Around The World In Minutes

The concept of the orbit, and of weightlessness, is one that is often misunderstood by laypeople who harbour the mistaken idea of there literally being no gravity up there. To understand how objects move in space, otherwise known as celestial mechanics, one has first to grasp the concept of freefall, because, for much of the time, that is the condition of everything in space. Our communications and weather satellites are in constant freefall around Earth, as is the Moon. Earth itself, along with the other planets, is in a permanent state of freefall around the Sun, which itself freefalls around our galaxy. Even the immense Milky Way galaxy that we inhabit is freefalling along with a collection of others in our local group of galaxies in an eternal gravitational dance that is essentially no different to the freefall experienced by a stone dropped off a bridge into a river.

The crucial ingredient that transforms freefall from a short-term descent that ends in a messy impact, into the essential element of an orbit, is speed - very high horizontal speed. A common thought experiment that explains the concept of the orbit is one that invokes a perfectly smooth, airless Earth with an imaginary tower. At the top of the tower is our intrepid imaginary experimenter - presumably wearing an imaginary spacesuit - whose task is to fling an object to the ground and watch how it travels before it impacts Earth's surface. Let us imagine that this object is a box containing beads, so that the effects of weightlessness can be observed from at least the perspective of our mind's eye.

In this thought situation, our experimenter begins by simply dropping the box from the tower. The box accelerates until it hits the ground. The beads within the box experience an identical acceleration such that not only is the box falling, but so are the beads. In our mind's eye, looking within the box, the beads can be seen freely floating around, and they appear to be weightless. Viewed from outside, however, they are falling with the box until they both meet their end directly below the point from which they were dropped.

The next incarnation of the thought experiment deals with what happens when, instead of just dropping the box, our experimenter throws it horizontally. For the few hundredths of a second that the throw is being executed, the beads are pushed against the back of the rapidly accelerating box, and experience whatever ^-forces the experimenter's arm can achieve until the throw is complete. Once it has left the thrower's hand and is coasting, the box follows a curved path to the ground that can be resolved into two components: the horizontal and the vertical. Following Newton's first law of motion, once horizontal velocity has been imparted by the throw, it is maintained until something causes it to change; and there is nothing in our thought experiment to do that because we have exorcised the effects of the atmosphere. In the vertical direction, however, the gravitational effect of Earth exerts the same force on the box as it did in the previous scenario, pulling the box and its beads to their untimely end on our imaginary airless surface. By combining these two velocities, we arrive at a curved path as the acceleration of gravity takes our subjects to their doom. Inside the box, the beads float around, apparently weightless and unaware of their fate. Although the box now follows a longer distance in its curved path to the surface, the time taken to reach the surface is essentially identical to the simple drop.

The next case for our thought experiment above our idealised Earth is where arrangements are made to throw the box at a far greater speed than is achievable by a human arm; say something of the order of a few thousand kilometres per hour. Traditionally, this can be achieved by an immense, imaginary cannon. Once the cannon has done its rather violent job of quickly accelerating the box, we see the same two influences affecting the box's flight. Gravity accelerates the box and its

The dynamics of falling objects in a thought experiment.

beads down to Earth while the constant horizontal speed takes it away towards the horizon, resulting in a curved path to the surface. Once the beads within the box have recovered from their sudden acceleration, they again float freely, exhibiting what we call weightlessness. However, on this occasion, the flight lasts rather longer than in the previous cases. The box's speed is so great, and so much horizontal distance is being gained as it drops, that by the time it has fallen the height of the tower, the curvature of Earth has dropped the surface level a little, and so the box has to fall further to reach the surface.

In successive versions of our thought experiment, we raise the power of the cannon higher and higher, reaching ever greater starting velocities. As we do so, we find that the effect of Earth's curvature becomes ever greater, increasing the time that the box coasts in freefall until impact. In every case, the beads gaily float around inside the box, appearing to be weightless to anyone who could look.

Eventually our thought experiment reaches a special case where the horizontal velocity of the box is so high that it manages to fall in a great ballistic arc all the way to the opposite side of our perfectly smooth, imaginary Earth without hitting it. You might think that it would simply travel a little further before meeting its doom but that is not what happens. By the time the box has reached the opposite side of the planet, the antipode, it not only has the horizontal velocity imparted by the cannon, but has also an additional momentum by virtue of the speed gained by its fall towards Earth. This momentum means that the box not only continues around Earth, but it also climbs back up to the altitude from which it was launched, much like a pendulum that, having fallen to the lowest point in its arc, has the momentum to continue to the top again. There is no case where the horizontally fired box will impact the surface beyond the antipodal point. In our idealised scenario, our experimenter had better watch out, because about 90 minutes after firing it from the cannon, his box will come whizzing by at the same speed, about 28,300 kilometres per hour, that it had when it was first set on its journey. The box has completed an orbit of Earth during which the beads within it experience the same weightless effects of freefall that they experienced in all the previous cases.

Having achieved an orbit, there are three further cases of orbital travel we can look at. The basic orbit just illustrated has two important features that are typical of nearly all orbits where a small body revolves around a much larger one. At the point where it just missed the surface on the opposite side of the planet, it was at its lowest altitude.

Freefall becomes an orbit in a thought experiment.

For an orbit around Earth, this is termed the perigee. The point at which it was launched was, in this case, the highest point in its Earthly orbit and is termed the apogee. This lop-sided trajectory around a large body is called an elliptical orbit.

Continuing with our thought experiment, there is a specific case with a slightly higher starting speed than the previous example, where the box maintains a constant altitude. The curvature of Earth's surface is falling away in exact sympathy with the box's path, making the two concentric and the orbit becomes circular. Again, the beads float around weightless within the box, and again, our space-suited experimenter needs to keep his head down as the box will whizz by in about 90 minutes.

Finally, we need to look at what happens when the experimenter adds even more charge to his hypothetical cannon and fires the box at an even higher starting velocity. In this situation, the box has more impetus than is needed for a circular orbit and this extra momentum straightens out the flight path a little, causing it to rise from Earth as it moves away from our imaginary tower. However, like a ball thrown vertically into the air, the box slows down as it rises away from the planet until it gets to the opposite side of Earth where it reaches an apogee. The box's vertical travel, i.e. A freefall thought experiment extends the orbit. its movement away from Earth's surface, has come to a stop and it gains no more height. Having passed apogee, it continues on its path, descending all the time and regaining all the speed it began with until, at the tower, it reaches its perigee, ready to repeat its elliptical orbit to apogee. In this, as in the previous cases, the beads within our box float around in the same state of apparent weightlessness as they felt when on their way to destruction in our first example. The orbit is simply a special case of freefall in a universe where gravity is king.

Applying this rather fun analysis to real life, the Saturn launch vehicle was both our cannon and our tower. It lifted our box, the Apollo spacecraft, to an altitude beyond the sensible atmosphere, accelerating it horizontally until it had enough speed to fall all the way around Earth. The beads represent the crewmen who found themselves floating around in their cabin, weightless, until another force pushed them back in their seats.

The elliptical nature of orbits was first worked out by Johannes Kepler in the early seventeenth century. His first law of planetary motion states that all planets move in ellipses with the Sun at one of the two foci of each ellipse. The same holds true for spacecraft orbits with Earth, the Moon or whichever planet is being orbited at one focus.

Second burn

Diagram showing how a spacecraft raises its orbit by a Hohmann transfer. Changing orbits

At this stage, having achieved an orbit, it is worth considering what a spacecraft does to change it. Our box of beads has no means of propulsion and once released into its orbit, it is doomed to revolve around our imaginary Earth to the end of time. Of course, real life isn't like that. Spacecraft usually have some kind of rocket motor, especially human-carrying ships that must return to the home planet.

Imagine, then, that our box of beads is a spacecraft with an engine. Let us assume that, having entered a circular orbit, we want to reach a higher orbit. To do so, we must increase speed further by firing the engine, aiming its nozzle rearwards. This would straighten out the flight path a little and cause the spacecraft to enter an elliptical orbit in which it would coast to an apogee on the other side of the planet. The point at which the burn was made becomes a perigee, and if the duration of the burn is appropriately timed, the spacecraft can be made to ascend to any desired apogee altitude. Half an orbit later, having slowed down considerably (just like any object tossed upward in a gravity field), it arrives at apogee but does not have enough momentum to stay at that altitude and falls back to its perigee as it continues around the planet. Apogee then becomes a good place to adjust the altitude of the orbit's perigee. By adding yet more speed at apogee with another burn, the flight path is straightened out further, so that the spacecraft does not fall quite so far as it descends to perigee. If enough extra speed is applied at apogee, the shape of the orbit can be made circular again, this time at the higher altitude. This method of transferring from one orbit to another involving a pair of burns performed 180 degrees apart is known as the Hohmann transfer orbit and was formulated in the early part of the twentieth century by Walter Hohmann, a member of the same German rocketry club as Wernher von Braun.

Now comes the counter-intuitive bit. Although our imaginary spacecraft's speed had been increased on two occasions, it ended up travelling much more slowly than when it was in the lower orbit. Its speed had been traded for height - a situation that will be familiar to all fighter pilots. There are all sorts of ramifications to this in terms of spaceflight operations, especially for Apollo. For example, if one spacecraft wanted to catch up with another further ahead in the same orbit, the wrong thing to do would be to aim towards the target, light the rockets and try to fly directly towards the quarry. This would make its orbit more elliptical, raise it to a higher apogee and it would therefore travel more slowly, thereby opening the range, which is exactly the opposite of the desired effect. The right thing to do would be to turn the craft around and fire to slow down, thereby making the orbit elliptical with a lower perigee, increasing the spacecraft's speed and closing the range. Then, to effect a rendezvous the spacecraft would need to turn around yet again and make another burn to rise back up to the target's orbit at just the right time.

Clearly, making large manoeuvres in space has to be done with careful forethought. Computers and radars are also indispensable tools. This was especially true during an Apollo flight where the success of the mission depended on the ability of two spacecraft to rendezvous successfully. It was also true from the point of view of their next major manoeuvre - the burn to set them on a path to the Moon, itself essentially a Hohmann transfer.

Immediately after the Apollo/Saturn stack had achieved orbit, Earth-based radars began tracking the vehicle and determining its trajectory as precisely as they could. From these measurements, Earth-based computers calculated a suitable Moon-bound trajectory and the details of a burn that would achieve it, given the constraints of what the S-IVB could manage. These details were transmitted to the computer within the instrument unit. Whatever information was relevant to the crew was passed on to them also in a list of numbers manually read up by the Capcom. Based on these calculations, and after a little more than 2% hours orbiting Earth, the S-IVB stage reignited and set the Apollo spacecraft on its path to the Moon.

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