ABSTRACT
In this thesis, theoretical studies are conducted to see whether plasma will detach from the magnetic field lines of the VASIMR thruster, and if so, at which location detachment takes place. A magnetic field similar to the field of the VASIMR VF-24 engine is used and ions of different speed and massare sent from various radial positions in the exhaust. Calculation with different values of the anomalous resistivity parameter ωτ is conducted and the sensitivity to this parameter is studied.
The validity of the method is studied by comparing results to previous work by Carl Wesslén. From the results it is concluded that using heavy ions sent at high speeds will achieve detachment and high thrust efficiency, even when assuming relatively high values of ωτ. Ejecting ions at a slower pace or using lighter ions will make the engine less efficient, requiring low ωτ which is difficult to achieve. For some combinations of mass and speed, detachment is not possible at all. Ions with heavy mass are recommended to use as propellant for this type of thruster.
DESCRIPTION OF PHYSICS
Figure 2: From Ilin et al.’s work
Figure 2: From Ilin et al.’s work. The top part of the picture shows the magnetic field setup and the ion trajectory. The lower part of the figure shows the energy in the axial and radial direction respectively. The transition lies between 1 and 2 m, where radial kinetic energy turns into axial kinetic energy. Figure 2 shows a single particle orbit of an ion in the magnetic field of the VASIMR thruster, from Ilin et al.This is the kind of orbit one isolated ion would follow if there is no plasma, and no electric fields that arise due to plasma effects.
Figure 3: Comparison of two magnetic field configurations
To validate the results, Ilin et al.’s work was used as a benchmark to compare ion detachment positions. This was to make sure that the results where reasonable, as calculation methods differ. The magnetic field configuration is shown in Figure 3. Comparison of two magnetic field configurations. The left is from work by Ilin et al. and the right is this work. The top shows the magnetic field in the RZ-plane and the lower part shows the absolute magnetic field strength at the Z-axis.
Figure 7: Overview of the physics behind plasma detachment
Figure 7: Overview of the physics behind plasma detachment. The last coil is the last part of the engine and the effective exhaust area is calculated at this point. The ions will follow the magnetic field lines and as the strength of the magnetic field decreases, the gyrating motion will turn into axial motion and the ions detach. After the ion detachment point, without the influence of the electrons, the ions would experience a perfect detachment, represented by the red curve and travel in a straight line.
If the ions instead follow the electrons that in turn follow the magnetic field lines they are considered trapped and this is represented by the blue curve. The green curve is the middle way called semi-detachment, used in the work by Wesslén. Here the ions are influenced by the electrons but not sufficiently enough to get trapped, resulting in a more curved trajectory than a perfect detachment.
DESCRIPTION OF CALCULATION METHOD
To calculate the ion trajectory, first the ion mass, exhaust velocity and the anomalous resistivity parameter need to be specified. For the calculations, different combinations of ωτ, ion mass and velocity are used and the ion path is iterated one step at a time.
First the magnetic field in every point needs to be calculated using equations. Then the ion detachment position is calculated using equation, where the direction and the velocity of the ions are known. The angle between the ion path and the magnetic field is then calculated using trigonometry. All these values are used in equation where we get the new velocity vector. The ion takes another step, the new radial and axial positions are noted and the iteration is repeated until a sufficiently long ion path is obtained.
APPROXIMATIONS
Figure 8: Relative error in percent for different timesteps. The radial position at the axial position of 30 m for different timesteps is compared with the timestep dt=10-11 s. The different lines represent ions originating from different radii at the exhaust coil
The effect of the timestep dt is investigated to ensure an accurate result. The process for calculating the trajectories are iterative so many calculations are made and therefore consume a lot of processing time. In order to get the results in a realistic computer time, the timestep used cannot be too small. To get the appropriate timestep, several trajectories are made using the same values but with different timesteps. The radial position when the particle has reached the axial position of z = 30 m is saved and plotted into a graph. Figure 8 shows the relative error of the radial position, given in percent, for different timesteps compared to the very small timestep dt = 10-11 s.
Figure 9: Relative error in percent of the efficiency compared to the efficiency at z = 1000 m as a function of distance from the exhaust for several trajectories. The trajectories originate from different radii in the exhaust. After about 1.6 m the relative error has dropped under 5 %
For the same reasons as choosing the timestep, it must be decided when it is acceptable to stop iterating. To be able to do this, ions are followed until they had reached 1000 m behind the engine. This result is used as a benchmark. The efficiency, as defined in section 2.4, is calculated along the trajectory and compared with the efficiency at the last iteration at the axial position of 1000 m. The result can be seen in Figure 9.
RESULTS
Figure 10: Image showing typical ion trajectories
Figure 10: Image showing typical ion trajectories, which is similar to the exhaust in a conventional rocket. The curvature is largest in the beginning and is straighten out after the magnetic field decreases with the distance. The ions that are near the z-axis will travel in a straighter path than the ions far out due to the structure of the magnetic field.
Figure 11: The efficiency of the thrust as function of the exhaust radius for different ion masses
Figure 11: The efficiency of the thrust as function of the exhaust radius for different ion masses. Heavier ions will be more efficient over the whole exhaust area. Masses under 20 u will only be able to use about 2.5 to 3.5 cm of the exhaust radius in order to be within the acceptable thrust loss limit of 30 %. Using helium will give negative thrust from about 4.2 cm. This result is for ions with exhaust velocity 50 km/s and ωτ =16.
Figure 14: Effective exhaust radii as function of mass and exhaust velocity with case A and B marked. This map shows the radius that can be used for each configuration to ensure an efficiency of 70 % or more
In order to know the effective exhaust radius, an efficiency limit, or acceptable loss of thrust is defined. This is set to 30 %. The ions are sent from a maximum radius of 5 cm. Efficiency graphs as function of exhaust radius are conducted for various masses and exhaust velocities. The exhaust radius at the efficiency limit is marked and then stored in an array. If the thrust loss at 5 cm radius is less than 30 %, meaning that the ions can be sent from even higher exhaust radius, they are still counted as being sent from 5 cm. The result is seen in Figure 14.
CONCLUSION
From the results we conclude that sending out ions with enough mass and speed will result in detachment and acceptable thrust. Slow and light ions can in many configurations detach too, but they will detach late resulting in a plume with a large angle and high thrust loss. If the ions are too slow or too light, using the whole exhaust area may result in that ions far out in the plume can detach almost perpendicular to the spacecraft velocity vector, giving no thrust at all, or detach backwards, giving thrust in opposite direction and decelerating the spacecraft. Some combinations of the key parameters mass, velocity, exhaust radius and anomalous resistivity parameter may result in the worst case scenario where the hot ions do not detach at all and smash into the spacecraft, possibly damage it.
The propellant used in the latest engine type, argon, will have an acceptable efficiency throughout the whole exhaust area if sent over 30 km/s. The operating exhaust velocity for this engine is 50 km/s so it’s well above the acceptable efficiency. The values of ωτ in order for detachment are within the expected interval for Bohm-diffusion. Using 30 km/s exhaust speed for deuterium will lead to that only about half the exhaust radius can be used efficiently. So instead of 5 cm, a radius of only about 2.5 cm gives acceptable thrust loss that we have defined to be 30 % or less. This means that in order to get efficiency out of the whole exhaust are, the lighter ions must be sent at a much greater speed.
However, the VASIMR is meant to vary its specific impulse and ions will be sent out with various kinetic energy and different speeds. With slower ions the specific impulse Isp will be lower, which means the spacecraft will be more energy efficient, less mass efficient and also less efficient in terms of having a wider angle of the plume. In order to minimize efficiency loss, the exhaust area could be varied as well when varying speed and potentially mass.
Source: KTH
Authors: Aleksander Slavic
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