Solar Navigation System
He Lin
Abstract AbstractThis project is focused on establishing a new satellite navigation system aimed at the positioning task of deep space missions. The proposed solar navigation system works with the same theory as GPS system and will have two navigation beacons at the SunMars L4/L5 Lagrange point to decrease geometry dilution of precision with solar sail as propulsion. This project paper introduced basic theories of Lagrange point, the GPS dilution of precision and solar sail. And a minimal solar sail trajectory to place the satellite beacon at Mars Lagrange point is simulated with Matlab. IntroductionAs we launch more and more probes in to the solar system, interstellar flight mission controls requires more and more precision in orbit determination. Even if future technology developments enable autonomous flight, orbit determination is still one crucial factor in space flights. Nowadays precise orbit determination largely relies on the Deep Space Network (DSN) which is a network of antennas that tracks interplanetary spacecraft missions. DSN angular accuracy is shown in Fig 1 below. We can see from the figure below that orbit determination accuracy at Mars orbit still has about 2.250 km (year 2000, estimated), which is not accurate enough to support precise maneuvers.
On the other hand, the fast development of Global Positioning System (GPS) has well proven its capability in nearearth orbital determination missions. Its nature of passive positioning brings more benefit as the system can support a very large group of users. 1.1 Lagrange pointThe Lagrange points are five stationary positions in a restricted three body system where the third body has negligible mass. A figure below demonstrates the layout of the problem.
In 1772, Lagrange discovered a set of equations that describes the dynamics of the third body.
The positions of the Lagrange points L1 ~ L5 in EarthSun system are shown in figure below.
And also from the potential curves (white lines in Fig 3) we can see that L4/L5 points are in the low potential areas, which means L4 and L5 are stable solutions. Objects on Earth orbit in the vicinity of L4/L5 points will have a tendency to stay there. This minimizes the need for altitude maneuvers to be carried out on navigation beacons in that position. Which means the space vehicles need to carry less fuel. 1.2 Global Positioning SystemThe Global Positioning System (GPS) is a satellite navigation system which uses a constellation of 24 Medium Earth Orbit satellites that transmit precise microwave signals, enabling GPS users to determine their current location, the time, and velocity. 1.2.1 Global Positioning System Dilution of PositionDilution of Position (DOP) is an important factor in calculating the position of the users. It denotes the impact of the geometry of positioning satellites on the position solution accuracy of the users. The initial guess of the position and clock bias of the user is
1.2.2 GPS DOP for farearth orbitsThe GPS system was designed for navigation on earth surface rather than interplanetary navigation so although GPS system has very high accuracy on earth, its position solution result accuracy for interplanetary missions are quite poor. Viewed from planets like Mars, all GPS beacons are clustered around the earth, which have almost the same position vector. The following figure shows the possible best DOP factor for space vehicles at different planet orbits. The result shows intolerable bad position solution which prevents the GPS system to be used in farearth missions.
1.2.3 Improving precision with beacon satellites far from EarthFrom the last section we can conclude that the layout of current GPS system greatly restricted the usage of GPS in interplanetary mission. Beacons satellites using the same technique of GPS but located at the SunMars Lagrange points will greatly contribute to the precision of the position solution.
This project only involves with deploying navigation beacons to the SunMars L4/L5 position, so the vertical DOP will not be improved. The following calculating only involves with the HDOP to demonstrate the improvement of horizontal DOP. For simplicity, calculation of HDOP is performed in the situation shown in the figure. The Earth is on the opposite position of the Sun to the Mars. Two navigation beacons are deployed at the L4/L5 points of Mars and the existing GPS orbiting Earth are considered one more source for the navigation solution. Calculations are performed on the positions of planets other than Earth and they are on the same line of EarthSunMars, on the same side of the Sun with Mars.
Fig 8 below shows the HDOP result of the projected solar navigation system. As seen from the figure, HDOP are greatly improved thanks to the more spread out configuration of navigation beacons. Considering the HDOP of solar navigation system at distance of Pluto is even less than the HDOP of the HDOP of existing GPS system at the distance of Moon (Fig 5), solar navigation system can well handle the navigation needs for outer space missions. Although future improvements in VDOP and GDOP are out of the reach of this project, simply put two beacons at a reasonable distance (several AUs) above/below the Sun with respect to the SunMars orbital plane could solve the problem of poor VDOP of this proposed solar navigation system.
1.3 Solar sail propulsionSolar sail propulsion uses a big thin film to collect the pressure from photons and henceforth generate the thrust. Although solar sail is often mentioned in science fictions in the past, the fundamental theory was by no means new. James Maxwell first demonstrated the pressure of light in 1873[3], and the solar sail concept was proposed by two Imperial Russian rocket scientists Konstantin Tsiolkovsky and Fridrich Sander in the 1920s. Solar sail was not tested until 1993, when a 20 meter wide solar sail was deployed from the Soviet space station Mir[4]. In 2004, Japan successfully deployed two solar sails from a sounding rocket. Although no successful thrust test had been performed up to date.
1.3.1 Solar Sail force modelThe main thrust force for solar sail is Sun Radiation Pressure (SRP). At distance r from the sun, the SRP is described with the function[5]:
Wright developed a higherorder force model for solar sails that uses a set of optical coefficients to construct the characteristics of the solar sail material. [6] The SRP force based on the above parameters is:
2 Simulations of a solar sail trajectory to Mars L4 point.2.1 Simplified SRP force modelAccording to Wright, a highly reflective aluminumcoated front side and with a highly emissive chromiumcoated back side solar sail has And calculated This denotes that the nonperfect terms of the solar radiation force is rather small. For simplicity we will use a force model of in the simulation. 2.2 Mission designThe mission consists of three main parts: after initial launch for earth, the space vehicle deploys its solar sail and start accelerating, gaining to escape from earth gravity field. After the space vehicle gains enough and reaches the Sphere of Influence (SOI) boundary of Earth, the space vehicle enters a heliocentric orbit and continues to gain In the last section of the orbit transfer the SV drops into the vicinity of the Mars orbit L4/L5 point and a minimal thrust is used to final push the SV into the L4/L5 Lagrange point. For simplicity, in the simulation Sun, Earth, Mars and the SV are all in a same orbital plane. 2.3 Earth departureFor this phase of the transfer, we use an ECI frame to calculate the accumulation of. There are two forces in the system which are gravitational force on the direction of and solar radiation force on the direction of. With this steering law, the solar sail is directly perpendicular to the sunSV vector and hence collects the most energy which shortens the time needed to reach Earth’s SOI.
A Matlab script was written to simulate this phase of orbit. In the simulation, satellite mass is 200 kg and solar sail area is 100000. For simplicity, Earth center mass and Sun center mass models are used. Using the Ode45 function in Matlab to integrate accelerations caused by gravity and solar radiation pressure we get the following diagram. From fig 12 we can see that at approximately 116.3 days from deployment of solar sail the space vehicle escapes the Earth’s SOI. Fig 13 shows the space vehicle orbit trajectory for 173 days.
2.4 Heliocentric transferAfter the SV leaves the SOI of Earth, the coordinate system is changed from ECI to Sun Inertial frame. The position vector and velocity vector are changed (See Fig 11).
The angleis the only one parameter in the heliocentric transfer stage. In order to minimize the needed reentering Mars orbit, several values are tried. seems to have the best result. The trajectory in Fig 15 is for a. In the figure we can see that the space vehicle slowly accelerates because of the solar pressure and reaches the Mars orbit.
When the SV has reached the Mars orbit, a will push the space vehicle into the Mars orbit. varies with . The table below shows several tested. Table 1. needed to decelerate the SV vs. solar sail angle
As can be seen that has the smallest requirement. Comparing to Hoffman Transfer from Earth orbit to Mars orbit, the solar sail has significantly smaller which is 712m/s comparing to 5594 m/s for Hoffman Transfer. And also, the low didn’t come along with very long transfer time. Transfer time for solar sail is 314.5 days comparing to 258.9 days for Hoffman transfer. ConclusionThe proposed solar navigation system will greatly enhance the position solution accuracy for space vehicles far from the Earth. And the special orbit of the beacons will reduce the need for altitude control. Simple simulation revealed the advantages of solar sail propulsion by means of both total needed and tolerable travel time. More precise simulations can be carried out by including 3body effects and perturbations introduced by Sun, Earth and Mars, and improved mission design can include the gravity assist of the Moon, but those are out of the scope of this class project. References[1] Martin Lo, Shane Ross, The Lunar LI Gateway: Portal to the Stars and Beyond Appendix Matlab Codes
Code download:
function dd = dop 'o',R(6),doph_improved(6),'o',R(7),doph_improved(7),'o',R(8),doph_improved(8),'o',R(9),doph_improved(9),'o');
function xv2 = simulator %% integragters function f = solarforce(A,r)
%% integragters alpha = atan2(r(2),r(1));
function [a, e] = elorb(R0, V0, miu) global Mr;
function deltav = test(xv) Mr = 227939186e3; dist = zeros(1,length(xv)); %calc distance from earth
