Sunday, January 22, 2017

Solar Sails

Solar sailing is a method of propulsion in space that utilizes solar radiation to accelerate a spacecraft, reducing the amount of fuel required for interplanetary missions.

The key to solar sailing is that light, though it has no mass, does have momentum! At first, this seems contradictory; the typical (Newtonian) definition of momentum that one first learns is that momentum equals mass times velocity, or p = mv (p denotes momentum). The mass m is simply a number indicating the quantity of matter in a given object, while p and v are vector quantities, having both magnitude and direction.

However, this definition of momentum is only approximate. Einstein's theory of special relativity holds that momentum, energy, and mass are all different aspects of a single quantity. The famous mass-energy equivalence E = mc2 (c is the speed of light) captures part of this relation. However, this equation is actually a special form of a more general expression for energy:

where p is momentum and m0 is the rest mass of an object (objects which are moving have additional mass and therefore additional energy by the mass energy relation). Photons, the particles of light, travel at the speed of light and are in fact never at rest. However, since objects with a nonzero rest mass can never reach the speed of light, it makes sense to classify photons as massless. Since m0 = 0, the equation reduces to E = cp, or p = E/c. Furthermore, light has energy, so it must have momentum. Different frequencies of light have different energies so photons of greater frequencies (such as X-ray or gamma ray photons) have correspondingly greater momentum.

Considering ordinary molecules for a moment, the macroscopic phenomenon of pressure (for example air pressure) emerges from individual collisions of particles with a surface such as the surface of a balloon. The average force that air molecules colliding with a surface exert is the pressure on that surface. Moreover, each of these collisions involves a transfer of momentum: a particle bouncing from a surface reverses the direction of its momentum vector so by the conservation of momentum the deflecting object also experiences a change in momentum. A similar momentum transfer occurs when light impacts a surface, creating what it known as radiation pressure.

The reason we do not feel radiation pressure whenever we enter sunlight is simply because this pressure is minute relative to the other forces we feel, dwarfed even by the force of a single tissue resting on a surface. The atmospheric pressure at sea level, around 100 pascals (Pa), is over ten billion times greater than the radiation pressure on a perfectly reflecting surface in direct sunlight on Earth (around 10 μPa = 10-5 Pa). Note that this phenomenon is distinct from what is called the solar wind, a term which refers to the stream of particles with mass constantly emanating from the Sun. These particles also exert a pressure when they collide with objects in space, but it is over a thousand times smaller than even the minute radiation pressure. Despite the apparent insignificance of radiation pressure, as in the case of ion propulsion, even small forces add to significant acceleration in space over time.

The concept of using radiation pressure as a means of propulsion is the foundation of the solar sail. Its design is simple: a large sheet of lightweight, reflective material surrounds the spacecraft payload (as in the artist's conception above). Notably, it is desirable for the sail material to reflect rather than absorb photons because this increases the acceleration of the sail.

The concept of a solar sail dates back to shortly after Maxwell's theory of electromagnetism was established in the 1860's in the works of Jules Verne. However, its first applications in spaceflight occurred almost 150 years later. Radiation pressure was used to save fuel in minor maneuvers on the MESSENGER mission and to compensate for a loss of maneuverability in the Kepler space telescope. However, the first true solar sail was IKAROS (Interplanetary Kite-craft Accelerated by Radiation of the Sun), a spacecraft launched by the Japanese Aerospace Exploration Agency (JAXA) in 2010 to demonstrate the technology.

IKAROS's solar sail measured 20 meters across the diagonal with a reflective film only 0.0075 mm thick that incorporated 0.025 mm thick solar cells to power the telemetry and steering instruments. The orange panels around the edges of the sail steered the craft by altering their reflectance with liquid crystal reflectors. For example, if one side of the sail were made more reflective then the opposite sides, the radiation forces would differ across the sail, causing it to rotate.

Launched on May 21, 2010, the IKAROS payload weighed only 310 kg and its cylindrical body measured on 1.6 meters in diameter and 0.8 meters in height. After reaching space, it followed the above procedure to release the sail (click to enlarge). By taking advantage of the centrifugal forces on four "tip masses" at each corner of the sail, the continually rotating apparatus can expand to full diameter and remain there without any rigid structure supporting the sail. The mission was a full success, demonstrating telemetry, propulsion, navigation, and attitude control for a solar sail.

Over the following years, NASA and the Planetary Society launched their own solar sails into Earth orbit for further testing demonstration of the technology, but IKAROS remained more significant as the first interplanetary solar sail. Once in space, craft employing solar sails do not have to carry any additional fuel, greatly reducing the amount of weight necessary for interplanetary missions. These sails may soon realize their potential as an inexpensive and efficient means of exploring the Solar System.


Sunday, January 1, 2017

Voronoi Diagrams and Metrics

In mathematics and visual art, a Voronoi diagram is a type of partition on a surface (usually a plane). Such a diagram is determined from some set of points (called "seeds") on the surface and a notion of distance on the surface by assigning each point a "cell," namely the region in the plane within which the given seed is closer than any other seed. The diagrams are named for the Ukrainian mathematician Georgy Voronoy.

Our first example of a Voronoi diagram consists of only two seeds (the black dots) and two cells, where the line connecting the two seeds is also shown. The maroon region contains the points in the plane closest to the left-hand seed, and the blue region the right. The divider between the two regions bisects the line between the two seeds (since the midpoint is by definition equidistant from the two endpoints) and is in particular the perpendicular bisector of this line. We now present a more complicated example.

In this image, the dots again represent the seeds, while the differently colored regions are the cells of the diagram. The inner region is bounded by a polygon (specifically a pentagon) whose sides are perpendicular bisectors of the lines connecting each of the outer seeds to the center seed. Note also that the central region is finite, since the center seed is surrounded by other seeds, while the other regions extend outward forever. Finally, each point at which three regions meet is the circumcenter of the triangle formed by three nearby seeds. The image below illustrates this fact for our example with six seeds.

Three of the seeds have been connected to form a triangle (white). The circumcenter of the triangle is the center of the circle containing the triangle's three vertices (black). By the definition of a circle, the circumcenter (red) is equidistant from the three seeds and is therefore the point at which the three neighboring regions meet.

Further, regular patterns of seeds produce correspondingly regular patterns of the cells. For example, a repeating square lattice of points produces a repeating pattern of square cells, as shown below.

The reader may experiment with different seed placements using the interactive feature found here. There are many ways to generalize the Voronoi Diagram concept beyond the two-dimensional plane. For example, it is possible to construct three-dimensional Voronoi diagrams, again using points as seeds, except that space will now be divided into three-dimensional cells instead of two.

The above image shows a number of seeds scattered in three-dimensional space and a single cell corresponding to the seed at the center. The lines connecting the center seed to the surrounding ones are also shown. Instead of a polygon, the cell is a polyhedron, bounded by faces which are sections of the planes that form the perpendicular bisectors of the line segments connecting the seeds.

In mathematics, Voronoi diagrams are useful for visualizing the notion of a metric. Metrics are generalizations of the familiar concept of distance to a number of different spaces in addition to the normal Euclidean plane and space (which we have worked with so far). For example, consider the surface of a sphere, such as the Earth. Typically, we define the distance between two points to be the length of the straight line connecting them (which in Euclidean space is the shortest path between the points). However, given two points on the Earth (a sphere), the line connecting them might go through the interior. When we speak of "distance" on the sphere, we want the shortest path along the surface between the two given points, or in other words the fastest travel route from one to the other!

The shortest distance between the points A and B above on the sphere is not the latitude line that they share (though this would be the straight path between them on the 2D map projection) but the arc of a circle passing through the sphere's center. These circles are known as great circles. The distance between two points on a sphere is defined to be the length of the great circle arc connecting them. This is also why planes take what appear to be inefficient paths on two-dimensional maps: they are in fact following a great circle (see below).

Having defined a metric for the sphere, we may choose some collection of points on it and create Voronoi diagrams, just as before. The diagram below takes major airports around the world as seeds and constructs a Voronoi diagram on the Earth's surface (which, of course, is nearly a sphere).

Voronoi diagrams also have a number of applications outside mathematics in settings where understanding distances from a fixed set of sources is important. They are used in modeling the spread of disease, the growth of forests, cell development, the distribution of minerals in the Earth's crust, and rainfall maps, among other things. They are a beautiful visual tool for comprehending the relative positions of points in a given space.