Magnus Effect On A Spinning Satellite In Low Earth Orbit-Books Pdf

Magnus Effect on a Spinning Satellite in Low Earth Orbit
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and to serve as platforms for in orbit demonstrations IOD In Low Earth Orbit LEO aerodynamic forces in the. form of atmospheric drag perturb orbital trajectories sometimes in an unpredictable fashion potentially leading to. premature reentry and end of life EOL However when satellite surfaces are configured for specific angles of. incidence relative to the freestream flow particularly for sufficiently long durations near the perigee aerodynamic lift. may have an appreciable effect on preserving orbital dynamics 1 For example a stabilized disk shaped satellite may. be inclined at the same angle to the air flow at the region of perigee where aerodynamic forces are greatest for. successive revolutions As a result the resulting lift would have an appreciable effect on the orbit for perigee heights. up to 500km 2 Furthermore a spinning body in a fluid creates a nonsymmetrical flow pattern that generates an. aerodynamic lift that is commonly described as the Magnus effect This effect has been the subject of great interest in. the history of fluid physics and is named after Professor Gustav Magnus who established that a lifting force is. developed by a spinning cylinder placed in an air flow 3 For example Newton observed that a transverse force acts. on a spinning sphere moving through a fluid and Robins observed a similar effect in the trajectory of cannon balls 4 . A description of the Magnus effect was done by Lord Rayleigh who predicted that the lift was proportional to the. speed of rotation and translation Some of the earliest inventions incorporating this effect included the Flettner rotor. which was a sailboat whose sail used a rotating cylinder which produced a Magnus lift thereby generating a thrust to. push the boat forward Similarly a popular invention was the Magnus airplane where the Magnus lift was produced. by wings made of rotating cylinders However the relatively large drag induced by the cylindrical form made the. design impractical Moreover the Magnus Effect can be seen in many applications including rotary sails propeller. blades wings wind turbines and in the movement of weather systems 4 For example in hurricane formation a. hurricane will stall over open water spinning in a clockwise direction If the wind comes from the West the storm . influenced by the Magnus Effect will move rapidly north On the other hand if the wind originates from the east the. Magnus Effect will propel the rotating air mass to the south . The Magnus force is a function of geometry air density spin rate and freestream velocity and therefore as the. altitude decays atmospheric density increases thereby increasing the magnitude of this force As a result this effect. could be significant for spacecraft in the low Ionosphere Thermosphere region This added maneuverability without. using conventional thrusters could improve efforts to perform in situ atmospheric research in the low Ionosphere . Thermosphere region Likewise this effect can be used to maintain a spacecraft s altitude and could possibly be used. to perform active controlled deorbiting to improve predictions of the impact location Thus this study investigates. the feasibility of using the Magnus effect to sustain a spacecraft s orbit at a low perigee altitude of 80km . Problem Description, A spinning body creates a nonsymmetrical flow pattern above and below the body that generates a Magnus effect. that yields an aerodynamic lift As fluid flows past a rotating body streamlines on the side moving in the same. direction as the flow will converge indicating a diminished pressure 4 The streamlines on the opposite side move. against the freestream and as a result become more widely spaced indicating an increase in pressure as shown in Fig . 1 This pressure differential causes a lifting force that will displace the body in a direction normal to the freestream. flow As a result of the dependence on density the expression for the Magnus force is different for the continuum and. free molecular regime , Figure 1 Lift on a Rotating Body in a Fluid Medium. American Institute of Aeronautics and Astronautics. A continuum or free molecular regime depends on the mean free path of the fluid If the mean free path is small. in comparison with the dimensions of the body then the fluid can be considered a continuum With this assumption . the fluid s density temperature and velocity has a definite value at each point in space However many modern. engineering applications including those for spaceflight occur at high altitudes where the mean free path is not. negligible when compared with the dimension of the body and therefore the effects of the discrete character of the. fluid must be taken into account when defining its properties 5 At these two different regimes the governing physics. and interaction of the molecules are different A widely recognized parameter that determines whether a fluid medium. is a continuum or free molecular is the Knudsen number Kn which is the ratio of the mean free path and the. macroscopic length scale of the physical system In other words the local Knudsen number is a measure of the degree. of rarefaction of a gas 6 As the local Knudsen number increases free molecular effects become more pronounced. and eventually the continuum assumption breaks down As the local Knudsen number decreases as a result of the. increase in atmospheric density the Magnus lift becomes more pronounced thereby enhancing the effect of a potential. Magnus maneuver As a result the work performed in this study is restricted to a spacecraft flying at a perigee of. 80km with the objective of investigating the feasibility of the Magnus effect in sustaining the altitude The magnitude. of this force on the orbital decay will be examined by varying the altitude of apogee spin rate and mass for a spherical. spacecraft having an initial mass of 25kg and a radius of 1m Continuum and free molecular theory will be used to. formulate the appropriate force as a function of the altitude . Motivation, Potential applications that intend to also use a lift perturbation to alter the spacecraft s trajectory include satellites. or space planes which will use an airfoil in the hypersonic flow regime to maneuver in Earth s atmosphere These. airfoils can be used for orbit maintenance by providing a lift vector normal to the orbit s velocity vector Thereby . active altitude adjustments using the proposed Magnus effect on a spinning spacecraft could serve as an orbital. maneuver capability without requiring conventional thrusters For example the added maneuverability of the Magnus. force can possibly allow the spacecraft to perform a skip reentry This reentry technique involves one or more. successive skips off the atmosphere to achieve greater entry range or to reduce the velocity of the spacecraft before. final entry which helps dissipate the heat at the surface For entry vehicles with a relatively low lift to drag ratio a. known strategy since the Apollo era for achieving long downrange is to allow the vehicle to skip out of the atmosphere. 7 In addition the Magnus phenomenon could be used to maintain a low perigee orbit and aid in performing in situ. atmospheric research in the low Ionosphere Thermosphere region This could be significantly more effective for. planets with higher atmospheric densities including Venus whose atmosphere is mostly made up carbon dioxide The. scenario of perigee maintenance allows for immediate benefits for Operationally Responsive Space ORS and Space. Reconstitution SR missions 8 Significantly to perform in situ research at low perigee altitudes a substantial. propulsion system would be needed to raise and lower the perigee 9 The Magnus maneuver could meet these. requirements , Equally important the Magnus effect could be used to perform active controlled deorbiting to improve predictions. of the impact location This could benefit satellites near EOL or for systems that fail to fully demise during reentry. allowing for a controlled deorbiting capability For example several events have previously occurred that illustrate. the importance of predicting the reentry location a priori For example in the article titled Assessing the Aviation Risk. from Space Debris and Meteoroids from the Space Safety Magazine it is explained how the uncontrolled reentry of. Russia s Phobos Grunt resulted in the closing of the European airspace for two hours Most importantly a study. conducted by the FAA following the disintegration of space shuttle Columbia in 2003 found that the probability of. an impact between Columbia debris and a general aviation aircraft was one in a hundred 10 According to the. Aerospace Corporation there are about 100 large man made objects that reenter the earth s atmosphere uncontrolled. each year 10 Furthermore if the predicted risk of human casualties exceeds a specified limit typically 01 per. reentry event a controlled reentry with prescribed reentry location has to be carried out 11 Similarly for satellites. partially surviving the reentry process destruction will occur forming a debris cloud where the time to impact the. ground or to reach the airspace can be short as shown in Figure 2 Thus there exists a need for improving knowledge. of the reentry impact location to mitigate the risk of collision between space debris and other ground stations which. could possibly be achieved using the Magnus force . American Institute of Aeronautics and Astronautics. Figure 2 The risk of collision between space debris and other ground stations 12 . II Literature Review, A literature review was first conducted to examine how an aerodynamic lift perturbation affects a satellite s orbit .
Ashenberg in 13 presents solutions for a flat plate satellite experiencing non constant aerodynamic coefficients by. using the Gaussian form for the Variation of Parameter VOP equations He describes that if a satellite has dominant. flat surfaces rotates at certain slow rates or has a large area to mass ratio the lift forces do not average out to zero . The lift perturbation is considered as a vector in the plane normal to the velocity pointing in any direction The. perturbations are projected in the normal direction given by hxV toward the inside of the orbit and calculations are. done assuming free molecular hyperthermal flow The orbit angular momentum is described by h whereas V is the. relative velocity of the satellite He describes how the lift acting in the orbital plane perturbs the eccentricity vector . while an orthogonal out of plane force perturbs the orientation of the orbital plane Significantly he states that since. the lifting force does not change the energy the semi major axis is perturbed by drag alone The general conclusion. is that time varying aerodynamic coefficients may cause various forms of secular orbital motion . Cook 2 explains that one can neglect the aerodynamic force for a satellite undergoing a rapid and uncontrolled. tumbling motion for most of their lifetime since the effects of the normal force to the velocity vector is averaged out. over one revolution However for satellites that remain stabilized for long intervals of time one must reexamine the. effect of the aerodynamic lift He assumes lift acts in the orbital plane and investigates two primary cases for a flat . plate satellite a constant lift to drag ratio followed by a trajectory that has a negative lift coefficient from perigee to. apogee and then a constant positive lift coefficient from apogee to perigee Similar to Ashenberg he considers the. simplest case of hyperthermal free molecule flow where the thermal accommodation coefficient 1 for which the. random thermal motion of the molecules is assumed negligible compared with the satellite s speed With complete. accommodation or with a thermal accommodation coefficient value of 1 the lift to drag ratio will be on the order of. 0 05 With no accommodation Cooke describes that the lift to drag ratio can be high as 2 3 and therefore the. importance of lift depends on the nature of the momentum exchange at the satellites surface Furthermore Cook goes. on to explain that since lift acts perpendicular to the satellite s velocity vector it can have no effect on the semi major. axis of the orbit Consequently one should only be concerned with variations of the eccentricity vector In order for. the orbital inclination to change a component of force normal to the orbital plane is required 14 For the constant. lift coefficient case Cook finds that the eccentricity remains constant and the only effect of lift is to rotate the major. axis For the discontinuous lift coefficient case Cook finds that the only secular perturbation is a decrease in the. eccentricity , Moore 15 also describes how satellites in stabilized attitudes may be subjected to steady or periodic lift giving. rise to perceptible perturbations in the orbital elements He uses the LaGrange equations of motion to study the effects. of lift and drag on the orbital elements and states that the precise determination of lift effects require either in situ. examination of the gas surface interaction or detailed analysis of orbital perturbations and spin rate data He describes. the hyperthermal free molecular flow as being where the mean free path of the molecules is very large compared with. the dimensions of the satellite and where the molecules have no random thermal motion Diffuse reflection is. American Institute of Aeronautics and Astronautics. significant at 200km 800km where atomic oxygen predominates and at higher altitudes the reflection mode may. theoretically approaches specular reflection , Hall 16 investigates multiple orbital schemes and maneuvers using electric propulsion along the satellite s. velocity vector to determine the feasibility of counteracting the drag force at a perigee of 100km He describes how. elliptic orbits utilizing a very low perigee can facilitate access to the surface and atmosphere at sub ionosphere. altitudes while counteracting drag using continuous electric propulsion Low perigee orbits has been studies for. interplanetary scientific missions and has a significant potential for remote sensing Similarly the current efforts in. this study is to counteract drag at a low perigee without using conventional thrusters or by using a continuous Magnus. effect Unnecessary consumption of fuel to maneuver the spacecraft for short term objectives may severely constrain. the life of the satellite 16 , Aerodynamic Lift on a Spinning Sphere. Due to the importance of lift on the gas surface interaction assumption a literature review was also done to. examine the expression for the Magnus force in the free molecular and continuum regime At high altitudes in orbit . the Knudsen is Kn 10 implying free molecule flow However as the satellite reenters to lower altitudes the Knudsen. becomes Kn 1 implying a continuous regime Wang in 17 determines the aerodynamic forces for free molecular. flow over a rotating sphere Most importantly he describe that in the free molecular regime the Magnus force exerts. a negative lift on the sphere Expressions are derived for the limiting case of hypersonic free molecular flow Wang. explains that if the temperature of the sphere is cold and the reflection is purely diffusive with complete. accommodation then the velocity of the reflected molecules is so small compared with the freestream that it may be. neglected Volkov 18 investigates the 3D rarefied gas flow past a spinning sphere in the transitional and near. continuum flow regimes numerically Volkov describes that in a rarefied gas flow in the absence of intermolecular. collisions the direction of the Magnus force is opposite to that in a continuum flow at small Reynolds numbers He. describes that the negative lift that occurs in the transitional region is attributable to the increase in the contribution of. the normal stresses to the Magnus force with a decrease in the Knudsen number The difference in the Magnus force. direction in the free molecular and continuum regime implies that in the transitional flow regime the Magnus force. depends significantly on the Knudsen number Moreover at a certain value of the Knudsen number this force. vanishes Volkov describes that with a decrease in the Knudsen number the Magnus lift coefficient should first. increase from 4 3 to the maximum value of 2 in the continuum flow regime at small Reynolds numbers and then. decrease to the limiting value corresponding to large Reynolds numbers . Rubinow et al 19 calculates the Magnus force in the continuum limit using the Navier Stokes equations assuming. small Reynolds number It is shown that at small Reynolds numbers the rotation of the sphere does not affect its drag. force coefficient In addition Rubinow et al states that in the continuum regime at small Reynolds numbers the. aerodynamic torque exerted on the spinning sphere is independent of the translational velocity of the sphere relative. to the fluid Thus performing this literature survey allows one to develop the appropriate expression for the Magnus. force in the free molecular limit and continuum regime To create a smooth transition between the lift coefficient from. 4 3 to the maximum value of 2 as the satellite descends into the atmosphere the hyperbolic tangent function was used. which will be discussed later , III Orbit Perturbations. To understand the orbital mechanics of satellites in LEO the equations of motion for the two body problem must. first be examined The physical motions of each planet was first addressed by Kepler where he summarized that 1 . the orbit of each planet is an ellipse with the sun at a focus 2 the line joining the planet to the sun sweeps out equal. areas in equal times and 3 the square of the period of a planet is proportional to the cube of its mean distance from. the sun 20 Newton then mathematically explained why planets and satellites followed an elliptical orbit by. combining his Law of Universal Gravitation and his Second Law of Motion resulting in Eq 1 This equation. describes the satellite s position vector as it orbits the earth and assumes that gravity is the only force acting on the. system the Earth is spherically symmetric the Earth s mass is much greater than the satellite s mass and the earth. and the satellite are the only two bodies in the system 21 To clarify r is the position of the satellite relative to Earth s. center and this differential equation is a second order nonlinear differential equation . 3 0 1 , A solution to the two body equation of motion for a satellite orbiting earth is the polar equation of a conic section.
21 In order to solve Eq 1 six constants of integration or initial conditions are required and thus one can define the. American Institute of Aeronautics and Astronautics. orbit with six classical orbital elements with one quantity varying with time as shown in Fig 3 A spacecraft s orbit. or trajectory is its path through space and an orbit is specified by a state vector which can be the position and velocity. of the spacecraft , Figure 3 Keplerian orbital elements of a satellite in an elliptic orbit 21 . A brief summary of the classical orbit elements as described in 21 includes . Semi Major Axis a defines the size of the orbit, Eccentricity e defines the shape of the orbit. Inclination i the angle between the angular momentum vector and unit vector Z. Right Ascension of the Ascending Node RAAN The angle from the vernal equinox to the ascending node . The ascending node is the point where the satellite passes through the equatorial plane moving from south to. Argument of Perigee The angle from the ascending node to the eccentricity vector measured in the. direction of the satellite s motion The eccentricity vector points from the center of the earth to perigee with. a magnitude equal to the eccentricity of the orbit . Mean anomaly M The fraction of an orbit period which has elapsed since perigee expressed as an angle . The mean anomaly equals the true anomaly for a circular orbit . Equations of Motion with Perturbations, A satellite will always remain in orbit and consequently its orbital elements will remain constant if gravitational. forces are the only force acting on it However when other perturbations are present the two body problem becomes. Eq 2 implying that orbital lifetime becomes finite . 2 , where is the resultant vector of all the perturbing accelerations Some of these perturbing acceleration terms include. atmospheric drag solar radiation pressure Earth s oblateness and other n body effect 20 In the solar system the. sum of the perturbing accelerations for all satellite orbits is at least 10 times smaller than the central force or two body. accelerations or aP 3 22 The non homogenous equation above implies that the semi major axis a orbit angular. momentum h and eccentricity e are not constants but satisfy 2 . American Institute of Aeronautics and Astronautics. 2a2 3 , , x 4 , x x 5 , Due to the non linear nature of Eq 2 with respect to r no closed form analytical solution exists and therefore it must.
be solved numerically to obtain and as functions of time . Methods of Solution, A perturbation is a deviation from the Keplerian motion and includes secular and periodic perturbations Secular. perturbations are those which the effects build up over time while periodic or cyclic perturbations are such that the. effects cancel after one cycle or orbit 22 Furthermore secular changes in a particular element very linearly over. time or proportionally to some power of time Periodic perturbations are either short or long term where short periodic. typically repeats on the order of the satellite s period or less and long periodic effects have cycles considerably larger. than one orbit period 22 Even in the absence of perturbations fast variables which change considerably during one. revolution include the mean true and eccentric anomalies However slow variables including semi major axis . eccentricity inclination node and argument of perigee change very little during one revolution 23 If there is no. perturbations all the slow variables would remain constant The largest perturbation is due to gravitation followed by. drag third body perturbations solar radiation pressure effects and smaller effects such as tides Third body effects. are perturbations caused by the attraction of the sun moon and other planets In addition solar radiation pressure is. when photons impact a satellite s surface and are reflected or absorbed . Techniques to solve the two body problem with perturbations encompass analytical and semi analytical methods . In using these methods the primary difference is whether one uses the satellite s position and velocity state vectors. or the orbital elements as the elements of state Typically analytical methods are faster but the expressions are truncated. to allow simpler expressions As a result the computational speed increases but accuracy decreases Numerical. approaches consists of numerically integrating the perturbing accelerations The numerical approach can also be. applied to the Variation of Parameter VOP equations in which case a set of orbital elements are numerically. integrated 22 , Furthermore the three main methods to solving the equations of motion with perturbations are special perturbation . general perturbation and semi analytic Special perturbation techniques including Cowell s method and Encke s. method uses numerical integration of the equations of motion including all perturbing accelerations 22 This. approach uses the position and velocity vectors of the satellite However the analytical approach uses the orbital. elements for integration while semi analytic methods uses a combination of numerical and analytical techniques Most. analytical and a few numerical approaches use the VOP form of the equations since the orbital elements in the two . body equation are changing Lagrange and Gauss both developed VOP methods to analyze perturbations Lagrange s. technique applies to conservative accelerations while Gauss s approach can also be implemented for non conservative. accelerations Conservative accelerations are explicitly a function of position only and there is no net transfer of energy. taking place and therefore the mean semi major axis of the orbit is constant However non conservative accelerations. are explicitly a function of both position and velocity including atmospheric drag outgassing and tidal friction effect. where energy transfer occurs thereby changing the semi major axis 22 Drag is a non conservative force and will. continuously reduce the energy of the orbit decreasing the semi major axis and period The orbit will become more. circular each revolution and will then rapidly spiral inwards due to the dense atmosphere Using the VOP technique . one can examine the effects of perturbation on specific orbital elements In the Gaussian VOP the rates of change of. the elements are explicitly expressed in terms of the disturbing forces Since a low perigee of 80km will be examined . the dominating perturbing force will be from drag and the Magnus effect allowing one to ignore other perturbations . Significantly one can see that the Magnus force will change as a function of time since it primarily depends on the. atmospheric density classifying it as a secular non conservative force . American Institute of Aeronautics and Astronautics. IV Implementation of Software, Numerical propagation of a satellite s trajectory using the Magnus effect would consist of many interacting. components including a numerical propagator that solves the equation of motion and a force model that evaluates the. effect of the Magnus force on the satellite Since the current study is examining the feasibility of the Magnus force . we decided to model its effect as a super efficient thruster for simplicity The Systems Tool Kit STK allows the user. to incorporate customer specific modeling into the computations by creating a plugin which provide a simple method. for customizing STK The equations of motion are integrated using the Runge Kutta Fehlberg method of 7th order. with 8th order error control 16 However before beginning to perform the simulations a test case was performed to. verify that the software implementation was correct The simulation validation case was taken from Hall 16 whom. used STK to examine the final altitude for a constant initial perigee altitude of 100km with an increasing apogee. altitude between 2 622km to 18 622km A 150mN of continuous thrust was fired along the velocity vector from. perigee to apogee The satellite was then allowed to coast back to perigee without the use of any thruster and this. sequence was repeated 100 times As shown in Figure 4 there is good agreement between the STK simulation and by. Hall giving confidence that the software was being executed correctly . Final Perigee, 20000 Final Apogee, Final Altitude km. 622 2622 4622 6622 8622 10622 12622 14622 16622 18622. Initial Apogee km, American Institute of Aeronautics and Astronautics.
Figure 4 STK Validation, Super Efficient Thruster Model. A plugin component is a user supplied software component called by the application at certain pre defined event. times within the computation cycle 24 The plugin is allowed to modify the computation by adding additional. considerations or modifying parameters A custom script is implemented using Visual Basic Scripting VBS that. pulls in the instantaneous density altitude and velocity to evaluate the magnitude of the Magnus force Ideally with. the real application of this concept the satellite will not be losing any mass To create a similar effect in STK an. exceedingly high specific impulse of 2x1012 s was created with a fuel mass of 5kg Therefore the mass of fuel. consumed for each simulation was negligible 3x10 13kg allowing one to approximately model the Magnus effect . Also theoretically one does not have the capability to incorporate a high spin rate on the actual satellite in STK Thus . a spin rate of 5000RPM and radius of 1m is assumed in order to evaluate the magnitude of the Magnus force which is. then implemented via a thruster , Three types of attitude motion include pure rotation coning and nutation The Magnus force is assumed to be in. pure rotation which is the limiting case where the rotation axis a principal axis and a geometrical axis are parallel or. anti parallel 25 As a result the angular momentum vector will lie along the same axis Thus with this assumption. the spin axis is assumed to lie perpendicular to the orbital plane resulting in the lift vector acting in the orbital plane . The Magnus force perturbation can be expressed as Eq 6 . 1 x , 3 , where , , 2 x 6 , where r is the radius of the sphere is the angular velocity V is the freestream velocity and is the freestream. density The lift coefficient for the Magnus force as described by 18 4 and 17 is negative in the free molecular. regime and depends on the momentum accommodation coefficient as shown below in Eq 7 . , To ensure that the Magnus force is always perpendicular to the satellite s velocity vector a new reference frame was. and its orbit angular, defined for the Magnus force that is based on the cross product of the velocity of the satellite .
American Institute of Aeronautics and Astronautics. momentum Please refer to the appendix which demonstrates that the orbit angular momentum is significantly. greater than the satellite s body angular momentum for a spin rate of 5000RPM . To create a realistic model and a smooth transition between the changing lift coefficients as the satellite s trajectory. descended from a free molecular regime to continuum regime the hyperbolic tangent function was used However in. order to decide what approximate altitudes the Magnus lift would change from negative to positive the Knudsen. number was first calculated using the expression below in Eq 8 taken from 26 . Kn 8 , 2 , where is the dynamic viscosity R is the specific gas constant D is the diameter of the sphere and T is the. temperature at a given altitude At altitudes higher than 100km free molecular conditions will prevail or when K n . 10 Subsequently the flow will move into transition flow as the Knudsen number decreases in the range of 0 1 K n. 10 and then into slip flow where the no slip boundary condition starts to break down or when 0 001 K n 0 1 . Examining Table 1 one can see that the continuum regime starts around 80km or where K n 0 001 assuming a. satellite radius of 1m As stated previously Volkov 18 describes that with a decrease in the Knudsen number the. value of Cl should first increase from 4 3 to the maximum value of 2 corresponding to the continuum flow regime at. small Reynolds number As a conservative approach the limiting case of hypersonic free molecular flow is assumed. and therefore the reflection is purely diffusive with complete accommodation 1 27 With a purely diffusive. assumption as opposed to a specular reflection the lift is small compared to drag and results in a conservative. approximation for the Magnus force , Table 1 Knudsen number at varying altitudes. Altitude km Kn, 100 0 0619, 86 0 0049, 80 0 0019, 70 0 0004. 66 0 0005,1 Hyperbolic Tangent Function, After calculating the altitude where continuity conditions prevailed the hyperbolic tangent function was developed to. create a smooth transition from the negative lift coefficient to the positive lift coefficient in the continuum regime as. described by the literature The developed function can be seen in Eq 9 and Fig 5 where x is the altitude of the. satellite , tanh 2 164 9 , American Institute of Aeronautics and Astronautics.
Figure 5 Hyperbolic Tangent Function creates a transition between Positive and Negative Magnus Lift . Thus the super efficient Magnus engine plugin is implemented using Eq 6 which evaluates the Magnus lift. coefficient based on the instantaneous altitude of the satellite in STK as seen in Eq 9 As a result the custom script. developed using Visual Basic Scripting VBS pulls in the instantaneous density and the altitude and velocity of the. satellite in order to evaluate the magnitude of the Magnus force described in Eq 6 . 2 STK Astrogator Settings, In the STK graphical user interface coefficients for solar radiation pressure are set to zero Drag is incorporated. into the simulations and is based upon the Jacchia Roberts Atmospheric density model . The coordinate system used in the simulations is the VNC Velocity Normal Conormal reference frame In this frame . the X axis is along the velocity vector V the Y axis is along the orbit normal or Y rxV and the z axis completes the. orthogonal triad The orbit epoch time is set to October 4th 2012 12 00 The Magnus thruster is modeled as a finite. maneuver which is effectively a propagate segment with thrust It uses the defined propagator to propagate the state. accounting for the acceleration due to thrust Each point calculated during the numerical simulation is added to the. satellite s ephemeris until a stopping condition is met 28 In STK s Astrogator two finite maneuvers are. implemented with the custom engine plugin to account for the change in the Magnus lift coefficient as the satellite. descends from a free molecular regime to a continuum regime Initially the satellite is not in the continuum regime. 84km and thereby the first maneuver puts the Magnus direction as equal to hxV to account for the negative Magnus. force Under 84km another finite maneuver is done to implement the thrust acting in the Vxh direction As previously. stated the mass is set to 20kg with 5kg of fuel defined with a cross sectional area of 3 14m 2 assuming a spherical. geometry with a radius of 1m As a note the mass of fuel needs to be defined in order for STK to perform the. simulations even if the fuel consumption is very low The drag coefficient is set of a value of 2 as described in 17 . that is based on the limiting case of hypersonic free molecular flow assumptions where one assumes the reflection is. purely diffusive with complete accommodation 1 Also for all simulations a decay altitude of 65km was used . 3 Correct Implementation of Formula, Before performing the required simulations a simple test case was performed to ensure that the custom engine. plugin was working correctly The Magnus thruster was programmed to pull in the atmospheric density altitude and. the velocity components of the satellite during each time step After pulling in the velocity components along the X . Y and Z axis the magnitude of the velocity vector was found With an assumed spin rate of 5000RPM and radius of. 1m and using the density and magnitude of the velocity output from STK one can plot the expected theoretical thrust. given by Eq 6 against the magnitude of the thrust output simulated in STK Looking at Fig 6 one can see that the. good agreement with the thrust ensures that the user defined thruster is accurately pulling in the density and velocity. as a function of time , American Institute of Aeronautics and Astronautics. 500, Theoretical, 850 870 890, 100 Time in Orbit min. Figure 6 Verifying Magnus Thruster Implementation is Correct. In addition as seen above in Fig 6 the lift coefficient is initially negative since the satellite is in the free molecular. regime However as the altitude decreases 84km we start to see a rapid increase in thrust and see that the thrust is. no longer negative This is expected since the satellite is now in the continuum regime One can see that the magnitude. of the thrust acting on the satellite starts to oscillate as shown in Fig 6 as a result of the interaction between the drag . gravitational force and Magnus effect To clarify as the satellite descents lower in the atmosphere the density. increases producing a larger Magnus thruster increasing the altitude of the satellite However as the altitude increases . the density decreases thereby reducing the effect of the Magnus thruster causing the thrust plot to have a sinusoidal. Spin Rate Required to Avoid Losing Height, After verifying that the Super efficient thruster was being implemented correctly a simple analysis based on 8 .
was then performed to examine the spin rate required for the satellite not to lose altitude using the equation for the. Magnus lift and drag as seen in Eq 10 and 11 This rough analysis gives one insight on the spin rate required. based on geometry mass and altitude For a conservative approximation the lift coefficient for a spinning sphere in. free molecular flow was used which can be found in 17 and 18 . 10 , 2 11 , where is the radius of the sphere is the density at a given altitude is the angular velocity of the sphere in. rad s V is the velocity of the sphere is the drag coefficient and A is the reference area Given a mass of 25kg and. a radius of 1m the angular velocity of the sphere is used as the independent variable in this example Assuming the. American Institute of Aeronautics and Astronautics. satellite travels through the atmosphere with a constant flight path angle of 10 the required spin rate as a function of. different radii at different altitudes can be found using the free body diagram of Fig 7 where the only assumed forces. acting on the satellite are lift drag and weight , Figure 7 Simplified Magnus Force Analysis in a Continuum Regime. Using the 1976 Standard Atmosphere Model a velocity of 7 5km s and a constant mass of 25kg the required spin. rate to avoid losing altitude is calculated by summing the forces in the x and y direction for different altitudes and is. plotted in Fig 8 As the angular velocity increases the radius required to produce the required lift to avoid losing. altitude decreases Also as the altitude increases the resulting low density requires an exceedingly large radius to. generate the required lift Examining this simple analysis might encourage one to believe that the Magnus effect is. impractical due to the high required spin rates However this study will demonstrate that in a low perigee altitude of. 80km a spin rate of 5000RPM and a 1m radius sphere is sufficient for delaying the reentry period assuming a decay. altitude of 65km , American Institute of Aeronautics and Astronautics. 40, 30 120km, 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000. Figure 8 Angular velocity and radius required to avoid losing height . V Results, To investigate the feasibility of using the Magnus effect on a spinning spacecraft to prolong its trajectory in a regime.
of considerable density we decided to first vary the altitude of apogee while keeping the altitude of perigee at 80km . Subsequently the next step involved changing the magnitude of the Magnus thruster by theoretically increasing the. rotational speed of the satellite Finally a simulation is conducted at different masses to examine the effect on the. feasibility of the Magnus effect , Maintaining Altitude of Perigee. Performing initial simulations in STK demonstrated that the Magnus effect was only effective at altitudes around. 80km due to the increase in atmospheric density Thus the first analysis examined the effect of holding the altitude. of perigee constant at 80km while increasing the altitude of apogee or eccentricity of the orbit as shown in Table 2 . Table 2 List of Orbital Elements for an Altitude of Perigee 80km. Apogee Altitude e i deg deg M deg , 145 18 0 005 40 0 0 180. 177 88 0 008 40 0 0 180, 210 74 0 010 40 0 0 180, 411 46 0 025 40 0 0 180. 760 08 0 050 40 0 0 180, 1127 54 0 075 40 0 0 180. 1515 42 0 100 40 0 0 180, 2359 62 0 150 40 0 0 180.
American Institute of Aeronautics and Astronautics. 3309 34 0 200 40 0 0 180, 4385 70 0 250 40 0 0 180. Examining the results in Fig 9 one can see that using 65km as the decay altitude the Magnus effect approximately. doubles the amount of time in orbit assuming a spin rate of 5000RPM Therefore there might be the possibility that. the Magnus effect could counteract drag at a low perigee without using conventional thrusters This extension of time. on orbit could possibly be used by a spacecraft to maneuver itself to an area that will reduce the impact of collisions. with the airspace Likewise the extension of time on orbit could be used to maintain a low perigee orbit and aid in. performing in situ atmospheric research in the low Ionosphere Thermosphere region Significantly this could be more. effective for planets with higher atmospheric densities . Time on Orbit min , 0 1000 2000 3000 4000 5000, Initial Apogee km. Figure 9 Amount of time on orbit with and without Magnus Thruster at 80km Perigee . Different RPM, Next the effect of changing the angular velocity in Eq 6 was performed to see the effect on the time in orbit . The first set of orbital parameters in Table 2 apogee 145 18km e 0 005 was chosen as the set to be analyzed . Examining Fig 10 without the Magnus thruster the time in orbit is around 20min However with the Magnus. Thruster enabled with a spin rate of 5000 RPM the time in orbit until decay is extended to 60min Furthermore as. the spin rate is increased to 10 000 RPM no decay is seen in the orbit within the allotted 20 000min simulation time. and the satellite is seen to oscillate at an altitude of 66km . American Institute of Aeronautics and Astronautics.


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