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User:Marshallsumter/Keynote lectures (draft)/Cosmic radiation astronomy

Cosmic rays are energetic charged subatomic particles, originating in outer space.

At right is an image indicating the range of cosmic-ray energies. The flux for the lowest energies (yellow zone) is mainly attributed to solar cosmic rays, intermediate energies (blue) to galactic cosmic rays, and highest energies (purple) to extragalactic cosmic rays.

“Cosmic ray astronomy attempts to identify and study the sources of ultrahigh energy cosmic rays. It is unique in its reliance on charged particles as the information carriers.”

Astronomy
"Astronomy based on cosmic rays with the highest energies [above 6 x 1019 electron volts] opens a new window on the nearby universe."

Cosmic rays
"Cosmic rays arise from galactic source accelerators."

Cosmic rays may be upwards of a ZeV (1021 eV).

About 89% of cosmic rays are simple protons or hydrogen nuclei, 10% are helium nuclei of alpha particles, and 1% are the nuclei of heavier elements. Solitary electrons constitute much of the remaining 1%.

Def. cosmic rays that originate from astrophysical sources are called primary cosmic rays.

Def. cosmic rays that are created when primary cosmic rays interact with interstellar matter are called secondary cosmic rays.

Def. low energy cosmic rays associated with solar flares are called solar cosmic rays.

Cosmic rays are not charge balanced; that is, positive ions heavily outnumber electrons. The positive ions are
 * 1) free protons,
 * 2) alpha particles (helium nuclei),
 * 3) lithium nuclei,
 * 4) beryllium nuclei, and
 * 5) boron nuclei.

Def. a nuclear reaction in which a nucleus fragments into many nucleons is called spallation.

Cosmic rays cause spallation when a ray particle (e.g. a proton) impacts with matter, including other cosmic rays. The result of the collision is the expulsion of large numbers of nucleons (protons and neutrons) from the object hit.

Carbon and oxygen nuclei collide with interstellar matter to form lithium, beryllium and boron in a process termed cosmic ray spallation. Spallation is also responsible for the abundances of scandium, titanium, vanadium, and manganese ions in cosmic rays produced by collisions of iron and nickel nuclei with interstellar matter.

Planetary sciences
"Production rates of 22Na (T1/2 = 2.6 years) from aluminium by the action of cosmic rays are measured at the Mont Blanc (altitude 4600 m), the Aiguille du Midi (3840 m), and the Col du Lautaret (2070 m). They are 2.3±0.5, 1.8±0.3, and 0. 77±0.18 atoms min−1 kg−1, respectively, in good agreement with the calculated production rates, 2.4, 1.7 and 0.6 atoms min−1 kg−1, respectively, at the three stations."

Colors
"[B]roadband optical photometry of Centaurs and Kuiper Belt objects from the Keck 10 m, the University of Hawaii 2.2 m, and the Cerro Tololo InterAmerican (CTIO) 1.5 m telescopes [shows] a wide dispersion in the optical colors of the objects, indicating nonuniform surface properties. The color dispersion [may] be understood in the context of the expected steady reddening due to bombardment by the ubiquitous flux of cosmic rays."

Larmor radius
The Larmor radius is the radius of the circular motion of a charged particle] in the presence of a uniform magnetic field. “[F]or a particle of energy E in EeV and charge Z in a magnetic field B in µG [the Larmor radius (RL)] is roughly”

 * $$R_L = 1 kpc \frac{E}{Z B}$$

where
 * $$R_L \ $$ is the Larmor radius,
 * $$E\ $$ is the energy of the particle in EeV
 * $$Z \ $$ is the charge of the particle, and
 * $$B \ $$ is the constant magnetic field.

Greisen-Zatsepin-Kuzmin limits
Notation: let the symbol GZK represent Greisen-Zatsepin-Kuzmin.

Based on interactions between cosmic rays and the photons of the cosmic microwave background radiation (CMB) cosmic rays with energies over the threshold energy of 5x1019 eV interact with cosmic microwave background photons $$\gamma_{\rm CMB}$$ to produce pions via the $$\Delta$$ resonance.

 * $$\gamma_{\rm CMB}+p\rightarrow\Delta^+\rightarrow p + \pi^0,$$

or

 * $$\gamma_{\rm CMB}+p\rightarrow\Delta^+\rightarrow n + \pi^+.$$

Pions produced in this manner proceed to decay in the standard pion channels—ultimately to photons for neutral pions, and photons, positrons, and various neutrinos for positive pions. Neutrons decay also to similar products, so that ultimately the energy of any cosmic ray proton is drained off by production of high energy photons plus (in some cases) high energy electron/positron pairs and neutrino pairs.

The pion production process begins at a higher energy than ordinary electron-positron pair production (lepton production) from protons impacting the CMB, which starts at cosmic ray proton energies of only about 1017eV. However, pion production events drain 20% of the energy of a cosmic ray proton as compared with only 0.1% of its energy for electron positron pair production. This factor of 200 is from two sources: the pion has only about ~130 times the mass of the leptons, but the extra energy appears as different kinetic energies of the pion or leptons, and results in relatively more kinetic energy transferred to a heavier product pion, in order to conserve momentum. The much larger total energy losses from pion production result in the pion production process becoming the limiting one to high energy cosmic ray travel, rather than the lower-energy light-lepton production process.

The pion production process continues until the cosmic ray energy falls below the pion production threshold. Due to the mean path associated with this interaction, extragalactic cosmic rays traveling over distances larger than 50 Mpc (163 Mly) and with energies greater than this threshold should never be observed on Earth. This distance is also known as GZK horizon.

Askaryan effects
The Askaryan effect is the phenomenon whereby a particle traveling faster than the phase velocity of light in a dense dielectric (such as salt, ice or the lunar regolith) produces a shower of secondary charged particles which contain a charge anisotropy and thus emits a cone of coherent radiation in the radio or microwave part of the electromagnetic spectrum. It is similar to the Cherenkov effect.

So far the effect has been observed in silica sand, rock salt, and ice.

Ultra-high energy cosmic rays
The Oh-My-God particle was observed on the evening of 15 October 1991 over Dugway Proving Ground, Utah. Its observation was a shock to astrophysicists, who estimated its energy to be approximately $3 eV$ (50 joules)—in other words, a subatomic particle with kinetic energy equal to that of a baseball (142 g or 5 oz) traveling at 100 km/h (60 mph).

It was most probably a proton with a speed very close to the speed of light (approximately 0.9999999999999999999999951c), so close that in a year-long race between light and the cosmic ray, the ray would fall behind only 46 nanometers (5 x 10-24 light-years), or 0.15 femtoseconds (1.5 x 10-16 s).

“The energy spectrum of cosmic rays extends to ~1020 eV (and smoothly to 1019).”

Minerals
"Bombardment by protostellar cosmic rays may make the rock precursors of [Calcium-aluminum-rich inclusions] CAIs and chondrules radioactive, producing radionuclides found in meteorites that are difficult to obtain with other mechanisms."

"The Earth is continually being bombarded by high-energy cosmic rays that originate predominantly from super nova explosions within our galaxy. Interactions between these high energy cosmic rays and the Earth's atmosphere creates secondary and tertiary cosmic rays, including neutrons and muons."

"When reaching the Earth's surface these high energy particles can penetrate meters into rock and sediment."

"Nuclear interactions between neutrons and muons and minerals [as in the diagram at the right] such as quartz, calcite, K-feldspar, and olivine, produce long-lived radionuclides such as Be-10, Al-26 and Cl-36."

"The production rates of these "in-situ produced terrestrial cosmogenic nuclides" are almost unimaginably small - a few atoms per gram of rock per year, however using accelerator mass spectrometry (AMS) we can detect and count cosmogenic nuclides down to levels of a few thousand atoms per gram (parts per million of parts per billion!)."

"The build-up of cosmogenic nuclides through time provides us with a way to measure exposure ages for rock surfaces such as fault scarps, lava flows and glacial pavements.Where surfaces are gradually evolving, cosmogenic nuclide measurements allow us to calculate erosion or soil accumulation rates.Where previously exposed rock or sediment is re-buried the relative decay between different cosmogenic nuclides can be used to date the burial time."

Theoretical cosmic-ray astronomy
"The phenomenology of cosmic ray cascades ... reflects in an essential way processes governed by the strong force."

"Full multiple scattering theory must take account of the angular dependence of hadron-nucleon scattering, which affects the degree of screening."

Objects
"Comparison with the chemical composition of various astrophysical objects, such as the Sun, the interstellar medium, supernovae or neutron stars, can give clues about the site at which cosmic rays are injected into the acceleration process."

Strong forces
"In field theory it is known that coupling constants “run”. This means that the values of the coupling constants that one measures depend on the energy at which the measurement is performed. [...] the three different coupling constants [one each for the strong force, electromagnetic force, and the weak force] of the standard model seem to converge to the same value at an energy scale of about 1016 GeV [...] This suggests that there is only one coupling constant at high energies and most likely only one symmetry group. [...] The current belief [is] that the electromagnetic, weak and strong forces [are] unified at about 1016 GeV [as such] one has to rely on [the] particle physics interactions which can lead to electromagnetic radiation and cosmic rays".

Weak forces
"The feature that makes deep inelastic lepton scattering and e+e- annihilation tractable is that these processes proceed via the electromagnetic and weak interactions."

Continua
Continuum "radiation ... diffuse gamma rays with energies above 10 MeV. In the galaxy these are produced primarily by bremsstrahlung from cosmic ray electrons and from decay in flight of π0's produced by interactions of cosmic ray protons."

Backgrounds
"The attenuation of photons in the microwave background via the process

 * $$\gamma + \gamma (3^o K) \rightarrow e^+ e^-$$

is strongly energy dependent, with a minimum attenuation length of ≈ 7 kpc around 2.5 PeV, as determined by the threshold for e+e- production (Gould and Schreder, 1966; Jelley, 1966)."

Meteors
"[T]he carbonaceous material [is] known from observation to dominate the terrestrial [micrometeorite (MM)] flux."

"Ureilites occur about half as often as eucrites (Krot et al. 2003), are relatively friable, have less a wide range of cosmic-ray exposure ages including two less than 1 Myr, and, like the dominant group of MM precursors, contain carbon."

Anomalous cosmic rays
"While interstellar plasma is kept outside the heliosphere by an interplanetary magnetic field, the interstellar neutral gas flows through the solar system like an interstellar wind, at a speed of 25 km/sec. When closer to the Sun, these atoms undergo the loss of one electron in photo-ionization or by charge exchange. Photo-ionization is when an electron is knocked off by a solar ultra-violet photon, and charge exchange involves giving up an electron to an ionized solar wind atom. Once these particles are charged, the Sun's magnetic field picks them up and carries them outward to the solar wind termination shock. They are called pickup ions during this part of their trip."

"The ions repeatedly collide with the termination shock, gaining energy in the process. This continues until they escape from the shock and diffuse toward the inner heliosphere. Those that are accelerated are then known as anomalous cosmic rays."

"ACRs [may] represent a sample of the very local interstellar medium. They are not thought to have experienced such violent processes as GCRs, and they have a lower speed and energy. ACRs include large quantities of helium, oxygen, neon, and other elements with high ionization potentials, that is, they require a great deal of energy to ionize, or form ions. ACRs are a tool for studying the movement of energetic particles within the solar system, for learning the general properties of the heliosphere, and for studying the nature of interstellar material itself."

Hadrons
The "problems plaguing (3 + 1)- dimensional quantum gravity quantization programs are solved by virtue of the fact that spacetime is dimensionally-reduced. Indeed, effective models of quantum gravity are plentiful in (2 + 1) and even (1 + 1) dimensions [11–13]. Similarly, the cosmological constant problem may be explained as a Casimir-type energy between two adjacent “foliations” of three-dimensional space as the scale size L > L4 opens up a fourth space dimension."

"What makes this proposal of evolving dimensions very attractive is that some evidence of the lower dimensional structure of our space-time at a TeV scale may already exist. Namely, alignment of the main energy fluxes in a target (transverse) plane has been observed in families of cosmic ray particles [18–20]. The fraction of events with alignment is statistically significant for families with energies higher than TeV and large number of hadrons. This can be interpreted as evidence for coplanar scattering of secondary hadrons produced in the early stages of the atmospheric cascade development."

In the image on the right, the "energy spectrum of hadrons in cosmic rays at sea level has been measured over the energy range 600 GeV - 8 TeV. The spectrum is found to be well represented in differential form by N(E)dE = AE-𝛄dE where 𝛄 = 2.74 ± 0.16 with no suggested anomalous behaviour over the whole energy range."

"Incident hadrons either interact in the lead (15 cm thick) or iron (15 cm thick) targets and the resulting cascade traverses the plastic scintillators [...] which are both 5 cm thick. Using a burst of size > 400 equivalent muons traversing either scintillator as a master trigger a high voltage pulse was applied to the flash tubes, which are photographed, after a time delay of 330 𝛍s. From the resulting photograph the projected angle of incidence of the incident hadron could be determined and a decision taken a to whether is was in the acceptance geometry as defined [...]. [...] In converting the burst spectrum measurements to an estimate of the incident hadron spectrum the hadrons have been assumed to be nucleons. If charged pions are assumed the energies shown in [the image on the right] should be reduced by 0.8."

Neutrons
Because free neutrons are unstable, they can be obtained only from nuclear disintegrations, nuclear reactions, and high-energy reactions (such as in cosmic radiation showers or accelerator collisions).

The Bonner Ball Neutron Detector "BBND ... determined that galactic cosmic rays were the major cause of secondary neutrons measured inside ISS. The neutron energy spectrum was measured from March 23, 2001 through November 14, 2001 in the U.S. Laboratory Module of the ISS. The time frame enabled neutron measurements to be made during a time of increased solar activity (solar maximum) as well as observe the results of a solar flare on November 4, 2001."

"BBND results show the overall neutron environment at the ISS orbital altitude is influenced by highly energetic galactic cosmic rays, except in the South Atlantic Anomaly (SAA) region where protons trapped in the Earth's magnetic field cause a more severe neutron environment. However, the number of particles measured per second per square cm per MeV obtained by BBND is consistently lower than that of the precursor investigations. The average dose-equivalent rate observed through the investigation was 3.9 micro Sv/hour or about 10 times the rate of radiological exposure to the average US citizen. In general, radiation damage to the human body is indicated by the amount of energy deposited in living tissue, modified by the type of radiation causing the damage; this is measured in units of Sieverts (Sv). The background radiation dose received by an average person in the United States is approximately 3.5 milliSv/year. Conversely, an exposure of 1 Sv can result in radiation poisoning and a dose of five Sv will result in death in 50 percent of exposed individuals. The average dose-equivalent rate observed through the BBND investigation is 3.9 micro Sv/hour, or about ten times the average US surface rate. The highest rate, 96 microSv/hour was observed in the SAA region."

Protons
About 89% of cosmic rays are simple protons or hydrogen nuclei.

The free proton is stable and is found naturally in a number of situations. Free protons exist in plasmas in which temperatures are too high to allow them to combine with electrons. Free protons of high energy and velocity make up 90% of cosmic rays, which propagate for interstellar distances.

"Proton astronomy [since protons are also most cosmic rays] should be possible; it may also provide indirect information on inter-galactic magnetic fields."

Antiprotons have been detected in cosmic rays for over 25 years, first by balloon-borne experiments and more recently by satellite-based detectors. The standard picture for their presence in cosmic rays is that they are produced in collisions of cosmic ray protons with nuclei in the interstellar medium, via the reaction, where A represents a nucleus:

 * + A → +  +  + A

The secondary antiprotons then propagate through the galaxy, confined by the galactic magnetic fields. Their energy spectrum is modified by collisions with other atoms in the interstellar medium. The antiproton cosmic ray energy spectrum is now measured reliably and is consistent with this standard picture of antiproton production by cosmic ray collisions.

Kosmos 60 measured the gamma-ray background flux density to be 1.7×104 quanta/(m2·s). As was seen by Ranger 3 and Lunas 10 & 12, the spectrum fell sharply up to 1.5 MeV and was flat for higher energies. Several peaks were observed in the spectra which were attributed to the inelastic interaction of cosmic protons with the materials in the satellite body.

Electrons
Solitary electrons constitute much of the remaining 1% of cosmic rays.

"The conventional procedure of delta-ray counting to measure charge (Powell, Fowler, and Perkins 1959), which was limited to resolution sigmaz = 1-2 because of uncertainties of the criterion of delta-ray ranges, has been significantly improved by the application of delta-ray range distribution measurements for 16O and 32S data of 200 GeV per nucleon (Takahashi 1988; Parnell et al. 1989)." Here, the delta-ray tracks in emulsion chambers have been used for "[d]irect measurements of cosmic-ray nuclei above 1 TeV/nucleon ... in a series of balloon-borne experiments".

Positrons
A few antiprotons and positrons are in primary cosmic rays.

"In the first 18 months of operations, AMS-02 [image under Cherenkov detectors] recorded 6.8 million positron (an antimatter particle with the mass of an electron but a positive charge) and electron events produced from cosmic ray collisions with the interstellar medium in the energy range between 0.5 giga-electron volt (GeV) and 350 GeV. These events were used to determine the positron fraction, the ratio of positrons to the total number of electrons and positrons. Below 10 GeV, the positron fraction decreased with increasing energy, as expected. However, the positron fraction increased steadily from 10 GeV to 250 GeV. This increase, seen previously though less precisely by instruments such as the Payload for Matter/antimatter Exploration and Light-nuclei Astrophysics (PAMELA) and the Fermi Gamma-ray Space Telescope, conflicts with the predicted decrease of the positron fraction and indicates the existence of a currently unidentified source of positrons, such as pulsars or the annihilation of dark matter particles. Furthermore, researchers observed an unexpected decrease in slope from 20 GeV to 250 GeV. The measured positron to electron ratio is isotropic, the same in all directions."

A High-Energy Antimatter Telescope (HEAT) has been developed and tested in the mid 1990s to measure the positron fraction in cosmic rays.

There is an "unexpected rise of the positron fraction, observed by HEAT and PAMELA experiments, for energies larger than a few GeVs."

"[T]he HEAT balloon experiment [30] ... has mildly indicated a possible positron excess at energies larger than 10 GeV ... In October 2008, the latest results of PAMELA experiment [36] have confirmed and extended this feature [37]."

Earlier measurements indicate that "the positron fraction, [f = ] e+/(e- + e+), increases with energy at energies above 10 GeV. Such an increase would require either the appearance of a new source of positrons or a depletion of primary electrons." All results taken together suggest a slight decrease with increasing energy from about 1 GeV to 10 GeV, but overall the fraction may be constant, per Figure 2.

Muons
At right is an image of the Moon's cosmic ray shadow, as seen in secondary muons generated by cosmic rays in the atmosphere, and detected 700 meters below ground, at the Soudan II detector.

"To reduce the background of ordinary cosmic ray showers, several large air shower experiments emphasize measurement of the muon content of the shower. Ironically, early indications are that the signal seems to have the same muon content as the background."

Neutrinos
Neutrinos are created as a result of certain types of radioactive decay, or nuclear reactions, or when cosmic rays hit atoms.

Cosmic "ray neutrinos of local origin are also the background for neutrino astronomy."

Gamma rays
"Over the last few years, the cold dark matter cosmogony has become a fiducial model for the formation of structure. [...] The problem with detecting dark matter using annihilation radiation gamma rays has been that the expected signal is comparable to the background (Stecker 1988) and it would be difficult to separate a "cosmic-ray halo" from a dark halo."

In gamma-ray astronomy, when cosmic rays [such as protons] interact with ordinary matter ... pair-production gamma rays at 511 keV [are produced that are included in] the gamma ray background.

X-rays
Some "of the possible sources of the ultra-high energy cosmic rays, such as very young supernova remnants and X-ray binaries, are associated with relatively dense concentrations of matter and would therefore be likely point sources of secondary photons and neutrinos."

Opticals
"An accelerator in this particular supernova [TeV range] could hardly be this powerful without having altered the behavior of the optical light curve, which was very successfully explained as being powered by the radioactive decay chain of 56Ni synthesized in the explosion (Pinto & Woosley, 1988)."

"Cygnus X-3 is obscured by the disk of the galaxy and is not visible in the optical. It is therefore impossible to determine unambiguously by what means it is powered."

Blues
"The Broad LAteral Non-imaging Cherenkov Array (BLANCA) takes advantage of the CASA-MIA particle array installation by augmenting it with 144 angle-integrating Cherenkov detectors. Located in Dugway, Utah at an atmospheric depth of 870 g cm−2, BLANCA uses the CASA trigger to collect Cherenkov light and records the Cherenkov lateral distribution from cosmic ray events in the energy range of the knee. The CASA trigger threshold imposes an energy threshold of ∼ 100 TeV on the Cherenkov array. However, BLANCA analysis uses events with a 200 TeV minimum to avoid composition bias introduced from the CASA trigger."

"Each BLANCA detector contains a large Winston cone [43] which concentrates the light striking an 880 cm2 entrance aperture onto a photomultiplier tube. The concentrator has a nominal half-angle of 12.5° and truncated length of 60 cm. The Winston cones were aligned vertically with ∼ 0.5° accuracy. A two-output preamplifier increases the dynamic range of the detector. The minimum detectable density of a typical BLANCA unit is approximately one blue photon per cm2."

Greens
"Babcock, using a Fabry and Perot interferometer, determined very accurately the wave-length of the auroral green line 5577. ... After a careful examination of all the results obtained in these reports, we may only say that the exact nature of the cosmical rays, responsible for the aurora, remains a mystery. ... The origin of the most prominent and interesting line of the auroral spectrum, the line 5577, has hitherto remained unexplained. Vegard* has recently obtained a luminescent band from solid nitrogen, that he supposes, under very special conditions, may coincide with the auroral green line. ... spectra of pure helium and of pure oxygen were taken at different pressures and with various excitations, but no trace of 5577 or of any other new lines was obtained. ... Mixtures of helium, oxygen and nitrogen were excited, and it was found that the line 5577 could be photographed on the same plate with the nitrogen band system, thus reproducing in the laboratory practically the entire auroral spectrum. In ... mixtures of neon and oxygen ... neon enhanced the line 5577 in the same manner as helium. ... From Plate 20 it will be seen that all the lines except 5577 have been identified as strong lines in the spectrum of helium, hydrogen, oxygen, or mercury. ... It has been shown that this line must be attributed to some hitherto unknown spectrum of oxygen, and that it is not a limiting member of the ordinary band spectrum of oxygen. It has been observed faintly in highly purified oxygen when currents of high density have been used."

Airglow is caused by various processes in the upper atmosphere, such as the recombination of ions which were photoionized by the sun during the day, luminescence caused by cosmic rays striking the upper atmosphere, and chemiluminescence caused mainly by oxygen and nitrogen reacting with hydroxyl ions at heights of a few hundred kilometers. It is not noticeable during the daytime because of the scattered light from the Sun.

Plasma objects
"When magnetic fields "reconnect" in a turbulent magnetohydrodynamic (MHD) plasma, electric fields are generated in which particles can be accelerated (Matthaeus et al., 1984; Sorrell, 1984)."

Berylliums
The "presence in ... cosmic radiation [is] of a much greater proportion of "secondary" nuclei, such as lithium, beryllium and boron, than is found generally in the universe."

Carbons
These "are nevertheless present in the cosmic radiation as spallation products of the abundant nuclei of carbon and oxygen (Li,Be,B) and of iron (Sc,Ti,V,Cr,Mn)."

Aluminums
"The dominant reactions for making 26Al by [cosmic-ray] proton and α bombardment of refractory rocks in impulsive flares are 27Al(p, pn)26Al (β=0.92), 26Mg(p, n)26Al (β=1.0), 24Mg(α, pn)26Al (β=2.5 and yCR = 0.1), 28Si(p, 2pn)26Al (β=0.10), and 28Si(α, αpn)26Al (β=0.41)."

Ions
"Energetic photons, ions and electrons from the solar wind, together with galactic and extragalactic cosmic rays, constantly bombard surfaces of planets, planetary satellites, dust particles, comets and asteroids."

Atmospheres
Atmospheric neutrinos result from the interaction of cosmic rays with atomic nuclei in the Earth's atmosphere, creating showers of particles, many of which are unstable and produce neutrinos when they decay. A collaboration of particle physicists from the Tata Institute of Fundamental Research (India), Osaka City University (Japan) and Durham University (UK) recorded the first cosmic ray neutrino interaction in an underground laboratory in Kolar Gold Fields in India in 1965.

"The major problems associated with the balloon borne positron measurements are (i) the unique identification against a vast background of protons, and (ii) corrections for the positrons produced in the residual atmosphere."

"[T]o account for the atmospheric corrections ... first [use] the instrument to determine the negative muon spectrum at float altitude. ... [Use this] spectrum ... to normalize the analytically determined atmospheric electron-positron spectra. ... most of the atmospheric electrons and positrons at small atmospheric depths are produced from muon decay at [the energies from 0.85 to 14 GeV]."

Materials
"The two groups of elements Li, Be, B and Sc, Ti, V, Cr, Mn are many orders of magnitude more abundant in the cosmic radiation than in solar system material."

Meteorites
Cosmic ray exposure history established with noble gas measurements have shown that all lunar meteorites were ejected from the Moon in the past 20 million years. Most left the Moon in the past 100,000 years.

Imaged at lower right is an igneous Martian shergottite meteorite. "The perimeter exhibits a fusion crust from the heat of entry into the Earth’s atmosphere. It is a fresh sample of NWA 6963, an igneous Martian shergottite meteorite found in September 2011 in Morocco. Meteorites are often labeled NWA for North West Africa, not because they land there more often, but because they are easy to spot as peculiar objects in the desert sands. From the geochemistry and presence of various isotopes, the origin and transit time is deduced. The 99 meteorites from Mars exhibit precise elemental and isotopic compositions similar to rocks and atmosphere gases analyzed by spacecraft on Mars, starting with the Viking lander in 1976. Compared to other meteorites, the Martians have younger formation ages, unique oxygen isotopic composition (consistent for Mars and not for Earth), and the presence of aqueous weathering products. A trapped gas analysis concluded that their origin was Mars quite recently, in the year 2000."

"The formation ages of meteorites often come from their cosmic-ray exposure (CRE), measured from the nuclear products of interactions of the meteorite in space with energetic cosmic ray particles. This one is particularly young, having crystallized only 180 million years ago, suggesting that volcanic activity was still present on Mars at that time. Volcanic flows are the youngest part of a planet, and this one happened to be hit by a meteor impact, ejecting" it from the youthful Mars.

Spectrometers
Using Germanium detectors - a crystal of hyperpure germanium that produces pulses proportional to the captured photon energy; while more sensitive, it has to be cooled to a low temperature, requiring a bulky cryogenic apparatus. When exposed to cosmic rays (charged particles in space that come from the stars, including our sun), chemical elements in soils and rocks emit uniquely identifiable signatures of energy in the form of gamma rays. The gamma ray spectrometer looks at these signatures, or energies, coming from the elements present in the target soil. By measuring gamma rays coming from the target body, it is possible to calculate the abundance of various elements and how they are distributed around the planet's surface. Gamma rays, emitted from the nuclei of atoms, show up as sharp emission lines on the instrument's spectrum output. While the energy represented in these emissions determines which elements are present, the intensity of the spectrum reveals the elements concentrations. Spectrometers are expected to add significantly to the growing understanding of the origin and evolution of planets like Mars and the processes shaping them today and in the past.

Sun
"A persistent problem of solar cosmic-ray research has been the lack of observations bearing on the timing and conditions in which protons that escape to the interplanetary medium are first accelerated in the corona."

"For solar cosmic-rays, the apparent lack of proton acceleration in the corona seems justified, in contrast to the electrons, proton bremsstrahlung and gyrosynchrotron emission are negligible. This suggests a transit time anomaly, ΔTA, defined as follows:

 * ΔTA = ΔTonset - 11 min,

where ΔTonset is the deduced Sun-Earth transit time for the first arriving relativistic protons and 11 min is the nominal transit time for a ~2 GeV proton traversing a 1.3 AU Archimedes spiral path."

"The solar wind is a stream of charged particles ejected from the upper atmosphere of the Sun. It mostly consists of electrons and protons with energies usually between 1.5 and 10 keV. ΔTA may have values from "7-19 min for a small sample of well-connected ... cosmic-ray flares." The transit time anomaly may be explained by a rise time associated with the ground-level events (GLEs). "The average GLE rise time ... for well-connected ... events ... defined to be the time from event onset to maximum as measured by the neutron monitor station showing the largest increase and whose asymptotic cone of acceptance ... includes the nominal direction of the Archimedean spiral path, is 21.3 min."

"Data from an extensive air shower detector of ultrahigh-energy cosmic rays shows shadowing of the cosmic-ray flux by the Moon and the Sun with significance of 4.9 standard deviations. This is the first observation of such shadowing."

"The ... solar proton flare on 20 April 1998 at W 90° and S 43° (9:38 UT) was measured by the GOES-9-satellite (Solar Geophysical Data 1998), as well as by other experiments on WIND ... and GEOTAIL. Protons were accelerated up to energies > 110 MeV and are therefore able to hit the surface of Mercury." {{clear}]

Coronal clouds
"[C]oronal magnetic bottles, produced by flares, [may] serve as temporary traps for solar cosmic rays ... It is the expansion of these bottles at velocities of 300–500 km/s which allows fast azimuthal propagation of solar cosmic rays independent of energy. A coronagraph on Os 7 observed a coronal cloud which was associated with bifurcation of the underlying coronal structure."

"A persistent problem of solar cosmic-ray research has been the lack of observations bearing on the timing and conditions in which protons that escape to the interplanetary medium are first accelerated in the corona."

Mercury
From the Mariner 10 observations in electron astronomy, it is concluded that "[d]ue to the limited shielding provided by its relatively weak magnetic dipole moment, the surface of Mercury is everywhere subject to bombardment by cosmic rays and solar energetic particles with energies greater than 1 MeV/nucleon."

"Galactic cosmic rays should have very similar fluxes on Mercury and the Moon." "Solar Cosmic Rays which result in the formation of particle tracks also increase by a factor of up to 10 when compared to the Moon. However, surface temperatures reach 700 K, which can result over millions of years in the annealing of irradiation effects."

Venus
Venus's small induced magnetosphere provides negligible protection to the atmosphere against cosmic radiation. This radiation may result in cloud-to-cloud lightning discharges.

Earth
The Earth's atmosphere is a relatively bright source of gamma rays produced in interactions of ordinary cosmic ray protons with air atoms.

When cosmic rays enter the Earth’s atmosphere they collide with molecules, mainly oxygen and nitrogen, to produce a cascade of billions of lighter particles, a so-called air shower.

An air shower is an extensive (many kilometres wide) cascade of ionized particles and electromagnetic radiation produced in the atmosphere when a primary cosmic ray (i.e. one of extraterrestrial origin) enters the atmosphere.

There is "a decrease in thunderstorms at the time of high cosmic rays and an increase in thunderstorms 2-4 days later."

It is believed that proton energies exceeding 50 MeV in the lower belts at lower altitudes are the result of the beta decay of neutrons created by cosmic ray collisions with nuclei of the upper atmosphere. The source of lower energy protons is believed to be proton diffusion due to changes in the magnetic field during geomagnetic storms.

The PAMELA experiment detected orders of magnitude higher levels of antiprotons than are expected from normal particle decays while passing through the SAA. This suggests the van Allen belts confine a significant flux of antiprotons produced by the interaction of the Earth's upper atmosphere with cosmic rays. The energy of the antiprotons has been measured in the range from 60 - 750 MeV.

Moon
“The lunar surface also lends itself well to cosmic ray astronomy (as it lies outside the Earth's magnetosphere) and other astronomies requiring large, bulky detectors (eg gamma-ray astronomy).”

Observations of the lunar shadowing of galactic cosmic rays (GCRs) has demonstrated that there does not appear to be an antiproton component of the galactic cosmic rays, but the antiprotons detected are instead produced by the GCR interaction with interstellar hydrogen gas.

The Compton Gamma Ray Observatory has imaged the Moon in gamma rays of energy greater than 20 MeV. These are produced by cosmic ray bombardment of its surface.

Gamma-ray spectrometers have been widely used for the elemental and isotopic analysis of airless bodies in the Solar System, especially the Moon These surfaces are subjected to a continual bombardment of high-energy cosmic rays, which excite nuclei in them to emit characteristic gamma-rays which can be detected from orbit. Thus an orbiting instrument can in principle map the surface distribution of the elements for an entire planet. They are able to measure the abundance and distribution of about 20 primary elements of the periodic table, including silicon, oxygen, iron, magnesium, potassium, aluminum, calcium, sulfur, and carbon. The chemical element thorium [is] mapped [by a GRS], with higher concentrations shown in yellow/orange/red in the left-hand side image shown on the left.

At right is the result of an all Moon survey by the Lunar Prospector using an onboard neutron spectrometer (NS). Cosmic rays impacting the lunar surface generate neutrons which in turn lose much of their energy in collisions with hydrogen atoms trapped within the Moon's surface. Some of these thermal neutrons collide with the helium atoms within the NS to yield an energy signature which is detected and counted. The NS aboard the Lunar Prospector has a surface resolution of 150 km.

Mars
"NASA's Curiosity rover ... Radiation Assessment Detector instrument, or RAD, collected data for about 3 1/2 hours on Wednesday (Aug. 8)". As the Sun was relatively quiet in the direction of Mars, most of the spikes in the collected, unprocessed temporal spectrum are considered to be from galactic cosmic-radiation.

"The data show that the radiation levels measured on Mars during this period of quiet solar activity are reduced from the average radiation detected in space during Curiosity's cruise to Mars. This is explained by the rover being on the planet versus out in space, where it would have more exposure to radiation from all directions. Red arrows point to spikes in the radiation dose rate from heavy ion particles, which would be the most dangerous to astronauts. ... RAD measures 26 kinds of charged particles as well as neutrons and gamma rays."

Interplanetary medium
The interplanetary medium includes interplanetary dust, cosmic rays and hot plasma from the solar wind. The temperature of the interplanetary medium varies. For dust particles within the asteroid belt, typical temperatures range from 200 K (−73 °C) at 2.2 AU down to 165 K (−108 °C) at 3.2 AU The density of the interplanetary medium is very low, about 5 particles per cubic centimeter in the vicinity of the Earth; it decreases with increasing distance from the sun, in inverse proportion to the square of the distance. It is variable, and may be affected by magnetic fields and events such as coronal mass ejections. It may rise to as high as 100 particles/cm³.

Uranus
"[F]or the regions of the giant planets, especially Uranus and Neptune, ... ionization is due mainly to cosmic rays."

Oort clouds
Cosmic "ray protons at energies up to 10 GeV [may be] able to build-up large amount of organic refractory material at depth of several meters in a comet during [its] long life in the Oort cloud (~4.6 x 107 yr). Ion bombardment might also lead to the formation of a substantial stable crust (Johnson et al., 1987)."

Heliospheres
"The sun emits a plasma wind with an embedded magnetic field that tends to exclude low energy galactic cosmic rays from the heliosphere."

The "observed cosmic ray flux at Earth is inversely correlated with solar activity. [...] At a period of high solar activity (for example in 1983), the flux below a GeV can be suppressed by as much as an order of magnitude."

The "flux of cosmic rays in the heliosphere varies with the eleven year solar cycle".

Interstellar medium
In astronomy, the interstellar medium (or ISM) is the matter that exists in the space between the star systems in a galaxy. This matter includes gas in ionic, atomic, and molecular form, dust, and cosmic rays. It fills interstellar space and blends smoothly into the surrounding [Intergalactic medium] intergalactic space.

In astronomy, the interstellar medium (or ISM) is the gas and cosmic dust that pervade interstellar space: the matter that exists between the star systems within a galaxy. It fills interstellar space and blends smoothly into the surrounding intergalactic medium. The interstellar medium consists of an extremely dilute (by terrestrial standards) mixture of ions, atoms, molecules, larger dust grains, cosmic rays, and (galactic) magnetic fields. The energy that occupies the same volume, in the form of electromagnetic radiation, is the interstellar radiation field.

The first gamma-ray telescope carried into orbit, on the Explorer 11 satellite in 1961, picked up fewer than 100 cosmic gamma-ray photons. They appeared to come from all directions in the Universe, implying some sort of uniform "gamma-ray background". Such a background would be expected from the interaction of cosmic rays (very energetic charged particles in space) with interstellar gas.

Milky Way
"As with solar system cosmic rays, it is likely that both extended and point sources play a role in acceleration of particles in the Galaxy."

Large Magellanic Cloud
Because neutrinos are only weakly interacting with other particles of matter, neutrino detectors must be very large in order to detect a significant number of neutrinos. Neutrino detectors are often built underground to isolate the detector from cosmic rays and other background radiation.

Active galactic nuclei
There is "a correlation between the arrival directions of cosmic rays with energy above 6 x 1019 electron volts and the positions of active galactic nuclei (AGN) lying within ~75 megaparsecs."

Some low energy cosmic rays originate or are associated with solar flares. Even these cosmic rays have too high an energy to originate from the solar photosphere. The coronal cloud in close proximity to the Sun may be a source or create them as it bombards the chromosphere from above.

"In particular we recognize a first trace of Vela, brightest gamma and radio galactic source, and smeared sources along Galactic Plane and Center [as a source of ultra high energy cosmic rays (UHECR)]."

"The main correlated map is the 408 MHz one. The first astronomical source that seem to correlate is the main multiplet along CenA. This AGN source, the nearest extragalactic one, sits in the same direction of a far Centaurus Cluster (part of the Super-Galactic Plane). The blurring by random galactic magnetic field might spread the nearest AGN event along the same Super-Galactic Plane, explaining the AUGER group miss-understanding [3]."

Locations on Earth
Ice cores contain thin nitrate-rich layers that can be analyzed to reconstruct a history of past events [such as solar cosmic ray events] before reliable observations; [this includes] data from Greenland ice cores and others. These show evidence that events of [the magnitude of the solar storm of 1859—as measured by high-energy proton radiation, not geomagnetic effect—occur approximately once per 500 years, with events at least one-fifth as large occurring several times per century. Less severe storms have occurred in 1921 and 1960, when widespread radio disruption was reported.

Technology
The Cherenkov telescopes do not actually detect the gamma rays directly but instead detect the flashes of visible light produced when gamma rays are absorbed by the Earth's atmosphere.

Fred Lawrence Whipple Observatory
The Fred Lawrence Whipple Observatory is an astronomical observatory owned and operated by the Smithsonian Astrophysical Observatory (SAO) with research activities that include imaging and spectroscopy of extragalactic, stellar, and planetary bodies, as well as gamma-ray and cosmic-ray astronomy.

Pierre Auger Observatory
The Pierre Auger Observatory is an international cosmic ray observatory designed to detect ultra-high-energy cosmic rays: single sub-atomic particles (protons or atomic nuclei) with energies beyond 1020 eV (about the energy of a tennis ball traveling at 80 km/h). These high energy particles have an estimated arrival rate of just 1 per km2 per century, therefore the Auger Observatory has created a detection area the size of Rhode Island — over 3,000 km2 (1,200 sq mi) — in order to record a large number of these events. It is located in western Argentina's Mendoza Province, in one of the South American Pampas.

The basic set-up consists of 1600 water tanks (water Cherenkov Detectors, similar to the Haverah Park experiment) distributed over 3,000 square kilometres (1,200 sq mi), along with four atmospheric fluorescence detectors (similar to the High Resolution Fly's Eye) overseeing the surface array.

Major Atmospheric Gamma-ray Imaging Cherenkov Telescopes
MAGIC (Major Atmospheric Gamma-ray Imaging Cherenkov Telescopes) is a system of two Imaging Atmospheric Cherenkov telescopes situated at the Roque de los Muchachos Observatory on [La Palma, one of the Canary Islands, at about 2200 m above sea level. MAGIC detects particle showers released by gamma rays, using the Cherenkov radiation, i.e., faint light radiated by the charged particles in the showers. With a diameter of 17 meters for the reflecting surface, it is the largest in the world. MAGIC is sensitive to cosmic gamma rays with energies between 50 GeV and 30 TeV due to its large mirror; other ground-based gamma-ray telescopes typically observe gamma energies above 200-300 GeV. Satellite-based detectors detect gamma-rays in the energy range from keV up to several GeV. MAGIC has found pulsed gamma-rays at energies higher than 25 GeV coming from the Crab Pulsar. The presence of such high energies indicates that the gamma-ray source is far out in the pulsar's magnetosphere, in contradiction with many models. A much more controversial observation is an energy dependence in the speed of light of cosmic rays coming from a short burst of the blazar Markarian 501 on July 9, 2005. Photons with energies between 1.2 and 10 TeV arrived 4 minutes after those in a band between .25 and .6 TeV. The average delay was .030±.012 seconds per GeV of energy of the photon. If the relation between the space velocity of a photon and its energy is linear, then this translates into the fractional difference in the speed of light being equal to minus the photon's energy divided by 2 x 1017 GeV.

Balloons
The various background effects OSO 1 encountered prompted the flight of similar detectors on a balloon to determine the cosmic-ray effects in the materials surrounding the detectors.

Measurements "of the cosmic-ray positron fraction as a function of energy have been made using the High-Energy Antimatter Telescope (HEAT) balloon-borne instrument."

"The first flight took place from Fort Sumner, New Mexico, [on May 3, 1994, with a total time at float altitude of 29.5 hr and a mean atmospheric overburden of 5.7 g cm-2] ... The second flight [is] from Lynn Lake, Manitoba, [on August 23, 1995, with a total time at float altitude of 26 hr, and a mean atmospheric overburden of 4.8 g cm-2]".

Orbital rocketry
Explorer 11 (also known as S15) was an American Earth-orbital satellite that carried the first space-borne gamma-ray telescope. This was the earliest beginning of space gamma-ray astronomy. Launched on April 27, 1961 by a Juno II rocket the satellite returned data until November 17, when power supply problems ended the science mission. During the spacecraft's seven month lifespan it detected twenty-two events from gamma-rays and approximately 22,000 events from cosmic radiation.

The HEAO 3 French-Danish C-2 experiment measured the relative composition of the isotopes of the primary cosmic rays between beryllium and iron (Z from 4 to 26) and the elemental abundances up to tin (Z=50). Cerenkov counters and hodoscopes, together with the Earth's magnetic field, formed a spectrometer. They determined charge and mass of cosmic rays to a precision of 10% for the most abundant elements over the momentum range from 2 to 25 GeV/c (c=speed of light).

The purpose of the HEAO 3 C-3 experiment was to measure the charge spectrum of cosmic-ray nuclei over the nuclear charge (Z) range from 17 to 120, in the energy interval 0.3 to 10 GeV/nucleon; to characterize cosmic ray sources; processes of nucleosynthesis, and propagation modes.

"The rigidity dependence of the escape length of cosmic rays in the galaxy has been derived in the framework of the leaky box model from the measured values of the B/C ratio."

For an interstellar medium "composed of 90% H and 10% He, [with a density of 0.3 atoms cm-3] and using the most recently measured cross sections (Webber, 1989; Ferrando et al., 1988b), the escape length has been found equal to 34βR-0.6 g cm-2 for rigidities R above 4.4 GV, and 14β g cm-2 below. ... where R and β are the interstellar values of the rigidity and the ratio of the velocity of the particle to the velocity of light."

Extreme Universe Space Observatory
The Extreme Universe Space Observatory (EUSO) is the first Space mission concept devoted to the investigation of cosmic rays and neutrinos of extreme energy (E > $5 eV$). Using the Earth's atmosphere as a giant detector, the detection is performed by looking at the streak of fluorescence produced when such a particle interacts with the Earth's atmosphere.

Heliocentric rocketry
Helios 1 and Helios 2 are a pair of probes launched into heliocentric orbit for the purpose of studying solar processes. The probes are notable for having set a maximum speed record among spacecraft at 252,792 km/h (157,078 mi/h or 43.63 mi/s or 70.22 km/s or 0.000234c). Helios 2 flew three million kilometers closer to the Sun than Helios 1, achieving perihelion on 17 April 1976 at a record distance of 0.29 AU (or 43.432 million kilometers), slightly inside the orbit of Mercury. Helios 2 was sent into orbit 13 months after the launch of Helios 1. The probes are no longer functional but still remain in their elliptical orbit around the Sun. On board, each probe carried an instrument for cosmic radiation investigation (the CRI) for measuring protons, electrons, and X-rays to determine the distribution of cosmic rays.

Exploratory rocketry
Pioneer 10 is a 258-kilogram robotic space probe that completed the first mission to the planet Jupiter and became the first spacecraft to achieve escape velocity from the Solar System.

Pioneer 10 was launched on March 2, 1972 by an Atlas-Centaur expendable vehicle from Cape Canaveral, Florida. Between July 15, 1972, and February 15, 1973, it became the first spacecraft to traverse the asteroid belt.

"In 1972, the return of the galactic cosmic rays in the inner solar system to solar minimum conditions and the launch of Pioneer 10 toward Jupiter coincided to make possible the measurements of the low-energy cosmic-ray charge spectra during solar quiet times." "Recent measurements using the Goddard-University of New Hampshire cosmic-ray telescope on the Pioneer 10 spacecraft have revealed an anomalous spectrum of nitrogen and oxygen nuclei relative to other nuclei such as He and C, in the energy range 3-30 MeV per nucleon."

"To eliminate [the solar cosmic-ray background] a very careful selection of times must be made to assure that solar cosmic rays are not obviously present [by] requiring that the 10-20 MeV proton intensity measured on the same experiment be essentially at background level."

The International Cometary Explorer (ICE) spacecraft was originally known as [the] International Sun/Earth Explorer 3 (ISEE-3) satellite.

ISEE-3 was launched on August 12, 1978. It was inserted into a "halo" orbit about the libration point some 240 Earth radii upstream between the Earth and Sun. ISEE-3 was renamed ICE (International Cometary Explorer) when, after completing its original mission in 1982, it was gravitationally maneuvered to intercept the comet P/Giacobini-Zinner. On September 11, 1985, the veteran NASA spacecraft flew through the tail of the comet. The X-ray spectrometer aboard ISEE-3 was designed to study both solar flares and cosmic gamma-ray bursts over the energy range 5-228 keV.

The instruments aboard ISEE-3 are designed to detect
 * 1) protons in the energy range 150 eV - 7 keV and electrons in the 10 eV - 1 keV range (Solar wind plasma experiment),
 * 2) Low, Medium and High-Energy Cosmic Rays (1-500 MeV/n, Z = 1-28, electrons 2-10 MeV, for Medium Energy; H to Ni, 20-500 MeV/n for High-energy),
 * 3) H-Fe 30 MeV/n - 15 GeV/n and electrons 5-400 MeV for the Cosmic-Ray Energy Spectrum experiment,
 * 4) 17 Hz - 100 kHz magnetic and electric field wave levels (Plasma Waves Spectrum Analyzer),
 * 5) low-energy solar proton acceleration and propagation processes in interplanetary space, Energetic Particle Anisotropy Spectrometer (EPAS),
 * 6) 2 keV to > 1 MeV interplanetary and solar electrons,
 * 7) radio mapping of solar wind disturbances (type III bursts) in 3-D, 30 kHz - 2 MHz,
 * 8) solar wind ion composition, 300-600 km/s, 840 eV/Q to 11.7 keV/Q, M/Q = 1.5 to 5.6,
 * 9) cosmic ray isotope spectrometer 5-250 MeV/n, Z=3-28, A=6-64 (Li-Ni),
 * 10) ground based solar studies with the Stanford ground-based solar telescope, and the comparison of these measurements with measurements of the interplanetary magnetic field and solar wind made by other experiments on this spacecraft,
 * 11) X- and gamma-ray bursts, 5-228 keV, and
 * 12) Gamma-ray bursts, 0.05-6.5 MeV direction, profile, spectrum.

The Voyager 1 spacecraft is a 722 kg space probe launched by NASA on September 5, 1977 to study the outer Solar System and interstellar medium.

The Cosmic Ray System (CRS) "[d]etermines the origin and acceleration process, life history, and dynamic contribution of interstellar cosmic rays, the nucleosynthesis of elements in cosmic-ray sources, the behavior of cosmic rays in the interplanetary medium, and the trapped planetary energetic-particle environment.

"Measurements from the spacecraft revealed a steady rise since May in collisions with high energy particles (above 70 MeV), which are believed to be cosmic rays emanating from supernova explosions far beyond the Solar System, with a sharp increase in these collisions in late August. At the same time, in late August, there was a dramatic drop in collisions with low-energy particles, which are thought to originate from the Sun.

Hypotheses

 * 1) Some cosmic rays are superluminals.


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Power Supply Testing/Tutorial to drill holes on the power supply shield

 * 1. Remove the power supply shield Power supply shield.JPG
 * 2. Put the power supply shield on the vise grip Shield in the grip.JPG
 * 3. Get a cup of oil. Oil.JPGCup of water.JPG
 * 4. Put a small drill bit in the drill. Adjust it so that is tight. Dip it in the cup of oil.
 * 5. Start drilling until one makes a hole. Small hole.JPG
 * 6. Change the drill bit. Increase the size. Dip it in oil
 * 7. Start drilling a bigger hole. Bigger hole.JPG
 * 8. Change the drill bit. Increase the size. Dip it in oil
 * 9. Start drilling bigger hole than previous one. Even bigger hole.JPG
 * 10. Change the drill bit. Increase the size. Dip it in oil
 * 11. Make the last hole (the hole should be as big to fit the bin post) Final hole.JPG
 * 12. Put the bin post Put the bin post.JPG
 * 13. Repeat steps 4-12 four more times to make the five holes. 5 bin post.JPG


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Power Supply Testing/Tutorial on when the drill stop working

 * 1. The drills are in the last cabinet of the table. Drills in the cabinet.JPG
 * 2. Get the drill and remove the battery. Removing battery.JPG
 * 3. Get the battery Get_the_battery.JPG
 * 4. Put the battery in the charger and allow it to sit several minutes to charge Charging the battery.JPG


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Power Supply Testing/Tutorial on charging the drill's battery

 * 1. The drills are in the last cabinet fo the table. Drills in the cabinet.JPG
 * 2. Get the drill and remove the battery. Removing battery.JPG
 * 3. Get the battery Get_the_battery.JPG
 * 4. Put the battery in the charger and allow it to sit several minutes to charge Charging the battery.JPG.


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Power Supply Testing/Tutorial on removing the power supply from the computer

 * 1. Get the computer case. Computer case with power supply.JPG
 * 2. Open the lid and disconnect all the wires that are attach to the power supply Removing cables.jpg
 * 3. Remove all screws attach to the power supply Remove screws.jpg
 * 4. Pull the power supply and your ready to work with it.


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User:Amokamorist

ABSTRACT SCIENCE

We feel. We are humans.we need others to understand us, we understand others. The way we feel is not only a psychological phenomenon but also an environmental impact. The place, relatives friends other people, institutions and our work place are some important parameters to which our FEEL sense is vulnerable to. The different degrees of FEEL sense can be calculated in a mathematical way, helping the psychologist or doctor or recruiters or management d real psych of the subject(person).to be continued...--Amokamorist (talk) 21:05, 28 March 2012 (UTC)  M.Milan

title: User:Amokamorist


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Power Supply Testing/Tutorial on measuring the voltage with the multimeter

 * 1. Turn off the power supply.
 * 2. With the wire strippers remove the insulation from two wires. (One of the wires must be the black wire because its voltage is 0). Removing_the_insaltion_from_the_black_wire.JPGRemove_insaltion_from_the_other_wire.JPG Wires with our insalation.JPG
 * 3. Turn on the multimeter to 20 in the DC.
 * 4. The black cable from the multimeter: place it near the black wire.
 * 5. The red cable from the multimeter: place it near the other wire.
 * 6. Tape them. Measuring the voltage the correct way.JPG
 * 7. Turn on the power supply Turn_on_the_power_supply.JPG
 * 8. Record your observation. Record observation.JPGMeasuring the voltage.jpg

Measurements

 * white and yellow wires: 12 volts
 * blue wire: -12 volts
 * red wires: 5 volts
 * orange wires 3.3 volts
 * black wires: 0 volts


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User:Egm4313.s12.team8.tclamb/R5

=R5.8 - Antiderivitave=

Problem Statement
Find the integral
 * $$\int x^n \log(x+1) \, dx$$

using integration by parts, and then with the help of the General Binomial Theorem:
 * $$(x + y)^n = \sum\limits_{k=0}^n \binom{n}{k} x^{n-k} y^k$$

Method 1
For the case of n=0, the integral becomes the integral of $$\log(1+x)$$ which is well known as the following, a result easily obtained by integration by parts: $$\int \log (x+1) \, dx = x \log (x+1) + \log (x+1) - x$$

Integrating by parts, we proceed as follows:

\begin{array}{rl} \int x^n \log (x+1) \, dx & = x^n \left( x \log(1+x) + \log(1+x) - x \right) - \int n x^n \left( x \log(1+x) + \log(1+x) - x \right) \, dx \\ & = x^{n+1} \log (x+1) + x^n \log (x+1) - x^{n+1} - n \int x^n \log (1+x) \, dx - n \int x^{n-1} \log (x+1) \, dx + n \int x^n \, dx \\ (n+1) \int x^n \log (x+1) \, dx & = x^{n+1} \log (x+1) + x^n \log (x+1) - x^{n+1} - n \int x^{n-1} \log (x+1) \, dx + \frac{n}{n+1} x^{n+1} \\ \int x^n \log (x+1) \, dx & = \frac{1}{n+1}\left(x^{n+1} \log (x+1) + x^n \log (x+1) - x^{n+1} - n \int x^{n-1} \log (x+1) \, dx + \frac{n}{n+1} x^{n+1}\right) \end{array} $$

This can be written as the sequence:

\begin{array}{l} R_0(x) = -x + \log (x+1) + x \log (x+1) \\ R_n(x) = \frac{1}{n+1}\left(x^{n+1} \log (x+1) + x^n \log (x+1) - x^{n+1} + \frac{n}{n+1} x^{n+1} - n R_{n-1}\right) \end{array} $$

The first few functions in the sequence are as follows:

\begin{array}{l} R_0(x) = -x + \log (x+1) + x \log (x+1) \\ R_1(x) = \frac12x - \frac14x^2 - \frac12\log(x+1) + \frac12x^2\log(x+1) \\ R_2(x) = -\frac13x + \frac16x^2 - \frac19x^3 + \frac13\log(x+1) + \frac13x^3\log(x+1) \\ R_3(x) = \frac14x - \frac18x^2 + \frac1{12}x^3 - \frac1{16}x^4 - \frac14\log(x+1) + \frac14x^4\log(x+1) \\ R_4(x) = -\frac15x + \frac1{10}x^2 - \frac1{15}x^3 + \frac1{20}x^4 - \frac1{25}x^5 + \frac15\log(x+1) + \frac15x^5\log(x+1) \end{array} $$

From this sequence, we can determine the following form of the solution:

\begin{array}{rl} \int x^n \log (x+1) \, dx & = (-1)^n \frac1{n+1} \log(x+1) + \frac1{n+1} x^{n+1} \log(x+1) + \sum\limits_{k=1}^{n+1}(-1)^{n+k}\frac{x^k}{k(n+1)} \\ & = \frac1{n+1}\left( \left( x^{n+1} + (-1)^n \right) \log (x+1) + \sum\limits_{k=1}^{n+1}\frac{(-1)^{n+k}}{k}x^k \right) \\ & = \frac1{n+1}\left( \left( x^{n+1} + (-1)^n \right) \log (x+1) - x \sum\limits_{k=0}^{n}\frac1{k+1}(-1)^{n-k}x^k \right) \end{array} $$

Unfortunately, the last term in the above solution has withstood every attempt to manipulate it into the form of the general binomial theorem.

Method 2
Beginning with the initial expression:
 * $$\int x^n\log (1+x)dx$$

Integrating by parts yields:
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\int x^{n+1}\frac{1}{1+x}dx$$

Substituting $$u=1+x$$:
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\int \frac{1}{u}(u-1)^{n+1}du$$

Expanding binomial with General Binomial Theorem:
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\int \frac{1}{u}\sum _{i=0}^{n+1} \binom{n+1}{i}u^i(-1)^{(n+1)-i}du$$
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\int \frac{1}{u}\sum _{i=0}^{n+1} \frac{(n+1)!}{i!((n+1)-i)!}u^i(-1)^{(n+1)-i}du$$

Moving in factor of $$\frac{1}{u}$$:
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\int \sum _{i=0}^{n+1} \frac{(n+1)!}{i!((n+1)-i)!}u^{i-1}(-1)^{(n+1)-i}du$$

Substituting back for x:
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\int \sum _{i=0}^{n+1} \frac{(n+1)!}{i!((n+1)-i)!}(1+x)^{i-1}(-1)^{(n+1)-i}dx$$

Pulling out the lowest term:
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\int \left((1+x)^{-1}(-1)^{n+1}+\sum _{i=1}^{n+1} \frac{(n+1)!}{i!((n+1)-i)!}(1+x)^{i-1}(-1)^{(n+1)-i}\right) \, dx$$

Integrating:
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)-\frac{1}{n+1}\left(\log (1+x)(-1)^{n+1}+\sum _{i=1}^{n+1} \frac{(n+1)!}{i!((n+1)-i)!}\frac{(1+x)^i}{i}(-1)^{(n+1)-i}\right)$$
 * $$\frac{1}{n+1}x^{n+1}\log (1+x)+\frac{1}{n+1}\log (1+x)(-1)^n-\frac{1}{n+1}\sum _{i=1}^{n+1} \frac{(n+1)!}{i!((n+1)-i)!}\frac{(1+x)^i}{i}(-1)^{(n+1)-i}$$
 * $$\frac{1}{n+1}\left(x^{n+1}+(-1)^n\right)\log (1+x)-\frac{1}{n+1}\sum _{i=1}^{n+1} \frac{(n+1)!}{i!((n+1)-i)!}\frac{(1+x)^i}{i}(-1)^{(n+1)-i}$$

Shifting the limits of summation:
 * $$\frac{1}{n+1}\left(x^{n+1}+(-1)^n\right)\log (1+x)-\frac{1}{n+1}\sum _{j=0}^n \frac{(n+1)!}{(j+1)!(n-j)!}\frac{(1+x)^{j+1}}{j+1}(-1)^{n-j}$$
 * $$\frac{1}{n+1}\left(x^{n+1}+(-1)^n\right)\log (1+x)-(1+x)\sum _{j=0}^n \frac{1}{(j+1)^2}\frac{n!}{j!(n-j)!}(1+x)^j(-1)^{n-j}$$

This gives us our final result:
 * $$\int x^n\log (1+x)dx = \frac{1}{n+1}\left(x^{n+1}+(-1)^n\right)\log (1+x)\,-\,(1+x)\sum _{j=0}^n \frac{1}{(j+1)^2}\binom{n}{j}(1+x)^j(-1)^{n-j}$$

Comparing this result to the one obtained above by evaluating for various n, we observe that these are both valid antiderivatives of $$\int x^n\log (1+x)dx$$, differing from each other by only a constant.

=R5.9 - Solve L2-ODE-CC with basis projection of excitation=

Problem Statement
Consider the L2-ODE-CC with $$\log(1+x)$$ as excitation:
 * $$y'' - 3y' + 2y = r(x)$$
 * $$r(x) = \log(1+x)$$

and the initial conditions:
 * $$y(-\frac{3}{4}) = 1, \, y'(-\frac{3}{4}) = 0$$

Part 1
Project the excitation $$r(x)$$ on the polynomial basis:
 * $$\{ b_i(x) = x^i, \, i=0,1,...,n \}$$

i.e. find $$d_i$$ such that:
 * $$r(x) \approx r_n(x) = \sum_{i=0}^n d_i \, x^i$$

for $$x$$ in $$\left[ -\frac{3}{4}, 3 \right]$$, and for n=3,6,9.

Plot $$r(x)$$ and $$r_n(x)$$ to show uniform approximation and convergence.

Note that $$\langle x^i, r \rangle = \int_a^b x^i \log(1+x) \, dx$$

In separate series of plots, compare the approximation of the function $$\log(1+x)$$ by 2 methods: Observe and discuss the pros and cons of each method.
 * Projection on a polynomial basis.
 * Taylor series expansion about $$\hat x = 0$$

Part 2
Find $$y_n(x)$$ such that:
 * $$y_{n}'' + a y_{n}' + b y_n = r_n(x)$$

with the above initial conditions.

Plot $$y_n(x)$$ for n=3,6,9, for x in $$\left[ -\frac{3}{4}, 3 \right]$$.

In a series of separate plots, compare the results obtained with the projected excitation on polynomial basis to those with truncated Taylor series of the excitation. Plot also the numerical solution as a baseline for comparison.

Part 1
To solve for the coefficients of the projection of a function onto an orthonormal basis, we use the following equation:
 * $$\mathbf{ \Gamma c } = \mathbf{ d }$$

Where $$\mathbf \Gamma$$, $$\mathbf c$$ and $$\mathbf d$$ are given by the following:
 * $$\mathbf \Gamma ( \{ b_i \} ) = \left[

\begin{array}{ccccc} \langle b_0, b_0 \rangle & \langle b_0 , b_1 \rangle & \cdots & \langle b_0 , b_{n - 1} \rangle & \langle b_0 , b_n \rangle \\ \langle b_1, b_0 \rangle & \langle b_1 , b_1 \rangle & \cdots & \langle b_1 , b_{n - 1} \rangle & \langle b_1 , b_n \rangle \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \langle b_{n-1}, b_0 \rangle & \langle b_{n-1} , b_1 \rangle & \cdots & \langle b_{n-1} , b_{n - 1} \rangle & \langle b_{n-1} , b_n \rangle \\ \langle b_n, b_0 \rangle & \langle b_n , b_1 \rangle & \cdots & \langle b_n , b_{n - 1} \rangle & \langle b_n , b_n \rangle \end{array} \right] $$

 * $$\mathbf c = \left[

\begin{array}{c} c_0 \\ c_1 \\ \vdots \\ c_{n-1} \\ c_n \end{array} \right] $$

 * $$\mathbf d = \left[

\begin{array}{c} \langle b_0, f \rangle \\ \langle b_1, f \rangle \\ \vdots \\ \langle b_{n-1}, f \rangle \\ \langle b_n, f \rangle \end{array} \right] $$

And where the scalar product for two functions is defined as:
 * $$\langle f, g \rangle := \int_a^b f(x) g(x) \, dx$$

Using the polynomial basis:
 * $$\{ b_i(x) = x^i, \, i=0,1,...,n \}$$

Over the region $$[a, \, b]$$ where:
 * $$a = -\frac{3}{4} \,, \; b = 3$$

We obtain the following matrix equation:
 * $$\left[

\begin{array}{ccccc} \langle 1, 1 \rangle & \langle 1 , x \rangle & \cdots & \langle 1 , x^{n-1} \rangle & \langle 1 , x^{n} \rangle \\ \langle x, 1 \rangle & \langle x , x \rangle & \cdots & \langle x , x^{n-1} \rangle & \langle x , x^{n} \rangle \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \langle x^{n-1}, 1 \rangle & \langle x^{n-1} , x \rangle & \cdots & \langle x^{n-1} , x^{n-1} \rangle & \langle x^{n-1} , x^{n} \rangle \\ \langle x^n, 1 \rangle & \langle x^n , x \rangle & \cdots & \langle x^n , x^{n-1} \rangle & \langle x^n , x^n \rangle \end{array} \right] \left[ \begin{array}{c} c_0 \\ c_1 \\ \vdots \\ c_{n-1} \\ c_n \end{array} \right] = \left[ \begin{array}{c} \langle 1, \ln(x+1) \rangle \\ \langle x, \ln(x+1) \rangle \\ \vdots \\ \langle x^{n-1}, \ln(x+1) \rangle \\ \langle x^n, \ln(x+1) \rangle \end{array} \right] $$

Solving with n = 3
Evaluating the scalar products gives us:
 * $$\left[

\begin{array}{cccc} 3.75 & 4.21875 & 9.14063 & 20.1709 \\ 4.21875 & 9.14063 & 20.1709 & 48.6475 \\ 9.14063 & 20.1709 & 48.6475 & 121.47 \\ 20.1709 & 48.6475 & 121.47 & 312.448 \end{array} \right] \mathbf c =\left[ \begin{array}{c} 2.14175 \\ 5.00755 \\ 10.3153 \\ 24.6142 \end{array} \right]$$

Solving for $$\mathbf c$$ then yields:
 * $$\mathbf c = \left[

\begin{array}{c} -0.0388344 \\ 1.12972 \\ -0.439081 \\ 0.0760925 \end{array} \right]$$

This results in the following polynomial:
 * $$r_3(x) = 0.0760925 x^3-0.439081 x^2+1.12972 x-0.0388344$$

Solving with n = 6
Evaluating the scalar products gives us:
 * $$\left[

\begin{array}{ccccccc} 3.75 & 4.21875 & 9.14063 & 20.1709 & 48.6475 & 121.47 & 312.448 \\ 4.21875 & 9.14063 & 20.1709 & 48.6475 & 121.47 & 312.448 & 820.112 \\ 9.14063 & 20.1709 & 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 \\ 20.1709 & 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 \\ 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 \\ 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 \\ 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 & 122640. \end{array} \right] \mathbf c =\left[ \begin{array}{c} 2.14175 \\ 5.00755 \\ 10.3153 \\ 24.6142 \\ 60.4329 \\ 153.96 \\ 400.898 \end{array} \right]$$

Solving for $$\mathbf c$$ then yields:
 * $$\mathbf c = \left(

\begin{array}{c} 0.00585919 \\ 0.982671 \\ -0.557953 \\ 0.464459 \\ -0.275708 \\ 0.0834434 \\ -0.0096586 \end{array} \right)$$

This results in the following polynomial:
 * $$r_6(x) = -0.0096586 x^6+0.0834434 x^5-0.275708 x^4+0.464459 x^3-0.557953 x^2+0.982671 x+0.00585919$$

Solving with n = 9
Evaluating the scalar products gives us:
 * $$\left[

\begin{array}{cccccccccc} 3.75 & 4.21875 & 9.14063 & 20.1709 & 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 \\ 4.21875 & 9.14063 & 20.1709 & 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 \\ 9.14063 & 20.1709 & 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 \\ 20.1709 & 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 & 122640. \\ 48.6475 & 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 & 122640. & 341641. \\ 121.47 & 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 & 122640. & 341641. & 956594. \\ 312.448 & 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 & 122640. & 341641. & 956594. & 2.69042\times 10^6 \\ 820.112 & 2187.01 & 5904.89 & 16104.3 & 44286.7 & 122640. & 341641. & 956594. & 2.69042\times 10^6 & 7.59648\times 10^6 \\ 2187.01 & 5904.89 & 16104.3 & 44286.7 & 122640. & 341641. & 956594. & 2.69042\times 10^6 & 7.59648\times 10^6 & 2.15234\times 10^7 \\ 5904.89 & 16104.3 & 44286.7 & 122640. & 341641. & 956594. & 2.69042\times 10^6 & 7.59648\times 10^6 & 2.15234\times 10^7 & 6.11717\times 10^7 \end{array} \right] \mathbf c =\left[ \begin{array}{c} 2.14175 \\ 5.00755 \\ 10.3153 \\ 24.6142 \\ 60.4329 \\ 153.96 \\ 400.898 \\ 1062.61 \\ 2854.9 \\ 7754.67 \end{array} \right]$$

Solving for $$\mathbf c$$ then yields:
 * $$\mathbf c = \left[

\begin{array}{c} -0.000966978 \\ 1.00131 \\ -0.477337 \\ 0.299654 \\ -0.304399 \\ 0.334633 \\ -0.237359 \\ 0.09511 \\ -0.0197817 \\ 0.00166379 \end{array} \right]$$

This results in the following polynomial:
 * $$r_9(x) = 0.00166379 x^9-0.0197817 x^8+0.09511 x^7-0.237359 x^6+0.334633 x^5-0.304399 x^4+0.299654 x^3-0.477337 x^2+1.00131 x-0.000966978$$

Comparison of different n
Plotting these three polynomials against the original function yields the following:
 * [[Image:R5.9 n=3,6,9.png]]

Closer inspection between 0.9 and 1.1 shows finer detail:
 * [[Image:R5.9 n=3,6,9 zoomed.png]]

Likewise, between -0.6 and -0.5:
 * [[Image:R5.9 n=3,6,9 zoomed 2.png]]

We can easily see the rapid convergence of this method to the function. Evaluated with n=3, there is an average error of 0.0297816, the difference easily visible. Moving up to n=6, there is only very small error between the function and the polynomial approximation, barely discernible in the unzoomed plot, with the average error being about 0.00365851. At n=9, the error is even smaller, imperceptible when unzoomed and difficult to see even when zoomed, with the average error at about 0.000551377. The average error is roughly halved from n to n+1.

Comparison to Taylor series
The Taylor series about x = 0 is given as follows:
 * $$\log(1+x) \approx \sum_{i=1}^n \frac{(-1)^{i+1}}{i} x^i$$

Generating polynomials for n = 3, 6, and 9:
 * $$\begin{align}

r_{taylor, 3}(x) &= \frac{x^3}{3}-\frac{x^2}{2}+x \\ r_{taylor, 6}(x) &= -\frac{x^6}{6}+\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x\\ r_{taylor, 9}(x) &= \frac{x^9}{9}-\frac{x^8}{8}+\frac{x^7}{7}-\frac{x^6}{6}+\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x \end{align}$$

Plotting these against their respective projections:
 * [[Image:R5.9 n=3+taylor.png]]
 * [[Image:R5.9 n=6+taylor.png]]
 * [[Image:R5.9 n=9+taylor.png]]

As we can see, this function is clearly better modeled within this domain by the polynomial obtained via projection than that from its Taylor expansion about 0. While both are accurate for x between -1 and 1, when we look at the graph past 1 for the Taylor polynomial approximations, they rapidly diverge from the actual value of the function. Therefore, while immensely easier to compute, Taylor polynomials are terrible for this range without extending them as in Report 4 by generating a new polynomial wherever the old one diverges and combining the functions piecewise. The major disadvantage of projections, their computational complexity, is easily mitigated nowadays with symbolic math software.

Part 2
From our work in Report 4, we know the homogeneous solution to the ODE is of the form:
 * $$y_h(x) = k_1 e^x + k_2 e^{2x}$$

Also from Report 4, we know that the particular solution to the ODE is of the form:
 * $$y_p(x) = \sum_{i=0}^{\infty} z_i x^i \approx \sum_{i=0}^{n} z_i x^i$$

From Report 4, we know we can solve for the coefficients by solving the matrix equation:
 * $$\mathbf{ A z } = \mathbf c$$

Where $$\mathbf A$$ and $$\mathbf \alpha$$ are given:
 * $$\mathbf A = \begin{bmatrix}

b &     a &  2     &      0 &       0 &         &       0 \\ 0 &     b & 2a     &      6 &       0 &         &       0 \\ 0 &     0 &  b     &     3a &      12 &         &       0 \\ &       &        & \ddots & \ddots  &  \ddots &       0 \\ 0 &     0 &      0 &      0 &       b & a (n-1) & n (n-1) \\ 0 &     0 &      0 &      0 &       0 &       b & a n     \\ 0 &     0 &      0 &      0 &       0 &       0 &   b \end{bmatrix}$$
 * $$\mathbf z = \left[

\begin{array}{c} z_0 \\ z_1 \\ \vdots \\ z_{n-1} \\ z_n \end{array}\right]$$

And $$\mathbf c$$, being the coefficients of our excitation function, determined above:
 * $$\mathbf c = \mathbf \Gamma ^{-1} \mathbf d = \left[

\begin{array}{ccccc} \langle 1, 1 \rangle & \langle 1 , x \rangle & \cdots & \langle 1 , x^{n - 1} \rangle & \langle 1 , x^n \rangle \\ \langle x, 1 \rangle & \langle x , x \rangle & \cdots & \langle x , x^{n - 1} \rangle & \langle x , x^n \rangle \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \langle x^{n-1}, 1 \rangle & \langle x^{n-1} , x \rangle & \cdots & \langle x^{n-1} , x^{n-1} \rangle & \langle x^{n-1} , x^n \rangle \\ \langle x^n, 1 \rangle & \langle x^n , x \rangle & \cdots & \langle x^n , x^{n-1} \rangle & \langle x^n , x^n \rangle \end{array} \right]^{-1} \left[ \begin{array}{c} \langle 1, \ln(x+1) \rangle \\ \langle x, \ln(x+1) \rangle \\ \vdots \\ \langle x^{n-1}, \ln(x+1) \rangle \\ \langle x^n, \ln(x+1) \rangle \end{array} \right] $$

From our ODE, we determine that $$a=-3$$ and $$b=2$$. Therefore, to solve for the the coefficients of $$y_p,n$$, we solve the following matrix equation:
 * $$\begin{bmatrix}

2 &     -3 &  2     &      0 &       0 &         &       0 \\      0 &      2 & -6     &      6 &       0 &         &       0 \\      0 &      0 &  2     &     -9 &      12 &         &       0 \\        &        &        & \ddots & \ddots  &  \ddots &       0 \\ 0 &     0 &      0 &      0 &       2 & -3 (n-1) & n (n-1) \\ 0 &     0 &      0 &      0 &       0 &       2 & -3 n     \\ 0 &     0 &      0 &      0 &       0 &       0 &   b \end{bmatrix} \mathbf z = \left[ \begin{array}{ccccc} \langle 1, 1 \rangle & \langle 1 , x \rangle & \cdots & \langle 1 , x^{n-1} \rangle & \langle 1 , x^n \rangle \\ \langle x, 1 \rangle & \langle x , x \rangle & \cdots & \langle x , x^{n-1} \rangle & \langle x , x^n \rangle \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \langle x^{n-1}, 1 \rangle & \langle x^{n-1} , x \rangle & \cdots & \langle x^{n-1} , x^{n-1} \rangle & \langle x^{n-1} , x^n \rangle \\ \langle x^n, 1 \rangle & \langle x^n , x \rangle & \cdots & \langle x^n , x^{n-1} \rangle & \langle x^n , x^n \rangle \end{array} \right]^{-1} \left[ \begin{array}{c} \langle 1, \ln(x+1) \rangle \\ \langle x, \ln(x+1) \rangle \\ \vdots \\ \langle x^{n-1}, \ln(x+1) \rangle \\ \langle x^n, \ln(x+1) \rangle \end{array} \right] $$

We will then apply the following initial conditions to our solution to solve for the constants in the homogeneous solution, thus giving us the final solution:
 * $$y(-\frac{3}{4}) = 1, \, y'(-\frac{3}{4}) = 0$$

Solving with n = 3
Generating our A matrix using the above definition and using our previous value for c, we see:
 * $$\begin{bmatrix}

2 & -3 & 2 &  0 \\  0 &  2 & -6 &  6 \\  0 &  0 &  2 & -9 \\ 0 &  0 &  0 &  2 \end{bmatrix} \mathbf z = \left[ \begin{array}{c} -0.0388344 \\ 1.12972 \\ -0.439081 \\ 0.0760925 \end{array} \right] $$

Then, solving for z, the coefficients of our particular solution:
 * $$\mathbf z = \left[

\begin{array}{c} 0.4875 \\ 0.305723 \\ -0.0483327 \\ 0.0380462 \end{array} \right]$$

We can now write our solution as the combination of the homogeneous and particular:
 * $$y_3(x) = k_1 e^x + k_2 e^{2 x} + 0.0380462 x^3-0.0483327 x^2+0.305723 x+0.4875$$

Applying initial conditions and solving for the constants gives us our final solution:
 * $$y_3(x) = 4.26043 e^x - 5.50107 e^{2 x} +0.0380462 x^3-0.0483327 x^2+0.305723 x+0.4875$$

Solving with n = 6
Generating our A matrix using the above definition and using our previous value for c, we see:
 * $$\begin{bmatrix}

2 & -3 & 2 &  0 &   0 &   0 &  0\\  0 &  2 & -6 &  6 &   0 &   0 &  0\\  0 &  0 &  2 & -9 &  12 &   0 &  0\\ 0 &  0 &  0 &  2 & -12 &  20 &  0\\ 0 &  0 &  0 &  0 &   2 & -15 & 30 \\ 0 &  0 &  0 &  0 &   0 &   2 &-18 \\ 0 & 0  &  0 & 0  &  0  &  0  &  2 \end{bmatrix} \mathbf z = \left[ \begin{array}{c} 0.00585919 \\ 0.982671 \\ -0.557953 \\ 0.464459 \\ -0.275708 \\ 0.0834434 \\ -0.0096586 \end{array} \right] $$

Then, solving for z, the coefficients of our particular solution:
 * $$\mathbf z = \left[

\begin{array}{c} -1.07721 \\ -1.25584 \\ -0.803619 \\ -0.221226 \\ -0.0784793 \\ -0.001742 \\ -0.0048293 \end{array} \right]$$

We can now write our solution as the combination of the homogeneous and particular:
 * $$y_6(x) = k_1 e^x+k_2 e^{2 x}-0.0048293 x^6-0.001742 x^5-0.0784793 x^4-0.221226 x^3-0.803619 x^2-1.25584 x-1.07721$$

Applying initial conditions and solving for the constants gives us our final solution:
 * $$y_6(x) = 5.82483 e^x - 5.52206 e^{2 x}-0.0048293 x^6-0.001742 x^5-0.0784793 x^4-0.221226 x^3-0.803619 x^2-1.25584 x-1.07721$$

Solving with n = 9
Generating our A matrix using the above definition and using our previous value for c, we see:
 * $$\begin{bmatrix}

2 & -3 & 2 &  0 &   0 &   0 &   0 &   0 &   0 & 0 \\  0 &  2 & -6 &  6 &   0 &   0 &   0 &   0 &   0 & 0\\  0 &  0 &  2 & -9 &  12 &   0 &   0 &   0 &   0 & 0\\ 0 &  0 &  0 &  2 & -12 &  20 &   0 &   0 &   0 & 0\\ 0 &  0 &  0 &  0 &   2 & -15 &  30 &   0 &   0 & 0\\ 0 &  0 &  0 &  0 &   0 &   2 & -18 &  42 &   0 & 0\\ 0 &  0 &  0 &  0 &   0 &   0 &   2 & -21 &  56 & 0\\ 0 &  0 &  0 &  0 &   0 &   0 &   0 &   2 & -24 & 72\\ 0 &  0 &  0 &  0 &   0 &   0 &   0 &   0 &   2 & -27 \\ 0&   0&   0&   0&     0&   0&   0&     0&     0& 2 \end{bmatrix} \mathbf z = \left[ \begin{array}{c} -0.000966978 \\ 1.00131 \\ -0.477337 \\ 0.299654 \\ -0.304399 \\ 0.334633 \\ -0.237359 \\ 0.09511 \\ -0.0197817 \\ 0.00166379 \end{array} \right] $$

Then, solving for z, the coefficients of our particular solution:
 * $$\mathbf z = \left[

\begin{array}{c} 149.094 \\ 148.883 \\ 74.23 \\ 24.7691 \\ 6.16541 \\ 1.23731 \\ 0.197483 \\ 0.0336832 \\ 0.0013397 \\ 0.000831895 \end{array} \right]$$

We can now write our solution as the combination of the homogeneous and particular:
 * $$y_9(x) = k_1 e^x+k_2 e^{2 x}+0.000831895 x^9+0.0013397 x^8+0.0336832 x^7+0.197483 x^6+1.23731 x^5+6.16541 x^4+24.7691 x^3+74.23 x^2+148.883 x+149.094$$

Applying initial conditions and solving for the constants gives us our final solution:
 * $$y_9(x) = -144.379 e^x - 5.48955 e^{2 x}+0.000831895 x^9+0.0013397 x^8+0.0336832 x^7+0.197483 x^6+1.23731 x^5+6.16541 x^4+24.7691 x^3+74.23 x^2+148.883 x+149.094$$

Comparison of different n
Using the following MATLAB code to obtain the numeric solution to y, we can then plot it versus our three projected solutions for y.
 * [[Image:R5 y=3,6,9.png]]

At this scale, there isn't much we can qualitatively say about the different approximations. They all appear to fit very well to the numerical solution. Zooming in on the graph at two points, we can begin to see the lines separate:
 * [[Image:R5 y=3,6,9 zoomed.png]]
 * [[Image:R5 y=3,6,9 zoomed 3.png]]

At this scale, we can see the separation between n=3 and n=6,9. This result is as we'd expect, that higher n converge more quickly to the solution. We can zoom even further on the clustered lines:
 * [[Image:R5 y=3,6,9 zoomed 2.png]]
 * [[Image:R5 y=3,6,9 zoomed 4.png]]

Here we observe, as expected, that again, the higher order polynomial more closely fits the graph. This remains true for the entire range of the function.

Comparison to corresponding Taylor series
To generate the Taylor series solutions for n=3,6,9, we reuse the following MATLAB code from Report 4:

Plotting this against the solutions determined above, we obtain the following plots for our three values of n:
 * [[Image:R5 y,n=3.png]]
 * [[Image:R5 y,n=6.png]]
 * [[Image:R5 y,n=9.png]]

It is difficult to determine which method finds a better solution to the ODE from this scale. We can make the following analysis, however: We can zoom in on portions of each graph to determine which method features better performance for a given n:
 * For n=3, we can see than both the projection-derived function and Taylor-derived function perform similarly for x approaching 3 in terms of error. That is to say, neither method remains too close to the numerical solution.
 * For n=6, we can see that the projection-derived function and Taylor-derived function diverge as x approaches 3. Looking at the n=3 graph, we can see that the Taylor-derived function hasn't converged toward the function, whereas the projection-derived function overlaps the numerical solution for this scale.
 * For n=9, we can see little change in the projection-derived function. However, the Taylor-derived function performs worse than for lower n, with large divergence for x > 2.4. This is expected, as the radius of convergence for the excitation function is -1 < x < 1.

Closer Comparison with n=3

 * [[Image:R5 y,n=3 zoomed.png]]
 * [[Image:R5 y,n=3 zoomed 2.png]]

In these plots, we can observe that the projection-derived function has marginally less error than the Taylor-derived function. As the function leaves that radius, the error increases, whereas the projection-derived function has roughly evenly distributed error within this range.

Closer Comparison with n=6

 * [[Image:R5 y,n=6 zoomed 2.png]]
 * [[Image:R5 y,n=6 zoomed.png]]

In both of these plots, we can observe that the error in the projection-derived function is much smaller than that of the Taylor-derived function.

Closer Comparison with n=9

 * [[Image:R5_y,n=9_zoomed.png]]

This is exactly what we expect considering the observed behavior for n=6. And as we observed above, the Taylor-derived function diverges greatly for x > 2.4. Here, we look at x around 0, to observe that even around where the Taylor series is evaluated for the excitation function, our projection-derived function performs better. This leads to a conclusion that projections, while more computationally expensive, perform better than Taylor series for a given domain of interest and the same given n.

title: User:Egm4313.s12.team8.tclamb/R5