Targeting Mars. 19th June University of Ljubljana. Faculty of Mathematic and Physics. Seminar: 4. year-university program

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1 University of Ljubljana Faculty of Mathematic and Physics Targeting Mars Seminar: 4. year-university program 19th June 016 Author: Jo²t Vrabi Koren Advisor: Prof. Dr. Tomaº Zwitter Ljubljana, June 016 ABSTRACT Spacetravel to Mars is an actual topic. I start with a two body problem, followed by a N-body problem. For a solution an integrator is needed. I describe the IAS15 integrator which was used for all calculations of the orbits. I then present an approximation for determining initial conditions for a rocket destined to travel to Mars and reach an areocentric orbit around Mars or even land.

2 Contents 1 Introduction Two body problem 3 Orbital elements 4 4 N-body problem and integrator 5 5 Targeting Mars 7 6 Results 10 7 Conclusion 10 1 Introduction The rst theoretical proposal of space travel using rockets was published in 1861 by Scottish astronomer and mathematician William Leitch. In the middle of the 0th century rst satellites were launched into space. Since then we have launched several thousand spacecrafts. Some of them have landed on other planets and some are orbiting other panets in our solar system. Typical journey of a spacecraft destined to other planets is consisting of launch from the Earth, a geocentric orbit, transfer to a heliocentric transfer orbit and transfer to a areocentric orbit (orbit around a planet). One of the main tasks of planning a mission is determining ight parameters (initial velocity, time of launch, end velocity,...) with n-body simulations using powerful n-body integrators. Mars is the next human landing destination. That is why it is extremely popular. It has several spacecraft already in its orbit, observing its surface and characteristics. Landing of a sonde on Mars was also already achieved, several times (Viking, Curiosity, Pathnder,...). Two body problem Figure 1: Particles with masses m 1 and m, distance r apart. Let us start with a conservative central force, where the potential is V(r), a function of r only, which means that the force is always along r. We take a particle with reduced mass m, moving about a xed center of force, which is taken as the origin of the coordinate system. Reduced mass is m = m1m m 1+m, [1] where m 1 and m are masses of our bodies (Figure 1). Since potential energy is dependent on radial distance only, the problem has spherical symmetry. Total angular momentum vector

3 L = r p, [] is then conserved. It follows that that r is always perpendicular to L, which has a xed direction in space. This can only be true if r lies in a plane with a normal parallel to L. Central force motion is thus always restricted to a plane. Motion of a particle in space is described by three coordinates. In spherical coordinate system, these are azimuth angle θ, the zenizh angle ψ and radial distance r. We choose a system with a polar axis in the direction of L so the motion is restricted to the plane perpendicular to the polar axis. ψ then has the constant value π / and can be dropped from the following sections. Conservation of total energy can be written as: Here we dene gravitational potential V (r) as: It can be shown that the equation of the orbit for our problem is: 1 l mṙ + 1 mr + V = konst. = E. [3] V (r) = k r. [4] 1 r = mk ( El² cos(θ l² mk² θ )). [5] where θ is one of the angles where the orbit turns. Eccentricity is dened as: e = 1 + El mk. [6] Orbits can be classied according to eccentricity e: e > 1 E > 0 hyperbola e = 1 E = 0 parabola e < 1 E < 0 ellipse e = 0 E = mk l circle Let us take a more detailed look at ellipse, as this is the orbit that is the most important for this seminar. r 1 in r are two apsidal distances (Figure ). Figure : Ellipse with apsidal distances r 1 and r and semimajor axis a. These two points corespond to angle θ where distance is the smallest. The radial velocity at these points is zero. It can be shown that the major axis depends solely upon the energy: 3

4 We can now rewrite the elliptical orbit equation [6] as a = r1+r = k E. [7] r = a(1 e²) 1+e cos(θ θ ). [8] Let us describe the motion of a particle on an elliptic orbit in time. The relation between the radial distance and the time is given by the enrgy equation [3], which we rewrite as ˆ m r t = dr k. [9] r l² +E This integral is most conveniently calculated with a new variable ψ. It is dened by the relation r 0 mr² r = a(1 e cos ψ). [10] By comparing this equation to equation [5], we see that ψ covers the interval 0 to π, as θ goes trough a complete revolution. Perihelion occurs at ψ = 0 (also θ = 0) and aphelion at ψ = π = θ. With this new variable (plus some algebra and substitutions), we can now rewrite equation [9] as t = ma³ k ψ (1 e cos ψ)dψ. [11] 0 If we integrate this equation over full range of π, we get the expression for the period of an elliptic movement τ = πa 3/ m k. [1] Let us remember that a particle moving around the Sun is a two body problem, so we have to take both masses into account. We substitute m with the reduced mass from equation [1], where mass m 1 is the particle mass and mass m Solar mass. Further, the gravitational law of attraction is Equation [1] then becomes f = G m1m r². [13] πa τ = 3/ πa3/ G(m1+m ) Gm, [14] if we neglect the mass of the particle compared to the Sun. In this seminar, as we are dealing with spacecrafts, this can be done. But planetary mass is not always negligible compared to the Sun - mass of Jupiter, for example, is approximately 0,1% of the mass of the Sun. 3 Orbital elements We need 6 parameters to determine an orbit. In cartesian coordinates, this are radial vector r and velocity vector v in a chosen system. In our case that is the Solar system with Sun in the center, with z coordinate equal to zero in Earths ecliptic plane and x axis pointing towards vernal equinox. In astronomy, an elliptical orbit is often described with orbital elements a, e, i, ω, Ω and T : ˆ Semimajor axis (a). 4

5 ˆ ˆ ˆ ˆ Eccentricity (e). Inclination (i) - vertical tilt (in degrees) of the ellipse with respect to the reference plane, measured at the ascending node (where the orbit passes upward through the reference plane). Longitude of the ascending node (Ω) - horizontally orients the ascending node of the ellipse (where the orbit passes upward through the reference plane) with respect to the reference frame's vernal point. Argument of periapsis (ω) - denes the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis. ˆ Time at epoch (T) - denes the position of the orbiting body along the ellipse at a specic time. Figure 3: Orbital elements. Note that true anomaly in our case is θ. True anomaly at epoch is related to Time at epoch (T). Source: 4 N-body problem and integrator The n-body problem considers N point masses m i, i=1,...n in an inertial reference frame in three dimensional space moving under the inuence of mutual gravitational attraction. Position vector q i is appointed to each mass m i. Gravitational force on mass m i by mass m j is given as and sum over al masses gives us the complete force on particle i: F i = F ij = Gmimj(qj qi) q j q i 3, [15] N j=1,j i Gm im j(q j q i) q j q i 3. (16] 5

6 This system cannot be solved analiticaly, so we have to use numerical integration. In calculations presented in later chapters, I used Rebound package for python that uses IAS15 integrator (Ref.[4]) for N-body problems. Here I will describe this integrator. Let us start with the equation y = F [y, y, t], (17) with y being the acceleration of a particle and F the specic force, depending on the velocity of the particle y, position of the particle y, and time t. Let us expand equation (17) into series: y [t] y 0 + a 0 t + a 1 t²+...+a 6 t 7. (18) The constant term is the force at the beginning of a timestep, y 0 = y [0] = F [t = 0]. By introducing the step size dt, h = t/dt and b k = a k dt k+1, the series becomes y [h] y 0 + b 0 h + b 1 h² b 6 h 7. (19) Coecients b k have dimensions of acceleration, since h is dimensionless. By integrating equation (19) y [h] y 0 + hdt(y 0 + h (b 0 + h 3 (b 1+...))), (0) we get an estimate of the velocity y during and at the end of the timstep. By integrationg once more y[h] y 0 + y 0hdt + h²dt² (y 0 + h 3 (b 0 + h (b 1+...))), (1) we get an estimate of the positions. The rst two terms are the position and the velocity at the beginning of the timestep. The approximation of this integral is very accurate as we choose Gauss- Radau spacing. It is closely related to Gaussian quadrature for approximating an integral and uses the starting point at h=0. We use a quadrature with 8 function evaluations to construct a 15th-order scheme. Now, we have to nd good estimates for the coecients b k. We can resolve this with a predictor corrector scheme. We get an estimate for the position and velocities to calculate the forces which is the predictor. In the rst iteration, we set all b k = 0, which corresponds to a particle moving along a path of constant acceleration. We then use these forces to calculate better estimates, which is the corrector. This process is iterated until the positions and velocities converge below machine precision. The number of predictor-corrector iterations is determined dynamically. The iterations converges with iterations in most cases. 1 iterations are set as the upper limit to prevent innite loops. If it has not converged by then, the timestep is too large. The timestep can be set to be choosen automatically. Authors claim this integrator is superior to all other integrators (symplectin and non-symplectic) and one of the main reasons is adaptive timestep (Ref. [4]). This is perticularly usefull when dealing with close encounters. 6

7 5 Targeting Mars Let us now imagine we have a rocket orbiting Earth on a circular ecliptic orbit. We want to increase its velocity at the right time, so we can escape Earth's gravitation and begin our journey to space. We want to choose the correct direction and magnitude of the velocity, so we can reach Mars. For this seminar, I simplied the problem. As we have seen in previous chapters, all orbits in a gravitational eld are conic sections in 3D space. The planets have almost elliptical orbits. First I asumed the orbits of Mars and Earth are circular. Eccentricites of Earth's and Mars's orbits are e Earth = and e Mars = We see that Earth's orbit is in fact almost circular (Fig [4]). Figure 4: Left: Comparison of real and circular approximation orbits of Earth and Mars in the same plane (i = 0 ). Right: Two possible starting positions and velocities of our rocket. Then I asumed the orbits are coplanar. Inclinations, in degrees, of the Earth's orbit is 0, and of the Mars's orbit i Mars = Orbital period of Mars is P Mars = days and of Earth P Earth = days. From equations [14], we see that semimajor axis is a = 3 GM Sun P ² 4π, () where M Sun is the mass of Sun. In the case of a circular orbit, a is radius of rotation r 0. By our asumption we get r 0Mars = a.e. and r 0Earth = a.e.. Our rocket is circling the Earth. We turn on the engines, when its velocity is maximum. This occurs when the rocket velocity vector is in the same direction as the Earth velocity vector v r v Earth. We have two dierent solutions (Fig [4]), depending on the rotation direction. Choosing magnitude is not so simple. We rewrite the energy equation [3] as ε = v rel² GM Mars r rel, [3] where ε is now energy per unit mass, v rel velocity of our rocket relative to Mars, r rel distance of the rocket from Mars and M Mars mass of Mars. We want the rocket to get close to Mars and dene close as F Mars F Sun > 10, [4] 7

8 where F Mars is the gravitational force of Mars magnitude and F Sun of the Sun. This way we can neglect the eects of the Sun when we are close and the equation [3] is then correct. The aproach orbit turns out to be nearly hyperbolic (with respect to Mars) and hyperbolic orbits have positive energy. We can then lower rocket's velocity, so the energy becomes negative and we get a closed orbit around Mars. So our desire is that the energy is as low as possible as we come close to Mars. This way we are closest to a closed orbit. The velocity magnitude of a particle on an elliptical orbit is greatest at perihelium and lowest at aphelium. That means we want to reach Mars at aphelium, so the rst term in equation [3] is minimal. Figure 5: Orbits representing velocities v a, v b and v n and Mars. Now, we choose two initial velocities. One so that aphelium is inside the Mars orbit and the other so that it is outside (Figure 4). The rst one is chosen to be the sum of Earth's escape velocity and Earth's velocity: GM v a = v escape + v Earth = Earth r Earth + v Earth, [5] where M Earth is Earth mass and r Earth distance of our rocket from the center of Earth. The second velocity is chosen to be slightly smaller than Sun escape velocity: GM v b < Sun r Sun, [6] where r Sun is the distance of our rocket to the center of Sun. Velocities v a and v b are chosen so they have a physical meaning. We could have ofcourse chosen two dierent velocities, as long as one would correspond to an orbit with an aphelion inside the Mars orbit and the other with aphelion outside. The following procedure is some kind of bisection. We appoint a new velocity v N = va+v b, [7] and send our spacecraft toward Mars. We nd the aphelium position and check if its distance from the Sun is greater or smaller than r 0Mars. If it is greater we assign v b = v N and if it is smaller v a = v N. We continue untill we achieve the desired accuracy for the initial velocity v 0 (Figure 5). This is all 8

9 carried out in a system with the Sun in the origin and consisting only of Earth and our spacecraft (no Mars). Figure 6: System without Mars showing paths of rockets, starting in a circular orbit around the Earth at 8000 km. Left one starts closer to the Sun and right one further from the Sun than Earth. Figure 7: System with Mars showing a rocket with initial condition v 0 and t, calculated by bisection. Rocket was red when it was further from the Sun than Earth. Let us now introduce Mars into our system. We start by putting it at the point of orbit intersection and then take it an angle ωt back on its orbit. t is the travel time the spacecraft. We have come close to Mars, but not close enough (Figure 6). To come closer, we have to solve a two-dimensional problem as we have two parameters now, initial velocity v 0 and travel time t, which also denes the angle of the position of Mars. It was shown before that we are searching for the energy minimum. We choose suciently small v 0 and t and start searching for the minimum. Eventually we get there (Figure 7). 9

10 Figure 8: Sheme of an energy minimum. We start with initial conditions v 0 and t and make our way to minimum. Note that we rst have to get close to Mars (Eq.[4]), before we start searching for the minimum. That means we rst have to get close to the gravitational minimum. This is done in the same fashion as our search for energy minimum. 6 Results I used Nasa's Horizons database to nd coordinates in time of Mars Atmosphere and Volatile EvolutioN Mission (MAVEN) and Mars Orbiter Mission (MOM). This are two Mars orbiting sondes, launched from Earth in November 013. They were rst in heliocentric orbits and then accelerated to reach v 0 and begin their journey toward Mars. When reaching v 0, MOM was approximately 7900km from the center of the Earth and MAVEN approximately 8000 km. Our rocket is at 8000 km. The main parameter I compared here is the change of velocity v final near Mars, needed to put the selected spacecraft on a closed orbit around Mars. This change denes fuel consumption at the end of the travel. MOM MAVEN Rocket under Earth Rocket above Earth time of travel 98 days 308 days 7 days 81 days v 0 40,1 km/s 40, km/s 40,5 km/s 40,6 km/s v final 0,31 km/s 0,39 km/s 0,48 km/s 0,46 km/s Table 1: Comparison of dierent spacecraft. v final was calculated via equation [3]. 7 Conclusion We have shown that by making several approximations to our problem, we still get comparable results. This was however expected, as the orbits of Earth and Mars are relatively circular and lying almost in the same plane. It also indicates that studying an approximated problem can lead us to initial conditions, which can be then used on real orbits in real time. 10

11 References [1] H. Goldstein, C. Poole and J. Safko, 00, CLASSICAL MECHANICS, third edition, Pearson Education, San Francisco, 638 p. [] [3] H. Rein and S. Liu, 011, REBOUND:An open-source multi-purpose N-body code for collisional dynamics, Astronomy & Astrophysics, Volume 537, id.a18, 10 pp. [4] H. Rein and D.S. Spiegel, 015, IAS15: A fast, adaptive, high order integrator for gravitational dynamics, accurate to machine precision over billion orbits, Monthly Notices of the Royal Astronomical Society, Volume 446, Issue, p [5] [6] [7]