Question

An ion thruster mounted in a satellite uses electric forces to eject xenon ions and produces a thrust of \(1.229 \cdot 10^{-2} \mathrm{~N}\). The rate of fuel consumption of the thruster is \(4.718 \cdot 10^{-7} \mathrm{~kg} / \mathrm{s}\). With what speed are the xenon ions ejected from the thruster?

Short answer

Answer: The speed at which xenon ions are ejected from the thruster is approximately \(2.6 \cdot 10^4 \mathrm{~m/s}\).

Step-by-step solution

Identify the given values

The given values in the exercise are: Thrust force (F): \(1.229 \cdot 10^{-2} \mathrm{~N}\) Rate of fuel consumption (mass flow rate, \(\dot{m}\)): \(4.718 \cdot 10^{-7} \mathrm{~kg} / \mathrm{s}\)

Derive the equation to calculate the speed of ejected xenon ions

The equation relating thrust force, mass flow rate, and exhaust velocity is given by: $$F = \dot{m} \cdot v_e$$ where \(F\) = thrust force, \(\dot{m}\) = mass flow rate, and \(v_e\) = exhaust velocity (the speed of ejected xenon ions). Now we need to solve the equation for exhaust velocity, \(v_e\): $$v_e = \frac{F}{\dot{m}}$$

Input the given values and calculate the exhaust velocity

Now that we have the equation to calculate the exhaust velocity, let's input the given values and solve for \(v_e\): $$v_e = \frac{1.229 \cdot 10^{-2} \mathrm{~N}}{4.718 \cdot 10^{-7} \mathrm{~kg} / \mathrm{s}}$$ Now, divide the two numbers to find the exhaust velocity: $$v_e \approx 2.6 \cdot 10^4 \mathrm{~m/s}$$ Therefore, the speed at which xenon ions are ejected from the thruster is approximately \(2.6 \cdot 10^4 \mathrm{~m/s}\).

Key concepts

Thrust Force Calculation

Understanding the calculation of thrust force is essential when exploring the dynamics of ion thrusters. Thrust is the force exerted by the ion thruster and is a result of Newton's third law: For every action, there is an equal and opposite reaction. In space propulsion, this principle manifests as the ions are pushed out of the thruster, thereby propelling the spacecraft in the opposite direction. To calculate the thrust force, one of the key equations used relates the force to the mass flow rate (the rate at which mass is expelled from the system) and the exhaust velocity (the speed at which the particles are ejected).

Mathematically, the equation is expressed as: \[ F = \text{mass flow rate} \times \text{exhaust velocity} \] where F represents the thrust force. In the context of the exercise problem, by rearranging the equation, we can solve for the other variables if the thrust force is known and one of the other two variables is given. The equation serves as the cornerstone for evaluating the performance of ion thrusters, dictating their efficiency and ultimately contributing to the spacecraft's trajectory and mission potential.

Mass Flow Rate

The mass flow rate in an ion thruster is a critical parameter that governs the efficiency and capacity of the propulsion system. The rate at which the ions—or any propellant—are expended determines not only the available thrust but also the lifespan of the thruster.

In other words, the mass flow rate is the quantity of mass passing per unit time, typically measured in kilograms per second (kg/s) in the context of propulsion systems. This measure directly affects the thrust because it's one portion of the total momentum change imparted to the vehicle. A higher mass flow rate will generally yield a higher thrust, provided the exhaust velocity is constant. However, an increase in mass flow rate results in a faster depletion of propellant, which might be undesirable for long-duration missions.

Consideration of the mass flow rate is vitally important when designing and evaluating propulsion systems, as it must be balanced with the overall mission requirements and the spacecraft's total mass.

Exhaust Velocity

Exhaust velocity, often represented as v_e, is a measure of how fast the propellant is being ejected from the propulsion system of a spacecraft. It is a pivotal factor in determining the thrust produced by an ion thruster. The higher the exhaust velocity, the more efficient the thruster is at converting propellant into thrust. This efficiency has major implications for the spacecraft's acceleration and the amount of propellant needed for a mission.

In the context of our ion thruster exercise, after calculating the thrust force and knowing the mass flow rate, we determined that the exhaust velocity was \(2.6 \times 10^4 \text{ m/s}\). This high exhaust velocity is characteristic of ion thrusters and is much greater than that of chemical rockets, enabling more thrifty fuel usage for long-duration space missions. The profound speed at which the ions are ejected speaks to the effectiveness of electric propulsion, which, despite its lower thrust, offers a higher specific impulse—a measure of the thrust-to-fuel consumption ratio—than traditional chemical propulsion.