## SMV M 5M 5v31

If these increments are small, the term SMSv may be ignored. This is the conservation of momentum for a brief increment in the rocket flight. From here in the calculus limit these increments become infinitesimal

This leads to a final velocity of the rocket with mass m after expending a mass M — m of burned fuel plus oxidant v = Vln(M/m). For those familiar with logarithms should sense that this is a terrible result! It's not terrible for being wrong, its terrible in its implications. If the initial rocket mass is ten times that of the final mass the velocity of that terminal part will be 2.3 times the velocity of the exhaust plume. This multiplication by 2 to 3 is a standard economic multiplication factor. For a rocket with a plume velocity of 3500 m/sec and a final payload mass of .1 the initial mass the final velocity will be 8050 m/sec, which will place the craft into Earth orbit. This is the primary reason that it takes a rocket some 100 feet tall to put a much smaller spacecraft into space. This becomes all the more the case for a round trip vehicle such as it the Saturn-Apollo ship, which started off at 365 foot tall and around 3000 tons, and what came back was a small capsule and its crew of three.

Essentially this result means a rocket has to carry its propellant up with it. Most of the energy and reaction mass, the rocket plume, expended is done so in order the carry the fuel along the way. Yet there is no escape from this reality. There is some talk of a space elevator, a colossal structure that reaches out into geosynchronous orbit, that will get around this problem. However, I have serious doubts about the feasibility of this. Engineers must then try to get the plume velocity as fast as possible. The measure of this is the specific impulse, which is the plume velocity divided by the Earth's gravity at the surface g = 9.8 m/sec2, s = V/g [3.1]. Liquid chemical rockets have an upper limit of s = 500seconds, and the space shuttle engines have s = 459 sec, which is close to the theoretical limit. Another physical quantity of use is thrust, which is the force of a rocket. This is T = Vdm/dt = sg dm/dt, where dm/dt is the rate that reaction mass is thrown out the nozzle of the rocket [3.1]. One strategy to reduce the impact of the rocket equation is to drop various rocket stages to eliminate lofting unneeded mass along the way. Obviously another is to reduce the mass of the rocket as much as possible. This constraint is a brutal limitation imposed on spaceflight.

Other methods of propulsion have been proposed. An ion rocket accelerates ions as the reaction mass. This can have a specific impulse of 3000 seconds. An ion propulsion rocket may be powered by either solar

Fig. 3.1. Schematic for the basic liquid chemical rocket.

photovoltaic panels or a nuclear reactor. This works well for propulsion of a craft already in space, but as its thrust is low ion propulsion will not work for lifting off the Earth. A nuclear propulsion system, where hydrogen runs through heat pipes in a nuclear reactor may have a similar specific impulse. This might be useful for launch vehicles. Yet concerns over accidents have posed a serious barrier to its implementation. Another propulsion method is the Variable Specific Impulse Magnetoplasma Rocket (VASIMR), which employs electromagnetic radiation to impart energy to hydrogen, ionize it and magnetic fields then guide the plasma out the rocket at very high velocity. This may theoretically have a specific impulse of up to 30,000 sec and higher thrust than the ion rocket. The VASIMR is the highest specific impulse propulsion system being seriously considered. Yet this is insufficient for sending a spacecraft to the stars. What will be primarily addressed is the photon rocket. The photon rocket employs some means, such as matter-antimatter reactions, to convert much of its initial mass into photons. Since the plume velocity is the speed of light c ~ 3 x 108 m/sec the specific impulse is s ~ 3. x 107 sec, which is the upper limit of specific impulse.

The basic features of each of these propulsion methods are considered. The simplest and oldest form is the solid rocket. Essentially this is a can with some rocket fuel in a solid form packed within and an exit port for the burned gases. The simplest fuel is gunpowder, which forms the basis for firework display rockets and model rockets. Another form is the sulphur-zinc rocket, which can be easily made, but great care is advised as this can be explosively dangerous. Various improvements in chemistry and configurations of the propellant have resulted in higher specific impulses, but the upper limit is s ~ 200 sec. Conversely solid propellant rockets do have high thrusts. They are mainly used for ballistic missiles, as they can be stored ready for launch over long time durations. Curiously they found spaceflight use with the space shuttle. The early pioneers of rocketry recognized that a superior method was required for effective space flight.

The liquid chemical rocket was first devised by Robert Goddard. It is comparatively simple in principle. It burns a fuel and an oxidant in a bottle with an exit nozzle, the thrust chamber. The fuel and oxidant are pumped into the thrust chamber by various means from separate tanks. A simple way of providing this feed is to pressurize the tanks. Other systems involve high pressure pumps which are powered by gasses channelled through pipes from the thrust chamber which drive a turbine which in turn powers a pump. This is remarkably simple, of course simple in principle. The complexities come with high temperatures and pressures in the thrust chamber. A crack or flaw can result in the rocket exploding, which early films indicate was a serious problem. To reduce heat damage often the oxidant is liquid oxygen, which is circulated around the thrust chamber before being injected in. Obviously other issues have to be addressed, such as oxidant and propellant vapors entering the rocket engine area causing explosions. The system for delivering fuel and oxidant into the thrust chamber needs to work at high pressure, where leaks could be explosive. Yet the chemical rocket is fairly reliable, and today their launches are comparatively routine.

Yet it is clear that the chemical rocket is marginally capable of lofting interplanetary spacecraft. Time of travel is measured in years. Further, chemical rockets are not able to loft very large spacecraft deep into interplanetary space. Chemical propulsion may well remain the method of choice for getting payloads off Earth and into space for some time. Yet once the craft is sent to escape velocity, v ~ 11 km/sec, it would be convenient to have a method of propulsion that can send a craft to 100 km/sec or even 1000 km/sec to cut mission times proportionately. This requires alternative forms of propulsion, such as the ion rocket, nuclear propulsion, or VASIMR. For interstellar rocketry the photon rocket will be discussed in detail in Chapter 7.

The ion rocket is the most developed of these three [3.2]. An ion thruster uses beams of ions for propulsion. There are various methods for accelerating the ions. This system employs the strong electromagnetic force to accelerate ions to very high velocities. Since the mass of the ions is small this results in a high specific impulse s ~ 3000 sec. Ion thrusters with a plume of high velocity ions have high specific impulse and a small amount of reaction mass constituting the plume. The small dm/dt means that the trust is very small. The power requirements for a high velocity ion plume is large when compared to chemical rockets. Since the thrust is low ion rockets can only be employed in space and are not effective for a launch vehicle.

The simplest ion rocket is essentially no different than a vacuum tube or a particle accelerator. Atoms are stripped of their electrons by electromagnetic means. The ions are then attracted to a highly negatively charged grid. There may be a series of these, where quadrupole magnets maintain ion beam stability. At the end of the process the electrons stripped from the ions are sent into the beam to recombine with the ions and the high velocity beam of ions, or recombined atoms, defines the rocket plume. This form of the ion drive is the electrostatic ion thruster.

Fig. 3.2. Schematic for an ion propulsion unit.

The typical thrust of an ion rocket is a fraction of a Newton of force. The accelerations of an ion rocket is 10~3 "gees," g = 9.8 m/sec2, or an acceleration around a centimeter per second squared. It is clear that this can't be used as a launch vehicle off the Earth's surface, it wouldn't lift off the ground. However, the exhaust plume travels at around 30 km/sec, where based on a 2.5 multiplication factor means the spacecraft may reach a final velocity of around 75 km/sec. Yet it is apparent that by v = at that it takes 7,500,000 sec or a quarter of a year for the craft to reach this terminal velocity. This can reduce a four year trip to Jupiter to a single year, with similar time reductions for any interplanetary exploration.

Ion thrusters have been used by the Russians for maintaining orbits, called station keeping, such as with geosynchronous satellites. The solar wind can perturb the orbit of such a satellite and the ion rocket is used to correct the orbit. NASA developed the electrostatic xenon ion engine called NSTAR for use in their interplanetary missions. The successful space probe Deep Space 1 employed this thruster. Deep Space 1 is a prototype of an interplanetary spacecraft propelled by an ion motor. Hughes Aerospace has developed the XIPS (Xenon Ion Propulsion System) for performing station keeping on geosynchronous satellites.

The Russians developed a Hall effect ion rocket motor. This employs the Hall effect, where electrons are caught in a circular orbit within a magnetic field, to trap electrons and use them to strip the electrons off of atoms, which are then electromagnetically ejected out for thrust. This system has been used for station keeping of satellites.

The High Power Electric Propulsion, or HiPEP was ground tested in 2003 by NASA [3.3]. The HiPEP engine produces xenon ions by a combination of microwave and magnetic fields to oscillate electrons in the propel-lant atoms, causing the energetic electrons to escape free of the propellant atoms, to produce positive ions. This Electron Cyclotron Resonance (ECR) is similar to the VASIMR process.

These are high impulse, low thrust engines that have comparatively high power requirements. Ion thrust is likely to be the next step for interplanetary spacecraft. This will require either solar power or nuclear power to drive them, where the latter power system choice will most likely be employed for spacecraft targeted to the gas giant planets due to low solar illumination in these more outer regions of the solar system. For the same reason the RTG type decay reactors were used on the Galileo and Cassini spacecrafts. To explore the outer reaches of the solar system nuclear power systems are an obvious necessity. This means that safety considerations for their deployment from Earth are critical issues.

NOZZLE SKIRT EXTENSION

NOZZLE SKIRT EXTENSION

Este Rhii reflector

Oise SHIELD

Fig. 3.3. Proposed design for the NERVA rocket engine.

Este Rhii reflector

Oise SHIELD

Fig. 3.3. Proposed design for the NERVA rocket engine.

The next form of propulsion beyond chemical rockets are nuclear propulsion units. The nuclear propelled rocket has some problematic safety and environmental issues, which have to be addressed. The essential idea is that liquid hydrogen is forced through heat pipes that run through a high temperature nuclear reactor [3.2]. The heat of the reactor is transferred to the hydrogen, which expands rapidly as the gas passes through the heat pipes to exit the engine out a nozzle at high velocity. The design of the NERVA rocket involved a solid core reactor. The solid-core can only be run at temperatures below the melting point of the materials used in the reactor core. This required lower temperature reduces thrust. A rocket engine is more efficient for high velocities of the plume, which means higher temperatures. A solid-core reactor design must be constructed of materials which remain strong at high temperatures. These material science limitations, where advanced materials melt at temperatures below temperatures which could be generated in a nuclear reaction, results in a loss of possible energy these reactions generate. Further, the system can't be run too close to the melting point, for otherwise while the core structure might not melt it will deform. An addition problem is that neutrons in the reactor core will induce damage to the crystalline structure of these materials, which will lower their tolerance to high temperatures. Generally the solid-core design is expected to deliver specific impulses on the order of 1000 sec, or about twice that of the space shuttle engines. However, the thrust of these rockets is comparatively low. The acceleration expected at this point is less than one gee, which means that nuclear propulsion on the engineering theory level is so far incapable of being a launch vehicle.

A crude form of this engine was actually built and tested in 1959, called Kiwi, where the designation indicated it was a static test system. This was essentially a hot running nuclear reactor of uranium oxide where liquid hydrogen was poured over the hot reactor. The system generated 70 megawatts (MW) and produced an plume at a temperature 2683K. This lead to the Phoebus series of nuclear propulsion static test devices. This involved a much larger nuclear reactor. The final test series in this program was conducted in 1968. The static test engine was run for over 12 minutes at 4,000 MW. This is the most powerful nuclear reactor ever built. The program was cancelled out and no nuclear propulsion test have been conducted since.

The best candidate for a high specific impulse nuclear propulsion system is the gas core reactor. However, some care combined with temerity would be involved with lighting this candle. Here the uranium is heated to a gas by its own nuclear reactivity. A configured gas of hydrogen is established to prevent the hot uranium from making contact with the wall of the reactor. This surrounding shell of hydrogen flows around this core and provides the rocket reaction mass. In effect this propulsion system is a sustained nuclear explosion. Quite obviously the rocket plume would drag a fair amount of the radionucleides out with them. To prevent or reduce this various shields have been proposed, such as a quartz container. However, so far there does not appear to be a material capable of such sustained containment. Further, a fair amount of detailed balance would be required to control this thruster. Yet this propulsion system could in principle deliver a specific impulse of s ~ 3000 sec. No testbed system of this type has been constructed or tested. Radiation safety and environmental concerns with this have blocked any serious consideration for its development.

In between the solid core and gaseous core nuclear propulsion options are various architectures for liquid core systems. These systems have specific impulses is the 1000 to 2000 sec range. This is also expected to have some radioactivity in its rocket plume. If this were developed it, as well as the gaseous core thruster would have to be run outside the magnetopause, a region near Earth of trapped charged particles in the planetary magnetic field, so as to prevent contamination on Earth. Again for safety and environmental concerns no serious program has been initiated for development of this.

There is then a form of nuclear propulsion that might be called outrageous. The earliest concept suggested by Stanislaw Ulam had small nuclear bombs exploding at the rear of a spacecraft. Project Orion was a design study carried out by General Atomics in the late 1950's and 60's. At the rear of the craft is a pusher plate, constructed with large amounts of steel, the explosion pushed on in accordance with Newton's third law of motion. A series of nuclear detonations would then push the craft forward in a sort of "putt-putt" fashion. The pusher plate would be attached to the body of the craft by large shock absorbers. For nuclear explosives as shaped charges that direct their explosion in a dipole fashion, instead of in a spherically symmetric release of energy, this could efficiently produce a specific impulse s = 6000 sec. The original design was an interplanetary ship with a crew of 100 to 200 that could reach the planets in a months instead of years.

While the system appeared workable in principle it was cancelled in 1965 due to the Treaty Banning Nuclear Weapon Tests in the Atmosphere, in Outer Space, and Under Water. Further, the use of fission devices means this spaceship would leave radioactive pollution in its wake, which near Earth would result in a measurable increase in radioactivity. Exponents of this idea propose that cleaner fusion devices, hydrogen bombs, be used instead. However, detonating a fusion bomb requires a fission bomb, so the problem remains. Currently this idea is politically moribund or dead.

A related concept was advanced with Project Daedalus [3.4]. The British Interplanetary Society conducted Project Daedalus to design a concept for an interstellar un-piloted spacecraft. The craft could reach a nearby star within a human lifetime. The system is similar in concept to Orion, but the nuclear explosions were due to the fusion of deuterium-lithium pellets.

This is a highly efficient fusion reaction to use, for there is no loss of energy in neutrons and all the energy is in charged species which can be manipulated electromagnetically. Large lasers or electron beams would implode lithium and deuterium to induce the fusion by inertial confinement. This lack of fission explosions removed the radionucleide problem with the Orion concept. The cavity for this fusion would employ magnetic fields to funnel the hot plasma helium out the back of the ship to prevent damage to the cavity walls.

This program was conducted at a time when nuclear fusion, in particular inertial confinement, appeared to have considerable promise. However, nuclear fusion power appears as remote now as it did then. The project never went beyond this theoretical level. Yet if nuclear fusion by inertial confinement is developed into a working power system this may reemerge as a reasonable propulsion system. It would likely provide an efficient propulsion

Fig. 3.4. Artist conception of the Orion spacecraft.

system for interplanetary exploration, with some prospects for interstellar probes.

The Variable Specific Impulse Magnetoplasma Rocket (VASIMR) is an extension of the ion thruster [3.5]. The thruster uses a plasma instead of a beam of ions. The neutral plasma will not suffer from beam separation issues seen with an ion beam. As such the density of the plasma beam can be much larger. This will permit higher thrusts than the ion rocket. A plasma is accelerated and controlled by electromagnetic and magnetic fields. By adjusting the density of the plasma the parameters of the thruster may be controlled. The specific impulse of the VASMIR is estimated to be in the 1,000-30,000 sec. The low end of this scale puts it in the range of the ion rocket, but for plasma density larger than an ion beam. This should provide several times the thrust of the ion rocket. Controlling the manner of heating and a magnetic choke for plasma flow, VASIMR adjusts the exhaust rate of flow and speed.

Fig. 3.5. Schematic for the Daedalus space probe.

The reaction mass or propellant is hydrogen. The application of an electromagnetic fields to H2 molecules excites the energy of electrons, causing them to become more energetic and leave the molecule. This produces a plasma of protons and electrons. The charged particles in the presence of a magnetic field will execute circular motion around the field lines given by the Lorentz equation

for q the electric charge, v the velocity of the electron and B the magnetic field vector. The cross product insures that the force on the electron is perpendicular to its velocity and the magnetic field. The use of the centripetal acceleration a = mu2r in Newton's second law determines the cyclotron frequency, the frequency the charge cycles around the magnetic field line, is then

mr for v± the component of velocity perpendicular to the magnetic field. An electromagnetic field tuned to this frequency increases the energy of the ions and heats them to 107 K.

Obviously this system must be supplied electricity by some power supply. This most likely would be from a nuclear reactor to meet the power

Reaction mass plume

Fig. 3.6. Schematic for a VASIMR propulsion unit.

Reaction mass plume

### Fig. 3.6. Schematic for a VASIMR propulsion unit.

demands of the VASIMR. So far such power requirements are formidable, requiring large power supply devices. So far this system is a laboratory system and has not seen any space application.

The final mode of propulsion is the solar sail [3.6]. This has been set for last, for this is technically not a rocket. The craft does not expel any reaction mass, nor does it need to generate large amounts of energy. As such these are beneficial aspects to this, and the physics is completely different than the previous propulsion systems. A large panel of thin material is impacted by photons from the sun. This will exert a pressure on the sail and provide an exterior thrust to the craft. Recently a test craft, Cosmos-1, failed to reach orbit due to failure of the conventional launch vehicle. However, it is likely there will be a subsequent test.

A photon has an energy given by E = hv, where h is the Planck unit of action and v is the frequency of the photon. Energy is defined by a force displaced through a distance. Force has units of N = kg-m/sec2, and energy defined as the displacement of a force E = f F • dx has the units of J = Nm, called a Joule. The momentum of this photon is given as p = E/c, which is stated now as a fact of special relativity to be investigated further on. Pressure is a force per unit area, where as seen in the discussion of Newton's law F = dp/dt. Thus a pressure due to photons is of the form

Ac dt

Ac dt

Here dE/dt is the power, with the unit of watts W = J/sec, and by dividing the power by an area A this energy is incident on this defines the energy flux density, or irradiance. The solar irradiance on the Earth is 1370 W/m2.

By dividing this by the speed of light the photon pressure on a surface perpendicular to the direction of light propagation P = 4.6 x 10~6 p, where p = N/m2 is a unit of pressure called a Pascal. This calculation assumes that the photons are absorbed by the surface instead of reflected. This means that a photon absorbing surface of one square meter will experience a tiny force of 4.6 x 10~6 N, and a surface of one square kilometer will experience of force of 4.6 N. If the photons are reflected by the surface Newton's laws indicate that the pressure on the surface is twice this amount. This is the basis of the Nichols radiometer, seen as those glass bulbs containing a vain balanced on a needle that rotates in the presence of light. Here one side of each arm of the vain is reflecting and the other side is colored to be absorbing. The pressure difference between the two sides of the thin metal piece is what causes the system to rotate. This is a windmill analogue of the solar sail.

Obviously a solar sail has to span a large area to capture as many photons as possible. Further, the material has to be made as thin as possible. A parachute designed craft that carries its payload behind it is the most obvious first idea here. For a craft sent to explore the near solar environment, a likely exploration use, a problem is that the payload is not shielded by the sail. An alternate design is to rigidify the sail with struts or masts. These may be placed opposite to the photon collecting side to protect them and the payload may be similarly shielded. The sail material is proposed to by micron thin sheets of mylar, aluminum or Kapton.

As a model situation assume solar sail made with micron thin material with a density of 1 grams per cm3. This means that the material will have a mass of 1 g (0.001 kg) for each square meter. Thus a sail with an area of 104 m2 would have a mass of 10 kg. This sail will then receive near Earth a force of 4.6 x 10~2 N. By Newton's second law F = ma it is easy to find that the acceleration would be 4.6 x 10~3 m/sec2 or 0.46 cm/sec2. This puts this solar sail in the same acceleration "ballpark" as the ion propulsion system. Of course closer to the sun this can be improved due to the higher solar irradiance. If by some means the Earthly solar irradiance may be sustained on the solar sail, say by a large space based Fresnel lens or a laser, Galileo's little equation v = at indicates that for 107 seconds the velocity increased by 104 m/sec or 10 km/sec, and so for three years of sustained acceleration a final velocity of ~ 100 km/sec is possible.

So far this has focused upon propulsion systems that obey Newtonian mechanics. The VASIMR thruster might be able to accelerate a craft to 5000 km/sec and the Daedalus probe might reach ten times this final

Fig. 3.7. A basic square solar sail craft depicted in the near Earth environment.

velocity. The upper range of these velocities relative to the Earth push the envelope of Newtonian mechanics, for small relativistic effects will manifest themselves. However, Newtonian mechanics works well enough in this domain. All of these options present far superior propulsion methods than chemical rockets for interplanetary exploration. Only the Daedalus concept approaches the prospect for interstellar exploration, which is the main topic of this book. Hence it is clear that propulsion technology that surpasses these will be required to send a spacecraft to another star. This spacecraft will have to approach some significant percentage of the speed of light in order to reach a star some 5-50 lightyears distant from Earth in a timely manner.