologies needed to gather, refine, and use resources in Outer Space. That means we must first create the infrastructure, both technologically and legally, needed for Outer Space mining. In addition, we will need to establish an outpost that can facilitate the production of spacecrafts. I believe the most likely destination for these facilities is the Moon; while each launch will still need to overcome Moon’s gravity, this escape cost is only 18% that of Earth’s. While space stations would eliminate this cost completely, they would require too much raw materials to construct and additional challenges will need to be solved such as orbital station-keeping measures.

While creating and operating an extraterrestrial output will require a massive undertaking, it is well worth the cost. Once the production and launch facilities become self-sustaining, it will open up a new gateway for Outer Space exploration. These energy-saving measures would enable us to reach and explore destinations that are currently out of our grasps. While laws of physics will always constraint our ability to set off from our mother planet, we should not be afraid of shedding our earthly bounds in taking the next step in transforming our space-faring infrastructure. Therefore, we must not solely look at Outer Space as a destination, but also as a part of our solution in working with the constraints of the rocket equation. Only by truly harassing the resources available in the broader universe, can humanity achieve its dream of becoming an interstellar species.