Efficient Orbit Transfer: Use of Electric Propulsion for Orbit Raising

Efficient Orbit Transfer: Use of Electric Propulsion for Orbit Raising Software Designed to Provide Preliminary Sizing of Power and Propulsion Systems 16.851 Satellite Engineering Motivation Software Module Electric propulsion systems offer the capability for mass-efficient orbit transfers. The specific impulse for electric propulsion is much higher than for chemical propulsion, ranging from 1,500 to 20,000 seconds;1 however, electric propulsion provides much lower thrust than chemical propulsion. This results in much longer spacecraft maneuver times for a given change in velocity. If short maneuver times are not critical, electric propulsion may lend itself to be used as the propulsion system for orbit transfer maneuvers. Test Case Script Requirements The MATLAB script electric_propulsion.m is used to simplify the use of the primary software modules, and to run pre-configured test cases to produce the results presented in this report. Description of the code The script initializes the orbit transfer and spacecraft properties, calls each of the primary software modules in turn, and plots the results. In summary, the script performs the following functions: Problem Statement Create a software module that sizes the power and electric propulsion systems for a spacecraft, given the spacecraft mass, initial and final circular orbit radii, and a specified transfer time, assuming constant tangential thrust. Use this module to characterize the dependence of propulsion and power system mass on orbit transfer requirements. Investigate this dependency for several different types of electric propulsion, such as pulsed plasma thrusters (PPT), Xenon ion propulsion, and Xenon Hall-effect propulsion. Introduction • Set inputs: spacecraft mass, propulsion type, transfer time, initial radius, final radius. • Get propulsion system properties from propulsion_properties.m. • Determine the thrust and the orbit characteristics for the transfer maneuver using ep_orbit.m. • Size the power and propulsion systems to provide the needed thrust and ∆v, using propulsion_power.m. • Plot the results. Typing electric_propulsion at the MATLAB prompt runs the test case and produces the output shown in this report. Inputs to the software module include the initial and final orbit radii and the orbit transfer time. The module determines the constant propulsive force required to move the spacecraft from the initial orbit to the final orbit in the specified time. Using this constant force, and given the type of propulsion system, the propulsion system is sized by determining the total propulsion system mass required for the orbit transfer. For this propulsion system, the batteries and solar arrays required to support the maneuver are sized. The performance results are compared for several types of electric propulsion systems. Propulsion Properties Module Requirements The MATLAB module propulsion_properties.m sets the propulsion system-specific values based on the type of propulsion system selected. Description of the code The desired type of propulsion system is passed to the module as a character string (e.g. ‘ion’). Propulsion system constants such as specific impulse and 2 efficiency for the specified type of electric propulsion system are returned in a data structure. This structure is a required input for the electric_propulsion.m and propulsion_properties.m modules. Outputs thrust [N]: constant thrust magnitude required to complete the specified maneuver. radii [m]: history of orbit radii. Inputs period [s]: history of orbital period. p_system: a string specifying which propulsion system is to be used. Valid values are ‘ion’ (Xenon Ion), ‘hall’ (Xenon Hall), and ‘ppt’ (Pulsed Plasma Thrusters). eclipse [s]: history of eclipse entry and exit times. deltav [m/s]: history of applied ∆v. Derivation of the spiral orbit 'v equation2 Outputs For thrust T<7000 hr PPT 1000 7 4000 N*s Propulsion System power.m_cells_req (kg): This output is the total mass of the solar arrays required to power the propulsion system as well as charge the batteries when in sunlight. power.m_batt_req (kg): This output is the total mass of the batteries needed to provide power to the propulsion system during the worst case eclipse (longest duration). propulsion.m_thruster (kg): This output is the mass of the thrusters in the chosen propulsion system. Table 2. Additional propulsion system constants4 Misc. PPU Thrust Thruster Propulsion Mass Mass Range5 Mass System (kg/kW) (kg/kW) (kg/kW) (N) Xe Ion .01 .20 4.5 8 10 Xe Hall .08 .20 2.5 8 10 PPT .001 .10 120 110 small propulsion.m_ppu (kg): This output is the mass of the power processing unit in the chosen propulsion system. propulsion.m_misc (kg): This output is the mass of miscellaneous components in the chosen propulsion system. propulsion.m_propellant (kg): This output is the mass of the propellant in the chosen propulsion system. Theory & Equations Two additional constants used are used in the calculations for sizing the power system. These constants, Xe and Xd, are the efficiencies of the electrical paths from the solar arrays through the batteries to the loads and the path directly from the Based on the inputs to the module, the first desired quantity to be calculated is the required power for the propulsion system. In order to calculate the required power, the mass flow rate of the propellant must be first determined. This is done using Equation 19 below.7 5 F m = I sp g equation is assumed to be an end-of-life (EOL) power for the solar arrays as far as the orbit raising maneuver is concerned. (19) ( p0 ) EOL = η cells (1358 mW2 ) I d In the above equation, F is the propulsion system constant thrust, Isp is the specific impulse of the propulsion system, and g is the gravitational acceleration constant. ,Q WKH DERYH HTXDWLRQ cells is the efficiency of the GaAs cells, which is approximately 18%. In addition, the quantity Id is the inherent degradation of the solar cells. This is assumed to be 0.77.9 Next, the required power for the propulsion system is determined in Equation 20.8 P= It should also be mentioned here that the solar arrays are assumed to track the sun and therefore it is assumed that the radiation from the sun is perfectly incident at all times on the solar arrays. This means that any reduced performance to any incidence angle to the solar radiation is ignored. 2 F 2m η (20) is the efficiency of the In the above equation, propulsion system, as reported in Table 1. Next, the required solar array area for each orbit during the transfer maneuver is determined. This is shown in the equations below. The next required quantity to be determined is the power required to be generated by the solar arrays for each orbit. This would include the power provided to the propulsion system as well as the power provided to the batteries for charging. The following equation is used to determine this required power for each “orbit” portion of the orbit transfer. Psai Td ⎞ ⎛ Te P⎜⎜ i + i ⎟⎟ Xe Xd ⎠ = ⎝ Tdi Acellsi = Psai ( p0 ) EOL A _ cells _ req = max( Acellsi ) where i = 1,2,... (22) (23) (24) Equation 24 show how module determines the required solar array area to be the maximum of all the areas calculated for all orbit portions of the spiral orbit raising maneuver. (21) In Equation 21, Te and Td are the durations of the eclipse and daylight times for each orbit portion of the spiral orbit transfer. Each orbit is determined by the time at which the satellite enters the earth’s eclipse until the time it next enters the eclipse. These orbits are denoted in the equation with the subscript i. It can be seen that only the eclipse and daylight times change with each successive orbit. Next, the required solar array mass can be determined. This is done using the following equation from SMAD.10 m _ cells _ req = 0.04 ∗ max( Psai ) (25) Next, the battery capacity is determined to allow for sizing of the batteries for the spacecraft. It is assumed that NiH2 batteries are being used with a specific energy density of approximately 50 W*hr/kg.11 It is also assumed that only a single battery is being used. Equation 26 below is used to determine the battery capacity required for each orbit eclipse during the orbit transfer. Normally, the numerator of Equation 21 has different power values for daytime and eclipse, but in this case, since the power required by the propulsion system does not vary from daylight to eclipse, this quantity is held constant at the value P, determined earlier. Next, the power generated per area by the solar arrays is determined. This calculation is shown in Equation 22. It should be noted here that Gallium Arsenide solar cells are being used for the purposes of this software module. In addition, it should be noted here that it is assumed that beyond the inherent inefficiencies in the solar cells, no additional degradation is assumed for the solar cells during the orbit transfer. This assumption is made because the orbit transfer is relatively short compared to the life of the satellite and the arrays will likely not degrade significantly during this orbit raising maneuver. Therefore, the power determined in the following C ri = PTei ( DOD)n (26) In the above equation, DOD is the depth-of-discharge of the nickel-hydrogen batteries. This assumed to be 75% since the number of cycles the batteries will need to be cycled for throughout the orbit transfer will not degrade the depth-of-discharge capability of the batteries to any significant amount.12 6 The quantity n is the power transfer efficiency to the batteries. This is assumed to be 0.9.13 24 22 Next, the required battery mass is determined. This is shown in the following two equations. 20 m _ batt i = C ri 50 Wkg*hr Orbital period [hours] 18 (27) 16 14 12 10 m _ batt _ req = max(m _ batt i ) (28) 8 6 Finally, the propulsion system mass is determined using information from Table 2. The required figures from the table are multiplied by the propulsion system power to determine the masses of all the components of the propulsion system. In addition, the mass of the propellant is determined from the following equation. m propellant = m * t _ transfer 4 0 100 200 300 Orbit number 400 500 600 Figure 1. Orbit period vs. orbit number The data in Figure 1 agree with what is expected for a spiraling orbit transfer, where the radius is constantly increasing. Figure 2 shows how the orbit radius varies with the orbit number for the test case. (29) Once the propellant mass is determined, the total propulsion system mass is calculated as the sum of the thruster, power processing unit, miscellaneous, and propellant masses. 4 4.5 x 10 4 Results 3.5 Radius [km] The software module was run for a specific test case of raising a satellite from a circular parking orbit at an altitude of 10,000 km to a geosynchronous orbit. In addition, the transfer time was varied by the user within the ranges allowed for Xenon ion and Xenon Hall propulsion systems. 3 2.5 2 It should be noted that the orbit number being used in many of the following charts is how the spiral orbit transfer is broken up into individual parts. Each orbit in the transfer is determined from the time the satellite enters the earth’s eclipse until it reenters the eclipse. 1.5 0 100 200 300 Orbit number 400 500 600 Figure 2. Orbit radius expansion with orbit number It can be seen in Figure 2 that the orbit radius increases from the initial parking orbit until it reaches the orbit radius for geosynchronous orbit. In Figure 1, it can be seen how the orbital period changes with the orbit number, as the radius of the orbit increases. Figure 3 shows the time spent in eclipse during each orbit of the spiral transfer, and Figure 4 shows the fraction of each orbit during which the satellite is in eclipse. 7 4200 16 4000 14 12 Single-orbit ∆ V [m/s] Time in eclipse [s] 3800 3600 3400 3200 10 8 6 3000 4 2800 2600 0 100 200 300 Orbit number 400 500 2 600 Figure 3. Eclipse time vs. orbit number 100 200 300 Orbit number 400 500 600 Figure 5. Single-orbit deltaV vs. orbit number As expected, the ∆v per orbit imparted by the electric propulsion system is continuously increasing as time elapses during the transfer orbit. This occurs because the orbital radius is constantly increasing throughout the orbit transfer due to the constant thrust provided by the electric propulsion system. 0.13 0.12 0.11 0.1 Eclipse fraction 0 0.09 Figure 6 shows a perspective view of the complete path traversed during the spiral orbit transfer, from the initial orbit to the final orbit. The color of the spiral orbit transfer path varies from red at the 10,000 km altitude parking orbit to blue at the final, geosynchronous orbit. 0.08 0.07 0.06 0.05 0.04 2.5 3 3.5 4 4.5 5 5.5 Normalized radius (r/RE) 6 6.5 7 Figure 4. Eclipse fraction vs. normalized radius 1 0 As seen in Figure 3, as the radius increases, the time spent in eclipse should increase; however, as seen in Figure 4, the ratio of time spent in eclipse to the orbital period should decrease as the orbit radius increases. It should also be noted from Figure 4 that the orbit transfer ends when the orbit radius is 6.5 times the radius of the Earth, in geosynchronous orbit. -1 6 4 5 2 0 0 -2 -4 -6 -5 Figure 6. Orbit transfer path Figure 5 displays how the ∆v per orbit provided by the propulsion system varies for each successive orbit during the transfer. Approximately 600 orbits are required to reach the final desired orbit, resulting in the densely-packed path shown above. Figure 7 shows a blown-up view of a portion of the spiral orbit track, in which individual spiral tracks are visible. 8 acceleration. This effect is likely due in part to the assumption that the initial spacecraft mass is independent of the type of propulsion system used. 4 x 10 3.2 3.1 Figure 9 shows how the total mass of the propulsion and power systems varies as a function of the specified orbit transfer time. Distance [km] 3 2.9 2.8 600 2.7 Ion Hall 580 Propulsion and power mass [kg] 2.6 2.5 2.5 2.6 2.7 2.8 2.9 Distance [km] 3 3.1 3.2 4 x 10 Figure 7. Blown-up view of spiral orbit path It can be seen from Figure 7 that the change in orbital radius in each successive orbit is very small as compared to the orbital radius. This behavior is expected for low-thrust, long term maneuvers. 560 540 520 500 480 460 3500 Figure 8 shows the required thrust from the propulsion system as a function of the user-specified transfer time. 4000 4500 5000 5500 Transfer time [hr] 6000 6500 7000 Figure 9. Total propulsion and power system mass as a function of transfer time It can be seen in Figure 9 that no designs exist for transfer times less than approximately 5000 hours. This is due to the fact that neither the Xenon Ion nor the Hall propulsion systems are capable of producing high enough thrust to complete the orbit transfer maneuver in less time than 5000 hours. The maximum thrust capability of the two propulsion systems is shown explicitly in Figure 8 with a red horizontal line. It should also be mentioned that the preceding figures do not show any data for the pulsed plasma propulsion system. It was determined during the testing of this software module that the PPT propulsion system does not have the capability to provide the ∆V necessary to perform the orbit transfer maneuver test case. Although PPT results are not shown, the software module is designed to handle such types of propulsion systems, and will return performance results given input requirements appropriate to the types of propulsion systems being considered. Figure 8. Dependency of thrust on transfer time As expected, the thrust required to complete the transfer in the specified time decreases as the transfer time increases, for both the Xenon Ion and Hall propulsion systems. This makes sense because the required acceleration (and therefore thrust) should decrease as the transfer time increases. Table 3 summarizes the final results for the sizing of the power and propulsion systems for the orbital transfer test case, using the Xenon Ion propulsion system. Table 4 summarizes the results for sizing the system using the Xenon Hall effect propulsion system. It can also be seen in Figure 8 that the required thrust for each transfer time differs for the ion and Hall propulsion systems. This is due to the differences in mass flow rate of propellant out of the thrusters. The Xenon Hall thrusters have a lower Isp than the Ion thrusters, which corresponds to a higher mass flow rate to produce a given thrust magnitude. The mass of the spacecraft decreases more rapidly with higher propellant mass flow rates, leading to more rapidly decreasing spacecraft mass, with the result that lower average thrust is requirement to achieve a given 9 Conclusion Table 3. Sizing results using Xenon Ion propulsion Solar cell area Solar cell mass Battery mass Thruster mass PPU mass Misc. mass Propellant mass Total mass 2 19.0 m 142.7 kg 81.5 kg 6.4 kg 20.4 kg 25.5 kg 223.3 kg 499.8 kg A software module was created to size the propulsion and power systems for a spacecraft that uses electric propulsion for simple orbital transfers. The module requires the spacecraft mass, the initial and final circular orbit radii, a transfer time, and a propulsion system type. The software then determines the constant tangential thrust required to complete the orbit transfer in the specified time, the path of the satellite during the transfer maneuver, and the masses of the power and propulsion systems. This tool is useful for preliminary sizing of propulsion and power requirements for a satellite when electric propulsion is used for orbital transfer. Although the tangential thrust approach to orbital transfer is suboptimal, it is useful for rapid comparison of the relative performance of different types of propulsion system. Table 4. Sizing results using Xenon Hall propulsion Solar cell area Solar cell mass Battery mass Thruster mass PPU mass Misc. mass Propellant mass Total mass 2 26.2 m 197.0 kg 112.6 kg 15.8 kg 28.1 kg 35.2 kg 130.9 kg 519.5 kg This tool can be easily extended to compare an arbitrary number of different types of electric propulsion system types. If the transfer scenario investigated is not within the capability of a propulsion system, results for that system are not displayed. Future Work Based on these results, the Ion and Hall propulsion systems appear to have similar performance for the test case transfer scenario. The Ion system is slightly less massive than the Hall-effect system, even though it requires 70% more propellant to accomplish the transfer. The mass savings is due to the lesser power requirement for the ion engine, which translates directly into smaller, less massive solar arrays and batteries. One major task that could be undertaken in the future to improve and expand the capability of this software module would be to allow for plane changes during the orbit transfer. Another task would be to implement the equations for optimal thrust. Either one of these tasks would involve a significant amount of work (appropriate for Master’s thesis-level research14), but would result in an extremely useful tool which could be used to provide detailed results for a greater number of orbit transfer scenarios. 10 Appendix A: MATLAB source code The MATLAB files containing the implementations of the equations and relations described in this document are listed below. To run the test case, simply type electric_propulsion at the MATLAB prompt. electric_propulsion.m % constants R_earth = 6378.1363; % [km] Earth average radius geo = 42164.169637; % [km] geostationary, from STK. fignum = 0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % begin user-configurable inputs % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % initial spacecraft mass m0 = 2000; % propulsion system type (’ion’, ’hall’, ’ppt’) prop_type = ’ion’; % orbital transfer time ttime = 3600*24*250; % initial orbital radius r0 = R_earth + 10000; % final orbital radius r1 = geo; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % end user-configurable inputs % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % get the propulsion system properties prop = propulsion_properties(prop_type); % get the orbital transfer properties [thrust, radii, period, eclipse, deltav] = ep_orbit(m0, prop, ttime, r0*1000, r1*1000); % size the propulsion and power systems [propulsion, power] = propulsion_power(thrust,eclipse,prop,ttime); %%%%%%%%%%%%%%%%%%% % display results % %%%%%%%%%%%%%%%%%%% radii = radii/1000; % change from [m] to [km] orbits = 1:length(radii); % orbit numbers etime = eclipse(2,:)-eclipse(1,:); % time spent in eclipse fignum=fignum+1; figure(fignum) plot(orbits, period/3600); xlabel(’Orbit number’) ylabel(’Orbital period [hours]’); fignum=fignum+1; figure(fignum) plot(orbits, radii); xlabel(’Orbit number’); 11 ylabel(’Radius [km]’); fignum=fignum+1; figure(fignum) plot(orbits, etime); xlabel(’Orbit number’); ylabel(’Time in eclipse [s]’); fignum=fignum+1; figure(fignum) plot(orbits, deltav); xlabel(’Orbit number’); ylabel(’Single-orbit {\Delta}V [m/s]’); fignum=fignum+1; figure(fignum) plot(radii/R_earth, etime./period); xlabel(’Normalized radius (r/R_E)’); ylabel(’Eclipse fraction’); % plot all the orbit tracks around the Earth fignum=fignum+1; figure(fignum) t = 0:pi/40:2*pi; x = cos(t); y = sin(t); z = zeros(size(x)); [sx,sy,sz] = sphere(20); colormap(’white’); h1 = surf(sx,sy,sz); set(h1,’EdgeColor’,[.5 1.0 .5]); set(h1,’FaceColor’,[.8 1.0 .8]); hold on c = ’bgrcmyk’; scale = radii/R_earth; for i=orbits h2 = plot3(scale(i)*x,scale(i)*y,scale(i)*z); r = 1.0 - i/orbits(end)/2; b = 0.5 + i/orbits(end)/2; set(h2, ’Color’, [r 0.3 b]); end hold off axis equal; % zoom in to see some of the tracks closer up fignum=fignum+1; figure(fignum) for i=orbits h3 = plot(radii(i)*x,radii(i)*y); r = 1.0 - i/orbits(end)/2; b = 0.5 + i/orbits(end)/2; set(h3, ’Color’, [r 0.3 b]); hold on end hold off axis equal; maxrad = max(radii); axis([maxrad/sqrt(3) 1.1*maxrad/sqrt(2) maxrad/sqrt(3) 1.1*maxrad/sqrt(2)]); xlabel(’Distance [km]’) ylabel(’Distance [km]’) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % trade over prop type and transfer time % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % propulsion system type (’ion’, ’hall’, ppt’) 12 prop_types = {’ion’, ’hall’}; % orbital transfer time ttimes = 3600*24*[150:20:290]; data = []; for i=1:length(prop_types) prop_type = prop_types{i}; for j=1:length(ttimes) ttime = ttimes(j); % get the propulsion system properties prop = propulsion_properties(prop_type); % get the orbital transfer properties [thrust, radii, period, eclipse, deltav] = ... ep_orbit(m0, prop, ttime, r0*1000, r1*1000); % size the propulsion and power systems [propulsion, power] = propulsion_power(thrust,eclipse,prop,ttime); % save the interesting data data(i,j).thrust = thrust; data(i,j).time = ttime; if ~isempty(propulsion) & ~isempty(power) data(i,j).mass = propulsion.m_thruster + propulsion.m_ppu + ... propulsion.m_misc + propulsion.m_propellant + ... power.m_cells_req + power.m_batt_req; else % no design was feasible for this case data(i,j).mass = NaN; end end end times = []; thrusts = []; masses = []; % sort the data into standard vectors for i=1:length(prop_types) times = [times cat(2,data(i,:).time)’]; thrusts = [thrusts cat(2,data(i,:).thrust)’]; masses = [masses cat(2,data(i,:).mass)’]; end fignum=fignum+1; figure(fignum) plot(times/3600,thrusts); xlabel(’Transfer time [hr]’); ylabel(’Required thrust [N]’); legend(’Ion’,’Hall’); fignum=fignum+1; figure(fignum) plot(times/3600,masses); xlabel(’Transfer time [hr]’); ylabel(’Propulsion and power mass [kg]’); legend(’Ion’,’Hall’); 13 propulsion_properties.m function [properties] = propulsion_properties(p_system) %PROPULSION_PROPERTIES sets propulsion system properties % % [PROPERTIES] = PROPULSION_PROPERTIES(P_SYSTEM); % Returns a structure containing properties for the selected % type of propulsion system. This structure is used as an % input to EP_ORBIT and PROPULSION_POWER. % % Inputs % P_SYSTEM A string describing the type of propulsion system % to use. Valid strings are ’ION’, ’HALL’, or ’PPT’. % % Outputs % PROPERTIES A structure containing propulsion system % characteristics in the following fields: % % type - descriptive string % Isp - specific impulse (s) % eta - efficiency % lifetime - lifetime (hours) % th_mass - thruster mass (kg/kW) % ppu_mass - power processor unit mass (kg/kW) % misc_mass - misc. prop. system mass (kg/kW) % max_thrust - Maximum thrust allowed (N) % min_thrust - Minimum thrust allowed (N) % % An empty set is returned if P_SYSTEM is not a % valid string. % William Nadir % 16.851: Satellite Engineering % 10/26/2003 properties = []; switch lower(p_system) case ’ion’ % type Isp eta lifetime th_mass ppu_mass misc_mass max_thrust min_thrust Xenon Ion propulsion system = ’ion’; = 2800; % specific impulse (s) = 0.65; % efficiency = 10000; % lifetime (hours) = 4.5; % thruster mass (kg/kW) = 8; % power processor unit mass (kg/kW) = 10; % miscellaneous prop system mass (kg/kW) = .2; % Maximum thrust allowed (N) = .01; % Minimum thrust allowed (N) case ’hall’ % Xenon Hall type = ’hall’; Isp = 1600; eta = 0.50; lifetime = 7000; th_mass = 2.5; ppu_mass = 8; misc_mass = 10; max_thrust = .2; min_thrust = .08; case ’ppt’ type Isp eta lifetime propulsion system % % % % % % % % % Pulsed plasma = ’ppt’; = 1000; % = 0.07; % = 4000; % specific impulse (s) efficiency lifetime (hours) thruster mass (kg/kW) power processor unit mass (kg/kW) miscellaneous prop system mass (kg/kW) Maximum thrust allowed (N) Minimum thrust allowed (N) (Teflon) propulsion system specific impulse (s) efficiency lifetime (N*s) 14 th_mass ppu_mass misc_mass max_thrust min_thrust = = = = = 120; 110; 0; .1; .001; % % % % % thruster mass (kg/kW) power processor unit mass (kg/kW) miscellaneous prop system mass (kg/kW) Maximum thrust allowed (N) Minimum thrust allowed (N) otherwise disp(’Invalid input for p_system’); return end properties.type = properties.Isp = properties.eta = properties.lifetime = properties.th_mass = properties.ppu_mass = properties.misc_mass = properties.max_thrust= properties.min_thrust= type; Isp; eta; lifetime; th_mass; ppu_mass; misc_mass; max_thrust; min_thrust; ep_orbit.m function [thrust, radii, per, eclipse, deltav] = ep_orbit(m0, prop, ttime, r0, r1) %EP_ORBIT Calculates circular orbit data for electric propulsion maneuvers % % [THRUST, RADII, PERIOD, ECLIPSE, DELTAV] = EP_ORBIT(MASS, PROP, TIME, R0, R1) % % Determines the constant tangential thrust needed to expand a circular orbit % of radius R0 to radius R1 in the specified time TTIME. Calculates the start % and end times of each eclipse period, assuming an orbit in the plane of the % ecliptic. Solves for the constant direction (tangential), constant thrust % case, rather than the optimal case, which is beyond the scope of this project. % % Inputs % MASS [kg] Initial spacecraft mass % PROP [-] Propulsion data structure (output of PROPULSION_PROPERTIES) % TIME [s] Required orbital transfer time % R0 [m] Initial circular orbit radius % R1 [m] Final circular orbit radius % % Outputs % THRUST [N] Constant thrust required to complete the maneuver % RADII [m] History of orbit radii % PERIOD [s] History of orbit period % ECLIPSE [s] Eclipse enter and exit times for each orbit % DELTAV [m/s] Delta V applied through each orbit % constants mu = 3.986004415e14; % [m^3/s^2] R = 6378136.3; % [m] Earth radius g = 9.80665; % [m/s^2] gravitational acceleration at Earth radius % propellant specific impulse and exit velocity Isp = prop.Isp; c = Isp*g; % check for invalid inputs if (m0 <= 0) error(’Initial mass must be positive’); end if (Isp <= 0) error(’Isp must be positive’); 15 end if (ttime <= 0) error(’Transfer time must be positive’); end if (r0 <= R) error([’Initial radius must be greater than Earth radius (’ num2str(R) ’ m)’]); end if (r1 < r0) error(’Final radius must be greater than or equal to initial radius’); end % constant thrust required to perform this maneuver in the specified time thrust = (m0*c/ttime)*(1-exp((sqrt(mu/r1)-sqrt(mu/r0))/c)); % initial timein = timesun = orbit = cdeltav = eclipse = deltav = r = r0; m = m0; values 0; 0; 0; 0; []; []; % iterate through the orbits, saving data while (r lifetime*3600 disp([’ERROR: transfer time (’ num2str(t_transfer/3600) ... ’ hr) exceeds life of propulsion system (’ num2str(lifetime) ’ hr).’]); validity = 0; end case {’ppt’} if t_transfer > lifetime/thrust disp([’ERROR: transfer time (’ num2str(t_transfer/3600)... 17 ’ hr) exceeds life of propulsion system (’ num2str(lifetime/thrust/3600)... ’ hr) at ’ num2str(thrust) ’ N thrust.’]); validity = 0; end end % is thrust within the allowed range? if thrust > max_thrust disp(’ERROR: thrust is over the capability range of propulsion system’) validity = 0; end if thrust < min_thrust disp(’ERROR: thrust is under the capability range of propulsion system’) validity = 0; end % this lets all error messages display before returning if (~validity) return end g = 9.81; % gravitational acceleration constant (m/s^2) % This is the constant power required for the electric propulsion system to % get the satellite from orbit A to orbit B mass_flow_rate = thrust / (Isp * g); % (kg/s) prop_power_continuous = (thrust^2) / (2 * mass_flow_rate * eta); % (Watts) % Constants for direct energy power control system Xe = 0.65; Xd = 0.85; eta_cells = 0.18; % cell efficiency for GaAs cells (conservative = 18%) I_degradation = 0.77; % inherent degradation of solar cells p_sun = 1358; % solar input power density (W/m^2) % Here we calculate the energy collection effectiveness of GaAs solar cells % assuming no degradation over the time of orbit transfer (short time span) power_per_area = eta_cells * p_sun * I_degradation; % (W/m^2) % Initialize parameters for for loop to utilize ’orbit’ vector data [A, B] = size(orbit); % The number of columns in the ’orbit’ array is the number of orbits + 1 N_orbit = B - 1; % Here the ’orbit’ array is analyzed to determine the total eclipse and day % times for each ’orbit’ in the spiral orbit transfer for I = 1:N_orbit eclipse_times(I) = orbit(2,I) - orbit(1,I); % (seconds) day_times(I) = orbit(1,I+1) - orbit(2,I); % (seconds) end % Here the depth-of-discharge (DOD) for the battery is defined (NiH2 % battery) DOD = .80; % Here the battery charging efficiency is defined (90%) charge_efficiency = 0.9; 18 % Here the battery specific energy density is defined for NiH2 batteries sp_energy_density = 50; % (W*hr/kg) % this for loop determines the power required for each portion of the % spiral orbit transfer as well as the solar array area and battery mass % required for each portion of the orbit transfer for M = 1:N_orbit % P_sa is the required power for the propulsion system for one entire % orbit (Watts) P_sa(M) = (((prop_power_continuous * eclipse_times(M)) / Xe) + ... ((prop_power_continuous * day_times(M)) / Xd)) / day_times(M); A_cells(M) = P_sa(M) / power_per_area; % required area of solar arrays (m^2) m_cells(M) = 0.04 * P_sa(M); % estimated mass of solar arrays (kg) % Here the required battery capacity is calculated for one battery with % an efficiency of 0.9 for each eclipse during the spiral orbit % transfer (W*hr) Cr(M) = (prop_power_continuous * (eclipse_times(M)/3600)) / ... (DOD * charge_efficiency); % Here the required battery mass is determined m_batt(M) = Cr(M) / sp_energy_density; % (kg) end % required solar cell area to provide power for the orbit transfer (m^2) A_cells_req = max(A_cells); % required solar array mass (kg) m_cells_req = max(m_cells); % required battery mass to provide power for the orbit transfer (kg) m_batt_req = max(m_batt); m_thruster = th_mass * prop_power_continuous / 1000; m_ppu = ppu_mass * prop_power_continuous / 1000; m_misc = misc_mass * prop_power_continuous / 1000; m_propellant = mass_flow_rate * t_transfer; % power output structure power.A_cells_req = A_cells_req; power.m_cells_req = m_cells_req; power.m_batt_req = m_batt_req; % propulsion output structure propulsion.m_thruster = m_thruster; propulsion.m_ppu = m_ppu; propulsion.m_misc = m_misc; propulsion.m_propellant = m_propellant; 19 % % % % (kg) (kg) (kg) (kg) References 1 http://web.mit.edu/dept/aeroastro/www/labs/SPL/electric.htm, MIT Space Propulsion Lab Website, 2002. 2 Course notes, AA420, University of Washington Dept. of Aeronautical and Astronautical Engineering, 1999. 3 Martinez-Sanchez, Manuel, Spacecraft Electric Propulsion – An Overview, Journal of Propulsion and Power, Vol. 14, No. 5, 9/98-10/98, p. 690. 4 ibid. 5 Tajmar, Martin, Advanced Space Propulsion Systems, Springer Wien, New York, 2003, p. 76. 6 Wertz, James, and Larson, Wiley, Space Mission Analysis and Design, 2nd Ed., Microcosm, Inc., 1997, p. 396. 7 Sellers, Jerry Jon, Understanding Space: An Introduction to Astronautics, 2nd Ed., McGraw-Hill, 2000, p. 537. 8 Tajmar, Martin, Advanced Space Propulsion Systems, Springer Wien, New York, 2003, p. 74. 9 Wertz, James, and Larson, Wiley, Space Mission Analysis and Design, 2nd Ed., Microcosm, Inc., 1997, p. 397. 10 ibid, p. 317. 11 ibid, p. 403. 12 ibid p. 404. 13 ibid, p. 405. 14 Kimbrel, Scott, Optimization of Electric Propulsion Orbit Raising, SM thesis, Massachusetts Institute of Technology, 2002. 20