The Mathematics of Super String Theory

 

 

The single most important equation in (first quantisized bosonic) string theory is the N-point scattering amplitude. This treats the incoming and outgoing strings as points, which in string theory are tachyons, with momentum k_i which connect to a string world surface at the surface points z_i. It is given by the following functional integral which integrates (sums) over all possible embeddings of this 2D surface in 26 dimensions.

The functional integral can be done because it is a Gaussian to become:

This is integrated over the various points z_i. Special care must be taken because two parts of this complex region may represent the same point on the 2D surface and you don't want to integrate over them twice. Also you need to make sure you are not integrating multiple times over different paramaterisations of the surface. When this is taken into account it can be used to calculate the 4-point scattering amplitude (the 3-point amplitude is simply a delta function):

Which is a beta function. It was this beta function which was apparently found before full string theory was developed. With superstrings the equations contain not only the 10D space-time coordinates X but also the grassman coordinates θ. Since there are various ways this can be done this leads to different string theories.

When integrating over surfaces such as the torus, we end up with equations in terms of theta functions and elliptic functions such as the Dedekind eta function. This is smooth everywhere, which it has to be to make physical sense, only when raised to the 24th power. This is the origin of needing 26 dimensions of space-time for bosonic string theory. The extra two dimensions arise as degrees of freedom of the string surface.

D-Branes

 

 

D-Branes are membrane-like objects in 10D string theory. They can be thought of as occurring as a result of a Kaluza-Klein compactification of 11D M-Theory which contains membranes. Because compactification of a geometric theory produces extra vector fields the D-branes can be included in the action by adding an extra U(1) vector field to the string action.






In type I open string theory, the ends of open strings are always attached to D-brane surfaces. A string theory with more gauge fields such as SU(2) gauge fields would then correspond to the compactification of some higher dimensional theory above 11 dimensions which is not thought to be possible to date.

Why Five Superstring Theories?

 

 

For a 10 dimensional supersymmetric theory we are allowed a 32-component Majorana spinor. This can be decomposed into a pair of 16-component Majorana-Weyl (chiral) spinors. There are then various ways to construct an invariant depending on whether these two spinors have the same or opposite chiralities:

Superstring Model	Invariant
Heterotic
IIA
IIB

The heterotic superstrings come in two types SO(32) and E8xE8 as indicated above and the type I superstrings include open strings.

Sources:

Wikipedia

SPACE POWER REACTORS Part III

"Energi Nuklir di masa yang akan datang menjadi kunci bagi bahan bakar penjelajahan manusia ke 

tempat-tempat yg sebelumnya belum dikunjungi"

~Arip~

This EOE article is adapted from an information paper published by the World Nuclear Association (WNA). WNA information papers are frequently updated, so for greater detail or more up to date numbers, please see the latest version on WNA website (link at end of article).

 

Heatpipe Power System

Heatpipe Power System (HPS) reactors are compact fast reactors producing up to 100 kWe for about ten years to power a spacecraft or planetary surface vehicle. They have been developed since 1994 at the Los Alamos National Laboratory in New Mexico as a robust and low technical-risk system with an emphasis on high reliability and safety. They employ heatpipes to transfer energy from the reactor core to make electricity using Stirling cycle or Brayton cycle converters. 

Energy from fission is conducted from the fuel pins to the heatpipes filled with sodium vapor that carry it to the heat exchangers and then in hot gas to the power conversion systems to make electricity. The gas is 72% helium and 28% xenon. 

The reactor itself contains a number of heatpipe modules with the fuel. Each module has its central heatpipe with rhenium-clad fuel sleeves arranged around it. They are the same diameter and contain 97% enriched uranium nitride fuel, all within the cladding of the module. The modules form a compact hexagonal core. 

Control is obtained by six stainless steel-clad beryllium drums, each 11 or 13 cm diameter, with boron carbide forming a 120 degree arc on each. The drums fit within the six sections of the beryllium radial neutron reflector surrounding the core, and rotate to effect control, moving the boron carbide in or out. Shielding is dependent on the mission or application, but lithium hydride in stainless steel cans is the main form of neutron shielding. 

The SAFE-400 space fission reactor (Safe Affordable Fission Engine) is a 400 kWt HPS producing 100 kWe to power a space vehicle using two Brayton power systems—gas turbines driven directly by the hot gas from the reactor. Heat exchanger outlet temperature is 880°C. The reactor has 127 identical heatpipe modules made of molybdenum, or niobium with 1% zirconium. Each has three fuel pins 1 cm diameter, nesting together into a compact hexagonal core 25 cm across. The fuel pins are 70 cm long (fuelled length 56 cm), the total heatpipe length is 145 cm, extending 75 cm above the core, where they are coupled with the heat exchangers. The core with reflector has a 51 cm diameter. The mass of the core is about 512 kg and each heat exchanger is 72 kg. SAFE has also been tested with an electric ion drive.

A smaller version of this kind of reactor is the HOMER-15—the Heatpipe-Operated Mars Exploration Reactor. It is a15 kW thermal unit similar to the larger SAFE model, and stands 2.4 meters tall including the heat exchanger and 3 kWe Stirling engine (see above). It operates at only 600°C and is therefore able to use stainless steel for fuel pins and heatpipes, which are 1.6 cm diameter. It has 19 sodium heatpipe modules with 102 fuel pins bonded to them, 4 or 6 per pipe, that hold a total of 72 kg of fuel. The heatpipes are 106 cm long and fuel height 36 cm. The core is hexagonal (18 cm across) with six beryllium oxide (BeO) pins in the corners. Total mass of reactor system is 214 kg, and diameter is 41 cm.

In the 1980s, the French ERATO program considered three 20 kWe turboelectric power systems for space operation. All used a Brayton cycle converter with a helium-xenon mix as working fluid. The first system was a sodium-cooled uranium dioxide-fuelled fast reactor operating at 670°C, the second a high-temperature gas-cooled reactor (thermal or epithermal neutron spectrum) working at 840°C, and the third a lithium-cooled uranium nitride-fuelled fast reactor working at 1150°C.

Project Prometheus 2003

In 2002, the U.S. National Aeronautics and Space Administration (NASA) announced its Nuclear Systems Initiative for space projects, and in 2003 this was renamed Project Prometheus and given increased funding. Its purpose is to enable a major step change in the capability of space missions. Nuclear-powered space travel will be much faster than is now possible, and will enable manned missions to Mars. 

One part of Prometheus, a NASA project with substantial involvement by the U.S. Department of Energy (DOE), is to develop the Multi-Mission Thermoelectric Generator and the Stirling Radioisotope Generator described in the RTG section above. 

A more radical objective of Prometheus is to produce a space fission reactor system such as those described above for both power and propulsion that is safe to launch and will operate for many years. This will have much greater power than RTGs. Power of 100 kW is envisaged for a nuclear electric propulsion system driven by plasma. 

The fiscal year 2004 budget proposal was US$279 million, with $3 billion to be spent over five years. This consists of $186 million ($1 billion over 5 years) building on last year's allocation, plus $93 million ($2 billion over five years) towards a first flight mission to Jupiter—the Jupiter Icy Moon Orbiter—expected to launch in 2017 and explore for a decade. Project Prometheus received $430 million in 200r budget. 

In 2003, NASA's Project Prometheus successfully tested a High Power Electric Propulsion (HiPEP) ion engine. This operates by ionizing xenon with microwaves. At the rear of the engine is a pair of rectangular metal grids that are charged with 6,000 volts of electric potential. The force of this electric field exerts a strong electrostatic pull on the xenon ions, accelerating them and producing the thrust needed to propel the spacecraft. The design was tested at up to 12 kW, though twice that is envisaged. The thruster is designed for a 7- to 10-year lifetime with high fuel efficiency, and to be powered by a small nuclear reactor.

Further Reading

• WNA paper on Nuclear reactors for space

Citation

World Nuclear Association, Ian Hore-Lacy (Lead Author);Cutler Cleveland (Topic Editor) "Nuclear reactors for space". In: Encyclopedia of Earth. Eds. Cutler J. Cleveland (Washington, D.C.: Environmental Information Coalition, National Council for Science and the Environment). [First published in the Encyclopedia of Earth December 6, 2009; Last revised Date December 6, 2009; Retrieved October 26, 2011 <http://www.eoearth.org/article/Nuclear_reactors_for_space>

The Authors

The World Nuclear Association is the global private-sector organization that seeks to promote the peaceful worldwide use of nuclear power as a sustainable energy resource for the coming centuries. Specifically, the WNA is concerned with nuclear power generation and all aspects of the nuclear fuel cycle, including mining, conversion, enrichment, fuel fabrication, plant manufacture, transport, and the safe disposition of spent fuel.The WNA serves its Members by facilitating their interaction on te ... (Full Bio)

Ian Hore-Lacy is Director for Public Communications at the World Nuclear Association, an international trade association based in London, and has been involved with studying uranium and nuclear power since 1995. His function is primarily focused on public information on nuclear power via the World Wide Web. He is a former biology teacher who joined the mining industry as an environmental scientist in 1974, with CRA (now Rio Tinto). He is author of Nuclear Electricity, the expanded eighth editi ... (Full Bio)

Sources:

1. Poston, D.I. 2002, Nuclear design of SAFE-400 space fission reactor, Nuclear News, Dec 2001.
2. Poston, D.I. 2002, Nuclear design of HOMER-15 Mars surface fission reactor, Nuclear News, Dec 2001.
3. Vrillon et al, 1990, ERATO article, Nuclear Europe Worldscan 11-12, 1990.
4. US DOE web site- space applications.
5. space.com 21/5/00, 16/6/00, 22/7/00, 17/1/03, 7/2/03.
6. www.nuclearspace.com
7. Delovy Mir 8/12/95.
8. G. Kulcinski, University of Wisconsin material on web.
9. Kleiner K. 2003, Fission Control, New Scientist 12/4/03.
10. OECD 1990, Emergency Preparedness for Nuclear-Powered Satellites.

11. NASA web site

Question About Gravitation Part I

"Kutemukan cinta dalam gemerlapnya bintang gemintang, kasih dalam terangnya cahaya rembulan
serta Keagungan-Nya dalam hangat cahaya Mentari"
~Arip~



Edited and Added By:

Arip Nurahman

Department of Physics

Faculty of Sciences and Mathematics, Indonesian University of Education
 
&

Follower Open Course Ware at MIT-Harvard University, M.A., U.S.A.

Gravitation is a natural phenomenon by which objects with mass attract one another. In everyday life, gravitation is most commonly thought of as the agency which lends weight to objects with mass. Gravitation causes dispersed matter to coalesce, thus accounting for the existence of the Earth, the Sun, and most of the macroscopic objects in the universe. It is responsible for keeping the Earth and the other planets in their orbits around the Sun; for keeping the Moon in its orbit around the Earth; for the formation of tides; for convection, by which fluid flow occurs under the influence of a density gradient and gravity; for heating the interiors of forming stars and planets to very high temperatures; and for various other phenomena observed on Earth. Modern physics describes gravitation using the general theory of relativity, in which gravitation is a consequence of the curvature of spacetime which governs the motion of inertial objects. The simpler Newton's law of universal gravitation provides an accurate approximation for most calculations.

(Wikipedia)

Q: What is gravity?

A: The answer cannot be found in today's theories. Newton only claimed that gravity was an attracting force between all objects because that's the way things appear objects fall to the Earth or approach each other when floating in outer space. So Newton understandably claimed that it must be some type of attracting force emanating from objects, but he gave no scientific explanation for this force. 

Why does it attract and not repel? How does it cause falling objects and orbiting planets without drawing on any known power source? Einstein was so dissatisfied with our lack of understanding about gravity even two centuries after Newton that he invented an entirely new theory of gravity, known as General Relativity. 

Yet this theory doesn’t solve these problems either, adding that since everything in 3-dimensional space takes time to occur we must include our time measurement as a literal 4th physical dimension of our universe’s structure hence "4-D space-time", which somehow warps 4-dimensionally in the presence of matter for still-unexplained reasons, presumably explaining gravity.

In addition to the increase in unanswered questions with General Relativity, it has been found to completely fail even to explain the motion of stars in galaxies. This has led to the further invention of exotic "Dark Matter", said to invisibly fill galaxies, rather than questioning Einstein’s theory and the often-repeated claim that it has been tested to extreme accuracy. 

Add to this the fact that there are still a half-dozen theories of gravity officially under consideration at the moment, all with different physical description of gravity, and it is no wonder many are still asking:

"What is gravity?" From "Nailing Down Gravity", Discover Magazine, Oct 2003: For Michael Martin Nieto, a theoretical physicist at Los Alamos National Laboratory in New Mexico, the mystery involves much more than a few hunks of spacefaring hardware; it reveals that there might be something wrong with our understanding of gravity, the most pervasive force in the universe.

"We don't know anything," he says. "Everything about gravity is mysterious".

To Be Continued

Sources:

1. http://www.thefinaltheory.com/scienceflaws.html
2. http://en.wikipedia.org/wiki/Gravitation