### About interplanetary travel

Space exploration is a fundamental endeavor of humanity, which, besides satisfying the inherent curiosity of individual human beings, can benefit all of humanity. The desire to explore this new frontier has led to scientific discoveries, new technologies, and new engineering solutions.

Traveling from Earth to other planets and the design of flyby, orbiting and landing missions is not a simple task. There are many challenges to meet, and it requires the integration of science, technology, engineering, art and mathematics (STEAM). The science part concerns the possible trajectories a body can assume under the gravitational force. Technology brings the feasibility of putting spacecraft into orbit around Earth and consequently to change its speed and transfer it to a different orbit. On the other hand, engineers will make specific plans for a specific mission and optimize energy requirements without compromising on necessities. Art can capture the essential aspects of a particular mission and produce artifacts, video, and images needed to communicate a mission effectively and aesthetically. Mathematics is instrumental in creating models needed to optimize processes within the required constraints.

(Click on the titles to access the simulations online)

As the name suggests, the aim of this simulation is to explore of two bodies under the influence of their own gravitational force. It is possible to set different values for the two bodies masses, and the position of the bodies. The application calculates and plots the force as a vector for the given positions of the bodies. The students, by setting different positions and masses, can become acquainted with the qualitative aspects of gravitational force, such as the occurrence in pairs with the same magnitude on each body and that the forces get smaller at greater distance.

Apart from the qualitative aspects one can be asked to compare the values of the gravitational force at two different distances; how does the ratio of the forces compare to the ratio of the distances? The question of the problem can be: given that there is a simple relation which connects the distance of the two bodies and the gravitational force, can you infer by comparing the ratio of the distances at two positions to the ratio of the resulted forces this relation and check if it holds to any pair of positions? It is up to the students to determine their strategy of how to choose sets of positions pairs and come up with some concrete numbers, produce the relation and check its validity to other pairs.

This particular is set to be used as an assessment exercise. It’s aim is two-fold: to assess the ability of students to measure distances on the screen using a ruler and to assess if they can determine the ratio of two distances that would result a give ratio of gravitational forces. The simulation gives two bodies at an arbitrary distance and shows the forces. Then the students are asked to find a new distance for the two bodies such that the force could be the one fourth of the force at the present distance. The student can either have a feeling of the qualitative aspect of the forces, i.e., that they become weaker as the distance increases, and she can try a larger distance until she succeeds the required force. If the distance is correct the simulation will respond whether the answer is correct or not. If the student has assimilated that the force is inversely proportional to the square of the distance then she can calculate the distance for the force to be the one fourth of what it was in the original distance, i.e. double the measured original distance. Many variants of this exercise can be set, such as for the student to determine the new distance for which the force would be four time the value of the original distance and so on.

The only solvable problem of the motion of bodies under the influence of the gravitational force is the so called two-body problem. It is a common approximation applicable to many cases to consider that one body has very large mass compared to the other body and thus effectively on can take into consideration only the motion of the small mass relative to the large mass. Cases that that applies are the planets orbiting the sun and satellites orbiting Earth and also the motion depends on the inertia reference frame of the observer. The scientific knowledge is that both bodies are in motion relative to the center of the mass. In a simulation it is possible to include this fact and there is no reason to omit it. Therefore, we have included in the parameter of this simulation the ratio of the masses of the two bodies along with the initial positions and velocities. The initial positions can be set by moving using the body on the simulation surface and similarly the initial vectors of the velocities by moving the end points of the default velocity vectors. In this way both the magnitude and the direction is determined. Additionally, the inertia frame of reference can be chosen for the students to be able to explore how the geometry of the orbit is different if the observer is on a different inertia frame. The above considerations lead to a simulation to study all aspects of the two-body problem and also check the validity of the case where one body has a relatively large mass compared to the other body.

This simulation is designed to allow the study of a body orbiting a star (a large mass body) such as the sun aiming to simulate the planetary motion. As before we can set the mass of the star, the mass of the second body, i.e., a planet, as well as, the position and its velocity, by moving the planet to the desired position and the velocity by fixing the end point of the velocity vector. Alternatively, one can set the values of the components of the position and of the velocity. On setting the planet free to move the simulation shows the orbit at each moment, the value of the force, the position vector and the velocity vector, the value of orbit eccentricity, and its energy. All the above can allow the simulation of any planetary system whether it exists or not. Since our planetary system is of particular interest, we have incorporated the option to choose an actual planet and automatically assume the appropriate parameters. For this case the simulation produces all the above information but for the particular planet. In addition, the scale of the simulation window can be chosen, as well as the scale of the vector quantities and the time scale. We can obtain the area swept by the planet in a given time interval, information from which we infer the perihelion and aphelion of the planetary orbit. A polar grid can be introduced to help estimate the distance of the planet from the sun at different points of the orbit.

This application draws an ellipse to visualize the shape of an ellipse for different values of the set of parameters scientists use to describe a planetary orbit. The available parameters are eccentricity, the large and small half axis, the focal distance, and the angle defined by the planet, the sun and the positive x axis of a cartesian system centered on the sun. This visualization is very helpful to recognize that most of the planets have elliptic orbits very close to circular orbits.

This application is 3D and enables the study of the motion of satellites having circular orbits. We can set a satellite at a distance from the surface of Earth and by observing its motion one can determine its period. This is very useful since we can describe many actual satellite orbits.

The aim of this application is to assess the amount of energy to use for the Hohmann transfer of a spacecraft from Earth to Mars. As mentioned before the orbits of Earth and Mars are almost circular and it is a good approximation to take the orbits to be circular. We can choose a change in the velocity for the spacecraft so to transfer it from circular orbit around the sun, which effectively performs while orbiting Earth to an elliptical orbit. This elliptical orbit will have its perihelion at the point of the Earth orbit where we change the velocity and its aphelion at a point of Mars orbit. Therefore, if the change in velocity and the time it happens are correct the spacecraft will meet at some later time Mars. By observing the path of the spacecraft we can assess if has achieved the correct change in velocity and the time it happens. When this happens, the orbit of Mars on its almost circular orbit and the perihelion of the spacecraft will coincide at a particular time. At this point we have to transfer from an elliptical orbit to the circular orbit of Mars, which can be achieved by increasing the speed of the spacecraft again. The cost per kilogram of the spacecraft in fuel is related to the two changes in the velocity of the spacecraft to complete the Hohmann transfer. The amount of fuel per kilogram used in each trial to accomplish the Hohmann transfer is calculated by the application and is presented by the fuel left in the predetermined amount of fuel for the Hohmann transfer.

The notion of relative velocity is central to the Hohmann transfer and a common student misconception. For the spacecraft to complete the Hohmann transfer it has to have the same orbital velocity as Mars, therefore zero relative velocity. To deal with the misconception we have prepared an application based on a plausible everyday experience, which appears in many movies, that of catching a train (car, bus) in motion. The application presents a train which is moving at a constant speed relative to a train station and a passenger who wants to get aboard the train. The passenger should acquire the speed of the train to achieve this safely. By setting different speeds for the passenger the student tries to match the unknown speed of the train and succeeds to board the train.

*Solar System*

https://seilias.gr/scientix/planets.html

This is an exact 3D replica of our planetary system with planets orbiting the sun as time passes by. We can set a date (year, month, and day) of the past, present or the future and observe positions of the planets and how the planets move from that date on. For each planet we can acquire a position a velocity and observe the relative position of the planets. The 3D character of the simulation is important for the students to realize that the orbits seem to have different shapes depending on the angle of observation of our planetary system. Also, the students can notice important qualitative aspects of the planets’ motion.

Based on the previous simulation, this simulation can help one inquire the motion of a spacecraft relative to the actual motion of the planets. After one determines the date of launch, the magnitude of the velocity and the angle of launch and then starts the simulation, it is possible to observe the position of the spacecraft as time goes on.

Click here to download the worksheets.

### The Partners

The project is supported by the Scientix STE(A)M Partnerships programme and its partnership consists of:

#### GFOSS Open Technologies Alliance

**Open Technologies Alliance**, GFOSS, a non-profit organization founded in 2008 which promotes Openness through the use and the development of Open Standards and Open Technologies in Education, Public Administration and Business in Greece. GFOSS has a framework agreement with the Ministry of Education in supporting educational actions related to STEM education.

#### Aristotle University of Thessaloniki

**Aristotle University of Thessaloniki**, Physics Department, is a multi-disciplinary, research-oriented public educational institution. The Didactics of Physics and Educational Technology graduate program of the Physics Department, trains students (MSc and PhD level) to conduct methodical, systematic, and theory-based research on areas encompassing inquiry based education, STEAM education, inclusive education, collaborative methods of learning, distance/blended synchronous/asynchronous learning with emphasis on 21st-century skills and sustainable development.

#### University of Crete

**University of Crete**, Department of Primary Education, Science Teaching Lab is a multi-disciplinary, research-oriented public educational institution. The Science Education lab focuses on research about the educational use of digital technologies and the integration of the educational innovations of ICT such as data loggers, virtual & augmented reality, and educational robotics in STEM teaching.