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Physics/Essays/Fedosin/Strong gravitational constant

Strong (nuclear) gravitation
In Astronomy the only one available characteristic empirical physical constant is the gravitational constant. Without completing the charge-mass unification or final unification: one cannot say, whether it is an ‘input to the unification’ or ‘output of unification’. The same idea can be applied to the atomic physical constants also. Sitting in a grand unified roof one cannot make an ‘absolute measurement’ but can make an ‘absolute finding’. Up till now, no atomic model has implemented the gravitational constant in the atomic or nuclear physics. Then, whatever may be its magnitude, measuring its value from existing atomic principles is impossible. Its value has been measured in the lab only within a range of 1 cm to a few metres, whereas the observed nuclear size is 1.2 fermi. Until one measures the value of the gravitational constant in microscopic physics, the debate of strong (nuclear) gravitation can be considered positively. The idea of strong gravitation originally referred specifically to mathematical approach of Abdus Salam of unification of gravitation and quantum chromodynamics, but is now often used for any particle level gravitation approach. Now many persons are working on this subject. A main advantage of this subject is: it couples black hole physics and particle physics.

Strong gravitational constant
The strong gravitational constant, denoted $$~\;\; \Gamma $$ or $$~G_s $$, is a grand unified physical constant of strong gravitation, involved in calculation of gravitational attraction at the level of elementary particles and atoms.

According to Newton's law of universal gravitation, the force of gravitational attraction between two massive points with masses $$ ~ m_1 $$ and $$ ~ m_2 $$, located at a distance $$ ~ R $$ between them, is:

 * $$F=G \frac{m_1 m_2}{R^2}.$$

The coefficient of proportionality $$ ~ G $$ in this expression is called gravitational constant. It is assumed, that in contrast to the usual force of gravity, at the level of elementary particles acts strong gravitation. In order to describe it $$ ~ G $$ in the formula for gravitational force must be replaced on $$ ~ \Gamma $$:

 * $$F_{sg}=\Gamma \frac{m_1 m_2}{R^2}.$$

Dimensions and magnitude
The dimensions assigned to the strong gravitational constant may be found from the equation above — length cubed, divided by mass and by time squared (in SI units, metres cubed per kilogram per second squared).

There are several ways to assess the value of $$ ~ \Gamma $$. J. Dufour, under the assumption that the strong gravitational constant depends on the type of objects, from the interaction of two deuterium nuclei determined, that $$ G' = 2.06 \times 10^{25} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

Based on the analogy between hadrons and Kerr-Newman black holes Sivaram, C. and Sinha, K.P, and Raut, Usha and Shina, KP accepted the value $$ \Gamma = 6.7 \times 10^{27} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

This value of the strong gravitational constant allowed estimating the strong spin-torsion interaction between spinning protons.

In paper of Mongan strong gravitational constant is $$ G_s = 1.1 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

According to Robert L. Oldershaw value of the strong gravitational constant is $$ G_s = 2.18 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

As in Oldershaw’ paper, strong gravitational constant could be related with the proton radius $$ ~ R_p $$, the proton mass $$ ~ m_p $$ and the speed of light $$~c $$:
 * $$sG_p= \frac{R_p c^2}{2 m_p }= 2.4 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

According to Tennakone who identified the electron and the proton as black holes in the strong gravitational field, strong gravitational constant is:
 * $$\Gamma = 3.9 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

Zane Andrea Quintili finds a strong gravitational constant based on the similarity between the Planck mass and radius, and accordingly the mass and radius of the proton:
 * $$ G_q = \frac {8\hbar c }{m^2_p} =9.04 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

Recami et al define strong gravitational constant through the mass of the pion $$ ~ m_{\pi} $$ as follows:

 * $$N\approx \frac{h c}{ m^2_{\pi} }= 3.2 \times 10^{30} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$,

where $$ ~ h $$ – Planck constant.

From this they derive constant of strong interaction of two nucleons in the following form:

 * $$ \frac{ N g^2}{\hbar c } \approx 14$$, where $$~g $$ indicates a strong charge, $$ ~ \hbar $$ is reduced Planck constant.

Stanislav Fisenko et all found a spectrum of steady states of the electron in proper gravitational field (0.511 MeV …0.681 MeV) on the base of strong coupling constant
 * $$N= 5.1 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

U. V. S. Seshavatharam and S. Lakshminarayana in determining $$ ~ G_s $$ repelled from the Fermi constant, which led them to the value $$ G_s = 6.94 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

In the paper strong gravitational constant equal to $$\Gamma =2.77 \times 10^{32} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$.

Sergey Fedosin entered the strong gravitational constant in 1999 on the basis of equality between the Coulomb electric force and gravitational force in the hydrogen atom on the Bohr radius. This leads to the following expression for the value of the strong gravitational constant:

 * $$\Gamma= \frac{e^2}{4 \pi \varepsilon_{0} m_p m_e }=1.514 \times 10^{29} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$,

where $$ ~ e $$ – elementary charge, $$ ~ \pi $$ – pi, $$ ~ \varepsilon_{0} $$ – electric constant, $$ ~ m_p $$ – the mass of proton, $$ ~ m_e $$ – the mass of electron.

It is assumed that strong gravitation, as a universal force, acts on the matter of nucleons, hadrons, electrons and elementary particles, regardless of the type of these particles. In contrast, the standard approach considers that strong interaction does not affect electrons and other leptons.

The small mass and large charge of matter do not allow the electron to be entirely in some small volume near the nucleus, and it gets disklike axisymmetric shape, which is limited by size of atom. In the hydrogen atom electrical forces between the nucleus and matter of the electron are attractive, but they are compensated by the repulsion of the intrinsic charge of the electron. There are the centripetal force of rotation of the electron around the nucleus, and the gravitational attraction between massive nucleus and matter of the electron. All these forces are equal in magnitude. From here follows that the action of strong gravitation between the masses of nucleus and electron on the one hand, and the electric force between charges of the nucleus and the electron, on the other hand, allows to estimate the value of $$ ~ \Gamma $$. If $$~ R_B = \frac {\hbar }{ m_e \alpha c } $$ is the Bohr radius, then the equality of forces gives:

 * $$ \frac {\Gamma m_p m_e }{R^2_B} = \frac{e^2}{4 \pi \varepsilon_{0} R^2_B } .$$

Fine structure constant is
 * $$ \alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}, $$

So that
 * $$ \Gamma= \frac{\alpha \hbar c }{m_p m_e }, \qquad \qquad \hbar = \frac{\Gamma m_p m_e }{ \alpha c }.$$

Bohr radius becomes equal
 * $$~ R_B = \frac{\Gamma m_p }{ \alpha^2 c^2 } = \frac{\Gamma m_p }{ V^2_B },$$

where $$~ V_B = \alpha c $$ is the orbital speed of the electron cloud at the first energy level.

Hence $$~ V^2_B = \frac{\Gamma m_p }{ R_B }$$, and the kinetic energy of the electron, taking into account determination of strong gravitational constant, is equal to:
 * $$~ K = \frac{m_e V^2_B }{ 2 } = \frac{\Gamma m_p m_e }{ 2 R_B }=\frac { e^2}{8 \pi \varepsilon_0 R_B } = - \frac {W}{2} ,$$

where $$~ W $$ is the potential energy of electron in the electric field of the nucleus of a hydrogen atom.

It turns out the virial theorem in the form $$~ K = - \frac {W}{2} $$. The total electron energy is also found at the first energy level:

 * $$~ E = K+W = \frac {W}{2} = -K = -13.6 $$ eV.

With the help of the constant $$ ~ \Gamma $$ the rest energy of proton in the form of a ball is equal to half of its potential energy of strong gravitational field in accordance with virial theorem, if we assume that the binding energy $$ ~ E_b $$ for the proton up to a sign is equal to the total energy of proton, and $$ ~ E_b $$ becomes very close to relativistic energy in the form of rest energy:

 * $$~ m_p c^2 \approx E_b = -\frac {W_p}{2} = \frac{ k \Gamma m^2_p }{ 2R_p},$$

where $$ ~ R_p =8.73 \times 10^{-16} $$ m is the proton radius, $$ ~ k=0.62 $$ (in the hypothetical case of a uniform mass density of the proton there must be $$ ~ k = 0.6 $$). This implies that the mass of nucleons is determined by the energy of the strong gravitation according to the principle of mass–energy equivalence.

If we assume that the magnetic moment of the proton is created by the maximum rotation of its positive charge distributed over the volume of the proton in the form of a ball, when the centripetal acceleration at the equator becomes equal to acceleration of strong gravitation, the formula for the magnetic moment is as follows:
 * $$ ~ P_m = \delta e \sqrt {\Gamma m_p R_p}, $$

where $$ ~ P_m = 1.41 \times 10^{-26} $$ J / T is the magnetic moment of the proton, $$ ~ \delta = 0.1875 $$ (in the case of uniform density and charge should be $$ ~ \delta = 0.2 $$).

From the formulas for the energy and the magnetic moment the radius of the proton is determined in the self-consistent model.

The strong gravitational constant is also included in the formula describing the nuclear force through strong gravitation and gravitational torsion field of rotating particles. A feature of the gravitational induction is that if two bodies rotate along one axis and come close by the force of gravitation, then these bodies will increase the angular velocity of its rotation. In this regard, it is assumed that the nucleons in atomic nuclei rotate at maximum speed. This may explain the equilibrium of the nucleons in atomic nuclei as a balance between the attractive force of strong gravitation and the strong force of the torsion field (of gravitomagnetic forces in gravitoelectromagnetism). In particular, the coupling constant is

 * $$\alpha_{pp}= \frac{\beta \Gamma m^2_p }{\hbar c }=13.4 \beta $$,

where $$ ~ \beta $$ is equal to 0.26 for the interaction of two nucleons, and tending to 1 for bodies with a lower mass density.

The constant $$~\alpha_{pp}$$ is close to coupling constant of strong interaction of two nucleons in Standard Model
 * $$\alpha_s= \frac{ g^2_{N \pi}}{4\pi\hbar c } \approx 14.6$$, where $$~g_{N \pi} $$ is the constant of the pseudoscalar nucleon-pionic interaction.

Fine structure constant is coupling constant of electromagnetic interaction and may be written so:

Role of squared Avogadro number
Considering Avogadro number $$N$$ as a scaling factor, U. V. S. Seshavatharam and S. Lakshminarayana finally arrived at a value of  $$ G_s = N^2G = 2.42 \times 10^{37} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}$$. It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N2G. This is a direct confirmation of the existence of the atomic or nuclear gravitational constant in nuclear physics. Therefore, this subject can now be considered as part of the mainstream research in quantum gravity.

The central idea is: for mole number of particles, strength of gravity is $$N.G$$ and force required to bind $$N$$ particles is $$\frac{c^4}{N.G}.$$ Force required to bind one particle is $$\frac{c^4}{N^2.G}.$$ By considering this force magnitude as the characteristic weak force magnitude, it is observed that, $$\ln \sqrt{\frac{e^2}{4 \pi \epsilon_0 G m_p^2}} \cong \sqrt{\frac{m_p}{m_e}-\ln\left(N^2\right)}$$ where $$m_p$$ is the rest mass of proton and $$m_e$$ is the rest mass of electron. Obtained value of $$ G\cong \; 6.{\rm 6}66270{\rm 1}79\times {\rm 1}0^{-{\rm 1}1} {\rm \; m}^ {\rm Kg}^ {\rm sec}^{{\rm -2}.}$$ Here the most important point to be emphasized is $$\frac{c^4}{G}$$ can be considered as the classical or upper limit of gravitational or electromagnetic force. It can be considered as the grand unified force. It is the origin of Planck scale and of the black hole astrophysics.

Connection with usual gravitational constant
With the help of similarity of matter levels and SPФ symmetry in Theory of Infinite Hierarchical Nesting of Matter the value of $$ ~ \Gamma $$ can also be defined in terms of coefficients of similarity and the gravitational constant:

 * $$\Gamma = G \frac{ \Phi }{ P S^2},$$

where $$ ~ \Phi =1.62 \times 10^{57} $$, $$ ~ P= 1.4 \times 10^{19} $$, $$ ~ S= 0.23 $$ are the coefficients of similarity in mass, size and speed, respectively, for the degenerate quantum objects at the atomic and stellar levels of matter. The powers of similarity coefficients in this equation correspond to dimension of gravitational constant according to dimensional analysis.

From the standpoint of Infinite Hierarchical Nesting of Matter and Le Sage's theory of gravitation, the presence of two gravitational constants $$ ~ \Gamma $$ and $$ ~ G $$ reflects the difference between the properties of gravitons and properties of matter at different levels of matter.

In particular, for the strong gravitational constant and the ordinary gravitational constant it is possible to write similar relations, in which these constants are expressed in terms of the corresponding energy densities of gravitons’ fluxes in electrogravitational vacuum and the parameters of the densest object of the corresponding level of matter:
 * $$~ \Gamma = \frac { \varepsilon_c \vartheta^2}{4 \pi M^2_n }, \qquad \qquad G = \frac { \varepsilon_{cs} \vartheta^2_s}{4 \pi M^2_s } , $$

where $$~ \varepsilon_c = 7.4 \cdot 10^{35}$$ J/m³ is the energy density of the graviton fluxes for cubic distribution; $$~ \vartheta = 2.67 \cdot 10^{-30} $$ m² is the cross-section of interaction of the charged particles of the electrogravitational vacuum (praons) with nucleons, which is very close in magnitude to the geometrical cross-section of the nucleon and is used to calculate the electric constant; $$~ M_n $$ is the mass of the nucleon; $$ \varepsilon_{cs} = \varepsilon_c \frac {\Phi S^2}{ P^3} = 2.3 \cdot 10^{34}$$ J/m³ is the energy density of the graviton fluxes at the stellar level for cubic distribution; $$~ \vartheta_s = \vartheta P^2 = 5.2 \cdot 10^{8} $$ m² is the cross-section of interaction between the gravitons and a neutron star; $$~ M_s = M_n \Phi = 2.7 \cdot 10^{30} $$ kg is the mass of the neutron star.

At the matter level of praons, its own strong gravitational constant $$~G_{pr} $$ must act. Considering that the coefficient of similarity in speed between the nucleon and praon levels of matter is $$~S \approx 1 $$, we can write:

 * $$ G_{pr} = \Gamma \frac{ \Phi }{ P S^2} = \frac{ q^2_{pr}\beta}{4 \pi \varepsilon_{0} m^2_{pr} } =1.752 \cdot 10^{67}$$ m3•s–2•kg–1,

where $$~ q_{pr} = 1.06 \cdot 10^{-57} $$ C is the charge of the praon, $$~ m_{pr} = 1 \cdot 10^{-84}$$ kg is the mass of the praon, $$~ \beta = \frac { m_p }{ m_e }= 1836.152$$ is the proton to electron mass ratio.

Connection with mass and unification of interaction
The main object of unification is to understand the origin of elementary particles mass, (Dirac) magnetic moments and their forces. Right now and till today ‘string theory’ with 4 + 6 extra dimensions not in a position to explain the unification of gravitational and non-gravitational forces. More clearly speaking it is not in a position to bring down the planck scale to the nuclear size. Physicists say – if strength of strong interaction is unity, with reference to the strong interaction, strength of gravitation is 10−39. The fundamental question to be answered is: is mass an inherent property of any elementary particle?

One can say: for any elementary particle mass is an induced property. This idea makes grand unification easy. Note that general relativity does not throw any light on the ‘mass generation’ of charged particles. It only suggests that space-time is curved near the massive celestial objects. More over it couples the cosmic (dust) matter with geometry. But how matter is created? Why and how elementary particle possesses both charge and mass? Such types of questions are not discussed in the frame work of general relativity.

The first step in unification is to understand the origin of the rest mass of a charged elementary particle. Second step is to understand the combined effects of its electromagnetic (or charged) and gravitational interactions. Third step is to understand its behavior with surroundings when it is created. Fourth step is to understand its behavior with cosmic space-time or other particles. Right from its birth to death, in all these steps the underlying fact is that whether it is a strongly interacting particle or weakly interacting particle, it is having some rest mass. To understand the first two steps somehow one can implement the gravitational constant in sub atomic physics.

To bring down the Planck mass scale to the observed elementary particles mass scale a large scale factor is required. Just like relative permeability and relative permittivity by any suitable reason in atomic space if one is able to increase the value of classical gravitational constant, it helps in four ways. Observed elementary particles mass can be generated and grand unification can be achieved. Third important application is characteristic building block of the cosmological dark matter can be quantified in terms of fundamental physical constants. Fourth important application is – no extra dimensions are required. Finally nuclear physics and quantum mechanics can be studied in the view of strong nuclear gravity where nuclear charge and atomic gravitational constant play a crucial role in the nuclear space-time curvature, quantum chromodynamics and quark confinement. Not only that cosmology and particle physics can be studied in a unified way. In this connection it is suggested that square root of ratio of atomic gravitational constant and classical gravitational constant is equal to the Avogadro number. The Avogadro constant expresses the number of elementary entities per mole of substance and it has the value mol–1. Avogadro's constant is a scaling factor between macroscopic and microscopic (atomic scale) observations of nature. It is an observed fact. The very unfortunate thing is that even though it is a large number it is neither implemented in cosmology nor implemented in grand unification.

Modern physics is having hardly 100 years of ‘strong nuclear’ back ground. By Einstein’s time very little information was available on nuclear strong and weak forces. Avogadro hypothesis was proposed in 1811. Compared to modern nuclear physics, Avogadro number is having 100 years of old history. Avogadro number may not be a fundamental physical constant but can be considered as a ‘scale factor’. But quantitatively it can be linked with the fundamental force ratios. Future thoughts and experiments may give some clue of it. Best present example is the ratio of planck mass and electron mass. Considering this ratio automatically N2 comes into picture. It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N2G. This is a direct confirmation of the existence of the atomoc or nuclear gravitational constant.

Here the very important question to be answered is – which is more fundamental either $$ G $$ or $$ G_s $$ ? It is proposed that both can be considered as the 'head' and 'tail' of matter coin. It can also be suggested that classical $$ G $$ is a consequence of the existence of atomic $$ G_s $$. It is known that there is a difference in between 'absolute findings' and 'absolute measurements'. Absolute findings can be understood where as 'absolute measurements' can not be made by nuclear experiments which are being conducted under the sky of universal gravity with unknown origin of elementary particles mass.

Till today there is no explanation for this fantastic and large difference between $$ G $$ or $$ G_s $$ or between gravitation and strong interaction, about 10−39. It can be supposed that elementary particles construction is much more fundamental than the black hole's construction. If one wishes to unify electroweak, strong and gravitational interactions it is a must to implement the classical gravitational constant $$ G $$ in the sub atomic physics. By any reason if one implements the Planck scale in elementary particle physics and nuclear physics automatically $$ G $$ comes into subatomic physics. Then a large arbitrary number has to be considered as a proportionality constant. After that its physical significance has to be analyzed. Alternatively its equivalent 'strong atomic gravitational constant' can also be assumed. Some attempts have been done in physics history.

Whether it may be real or an equivalent if it is existing as a 'single constant' its physical significance can be understood. Nuclear size can be fitted with 'nuclear Schwarzschild radius'. Nucleus can be considered as 'strong nuclear black hole'. This idea requires a basic nuclear fermion! Nuclear binding energy constants can be generated directly. Proton-neutron stability can be studied. Origin of strong coupling constant and Fermi's weak coupling constant can be understood. Charged lepton masses can be fitted. Such applications can be considered favorable for the proposed assumptions and further analysis can be carried out positively for understanding and developing this proposed 'Avogadro's strong nuclear gravity'.

Unification means: finding the similarities, finding the limiting physical constants, finding the key numbers, coupling the key physical constants, coupling the key physical concepts, coupling the key physical properties, minimizing the number of dimensions, minimizing the number of inputs and implementing the key physical constant or key number in different branches of physics. This is a very lengthy process. In all these cases observations, interpretations, experiments and imagination play a key role. The main difficulty is with interpretations and observations. As the interpretation changes physical concept changes, physical equation changes and finally the destiny changes.

Note that human beings are part of this universal gravity. There are some natural restrictions to experiments. Seeing a black hole is highly speculative. But indirectly its significances can be well understood. In the similar way in nuclear and particle physics: any experimental setup which is being run under the influence of the proposed strong nuclear gravity, without knowing the probing particle’s massive origin, without knowing the massive origin of the nucleus: based on ‘grand unified scheme’ one may not be able to unearth the absolute findings. Note that observer, experimental setup and the probing particle all are under the same influence of universal gravity. When searching for an experimental proof in grand/final unification scheme or dark matter projects this fact may be considered positively for further analysis.

To conclude it can be suggested that – existence of strong gravitational constant as Atomic gravitational constant is true and its consequences can be understood easily and can be implemented easily in grand unification program and dark matter projects.


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User:Eml5526.s11.team2.greatsemesterHW7

=Problem 7.1: Two Dimensional Heat problem with fixed boundary considering static and transient case =

Given: The Following 2D Data Set
Given the data set provided below.

Domain: Bi-Unit Square
1) The following domain information $$ \Omega = \bar \omega  = \square $$ which is for a bi-unit square

Equation: PDE, K, f, and Initial Condition
2) The partial differential equation of the form $$ \displaystyle \operatorname{div} \left ( K  \cdot \nabla{u} \right ) + f = \rho c\frac $$ 2.a) $$  {K}= {I}=I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} $$

Find: 2-D Lagrange Interpolation Basis Functions (LIBF), Linear Lagrange Interpolation Basis Functions(LLEBF) and Solve the PDE for $$ m = n = 2,4,6,8 $$
Use 2-dimensional Lagrange Interpolation Basis Functions (LIBF) and Linear Lagrange Interpolation Basis Functions(LLEBF) with $$ \displaystyle m = n = 2,4,6,8 $$ nodes per element to solve the data set above. Ensure the accuracy of the approximate solution $$ \displaystyle u^h(x = 0, y = 0) = 10^{-6} $$ at the center of the element $$ \displaystyle (x,y) = (0,0) $$

1) Static case (steady state)
$$f\left(\boldsymbol{x} \right)=1$$ in $$\Omega =\square $$

2) Dynamic Case(Transient)
$$\rho c=3$$

initial condition: $$u\left( x,t=0 \right)=xy$$, $$\forall x\in \Omega $$

(a) using 2D LIBF
$$f\left(\boldsymbol{x},t \right)=0$$, $$\forall \boldsymbol{x}\in \Omega $$, $$\forall t>0$$

$$g\left( \boldsymbol{x},t \right)=2$$ on $${{\Gamma }_{g}}=\partial \Omega $$

(b) With new conditions:
$$f\left( \boldsymbol{x},t \right)=1$$, $$\forall \boldsymbol{x}\in \Omega $$, $$\forall t>0$$

$$g\left( \boldsymbol{x},t \right)=2$$ on $${{\Gamma }_{g}}=\partial \Omega $$

Solution
The 2-D LIBF can be defined as in (6.5.1)

Where $$ \displaystyle L_{i,m}(x) $$ and  $$ \displaystyle L_{j,n}(y) $$ represent Lagrange Interpolation Basis Functions

For LLEBF we will have only four basis function for each element, which can be constructed from the coordinates of that particular element in the same way as LIBF are constructed.

Initial condition is given for transient case as

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$$  \displaystyle \begin{matrix} u(x,t=0)=u_{o}(x)=xy & \forall x\in \Omega \end{matrix} $$     (7.1.3) Now in order to find values at nodes at time zero we have to use projection method to get matrices.
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$$  \displaystyle = $$     (7.1.4) Our goal is
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Find $$\displaystyle \begin{matrix} d_{j}(0) & st \forall x_{j} \in \Omega \end{matrix}$$

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$$  \displaystyle \sum_{i} c_{i}\left [ \sum_{j}- \right ]=0 $$     (7.1.5)
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$$  \displaystyle \begin{matrix} \forall c_{i} & \forall x_{i}\in \Omega =c_{F} \end{matrix} $$     (7.1.6)
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$$  \displaystyle G_{i}= $$     (7.1.7)
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$$  \displaystyle c_{F}[\Gamma _{FF}d_{F}-G]=0 $$     (7.1.8)
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$$  \displaystyle d(0)=d_{F}(0)=\Gamma _{FF}^{-1}G $$     (7.1.9)
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$$  \displaystyle \Gamma _{FF}=[,i,j\in \eta _{F}] $$     (7.1.10)
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Where $$\displaystyle\eta _{F}$$ is a set of global node numbers not on essential boundary.

DWF form was given in Meeting 23

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$$  \displaystyle c_{E}\left [ (M_{EE}g^{(s)}+K_{EE}g)+M_{EF}d_{F}^{(s)}+K_{EF}d_{F}-F_{E} \right ]=0 $$     (7.1.11)
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$$  \displaystyle c_{F}\left [ (M_{FE}g^{(s)}+K_{FE}g)+M_{FF}d_{F}^{(s)}+K_{FF}d_{F}-F_{F} \right ]=0 $$     (7.1.12)
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$$  \displaystyle M_{FF}d_{F}^{(s)}+K_{FF}d_{F}=F_{F}-(M_{FE}g^{(s)}+K_{FE}g) $$     (7.1.13)
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$$  \displaystyle Md^{(s)}+Kd=F $$     (7.1.14) For static case
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$$  \displaystyle \begin{matrix} Kd=F\\ where\\ F=F_{F}-K_{FE}g \end{matrix} $$     (7.1.15) For transient heat transfer case
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$$  \displaystyle \displaystyle \dot{d}=M^{-1}[F-Kd] $$     (7.1.16)
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1(a) using 2D LIBF

Temperature (u) at the center when m=n=5. This value is taken as accurate and error is calculated with reference to this value

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$$  \displaystyle
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{T_{center}} = {\text{2}}{\text{.294921821297876}} $$     (N)
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1(b) using 2D LLEBF

comment
The plot of $$u-u_h$$ is zig-zag in this case because in this plot both odd number of global node(3*3,5*5,7*7) and even number of global nodes (4*4,6*6) are considered together. If we consider only odd numbers and even numbers separately, curve will smoothly approach to accurate solution (Tcenter = 2.294921821297876).

2(a) using 2D LIBF

first five plot represent the state of temperature with respect to time. Last plot represent system approaching steady state with time. In this case as heat source is zero system should approach to (u=2) at center with time

comment
First five plot represent the state of temperature with respect to time. Last plot represent system approaching steady state with time. In this case as heat source is zero system should approach to (u=2) at center with time.

2(b)1 using 2D LIBF

comment
First three plot represent the state of temperature with respect to time. Last plot represent system approaching steady state with time. In this case as heat source is one system should approach to (u=2.294921821297876) at center with time.

2(b)2 using 2D LLEBF

comment
First three plot represent the state of temperature with respect to time. Last plot represent system approaching steady state with time. In this case as heat source is one system should approach to (u=2.294921821297876) at center with time

=Problem 7.2: 2-D Vibrating Membrane Using Static and Dynamic LIBF Analysis=

Given: The 2-D Vibrating Membrane Data Set
Given the data set provided below.

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$$ \begin{align} \displaystyle T \cdot div(\nabla \mathbf{u}) + \mathbf{f} &= \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} \\ T &:= \mathbf{Tension} \left( \frac{force}{length} \right) \\ \mathbf{u}(\mathbf{x},t) &:= \mathbf{Transverse \quad Displacement} \\ \mathbf{f}(\mathbf{x},t) &:= \mathbf{Distribute \quad Transverse \quad Force} \\ \rho(\mathbf{x}) &:= \mathbf{Mass \quad Density}\left( \frac{mass}{area}\right) \\ \Omega &:= \mathbf{Membrane \quad Boundary}\\ d\Omega &:= \mathbf{Displacement \quad on \quad the \quad Membrane \quad Boundary}\\ \end{align} $$     (7.2.1)
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1) Static (Steady State):

 * Find the solution for the static vibrating membrane $$ \displaystyle \mathbf{u}_s^{h} $$ to $$ 10^{-6} $$ accuracy for the problem formulated below.

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$$ \begin{align} \displaystyle T \cdot div(\nabla \mathbf{u}) + \mathbf{f} &= 0 \\ T &= 4 \\ \mathbf{f}(\mathbf{x},t) &= 1\quad in \quad \Omega=\square \\ g &= 0\quad on \quad d\Omega \implies \frac{d^{2}\mathbf{u}}{dt^{2}}= 0 \\ \end{align} $$     (7.2.2)
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2) Dynamic (Transient): $$ \rho=3 $$

 * Find the solution for the dynamic vibrating membrane $$ \displaystyle \mathbf{u}_d^{h} $$ to $$ 10^{-6} $$ accuracy for the problem formulated below.

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$$ \begin{align} \displaystyle T \cdot div(\nabla \mathbf{u}) + \mathbf{f} &= \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} \\ T &= 4 \\ \rho &= 3 \\ \mathbf{f}(\mathbf{x},t) &= 0\quad in \quad \Omega=\square \\ \mathbf{u}(\mathbf{x},t=0) &= 0 \quad \forall \mathbf{x} \in \Omega\\ \mathbf{\dot{u}}(\mathbf{x},t=0) &= 0 \quad \forall \mathbf{x} \in \Omega\\ \end{align} $$     (7.2.3)
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2a) Free Vibration: Solve General Eigenvalue Problem (EVP)

 * The general EVP is given by,
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$$ \displaystyle \mathbf{K \Phi} = \lambda \mathbf{M \Phi} $$     (7.2.4)
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 * Where each of the component to the EVP are defined here,
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$$ \begin{align} \displaystyle \quad \mathbf{\Phi} &:= \mathbf{Eigenvector} \\ \mathbf{\lambda}&:= \mathbf{Eigenvalue} \\ \mathbf{M}&:= \mathbf{Mass(Inertial)} \ \mathbf{Matrix} \\ \mathbf{K}&:= \mathbf{Stiffness} \ \mathbf{Matrix} \\ \end{align} $$
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 * Determine the first 3 eigenpairs $$\displaystyle \left( \lambda_{i},\mathbf{\phi _i} \right) $$ where the fundamental frequency $$\displaystyle \omega_1 $$ and fundamental period $$\displaystyle T_1 $$ are given by

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$$ \begin{align} \displaystyle \omega _1 &= \left( \lambda _1 \right)^{\frac{1}{2}} \quad s.t. \quad {f_1} = \frac \\ and \quad {T_1} &= \frac{1} \end{align} $$     (7.2.5)
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 * Plot $$\displaystyle \mathbf{u}^{h}(\mathbf{0},t)$$ vs. t for $$\displaystyle t \in [0,2T_1] $$.
 * Produce a movie of the vibrating membrane.

3) Dynamic (Transient): Similar to Part 2 but Using Asymmetric $$ u_0 $$

 * Find the solution for the dynamic vibrating membrane $$ \displaystyle \mathbf{u}_d^{h} $$ to $$ 10^{-6} $$ accuracy for the problem formulated below.

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$$ \begin{align} \displaystyle T \cdot div(\nabla \mathbf{u}) + \mathbf{f} &= \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} \\ T &= 4 \\ \rho &= 3 \\ \mathbf{f}(\mathbf{x},t) &= 0\quad in \quad \Omega=\square \\ \mathbf{u}(\mathbf{x},t=0) &= (x+1)\left( y+\frac{1}{2} \right)cos\left( \frac{\pi}{2}x\right)cos\left( \frac{\pi}{2}y\right) \quad \forall \mathbf{x} \in \Omega\\ \mathbf{\dot{u}}(\mathbf{x},t=0) &= 0 \quad \forall \mathbf{x} \in \Omega\\ \end{align} $$     (7.2.6)
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3a) Free Vibration: Solve General Eigenvalue Problem (EVP)

 * The general EVP given in (7.2.4).
 * Determine the first 3 eigenpairs $$\displaystyle \left( \lambda_{i},\mathbf{\phi _i} \right) $$ where the fundamental frequency $$\displaystyle \omega_1 $$ and fundamental period $$\displaystyle T_1 $$ are given in (7.2.5).
 * Plot $$\displaystyle \mathbf{u}^{h}(\mathbf{0},t)$$ vs. t for $$\displaystyle t \in [0,2T_1] $$.
 * Produce a movie of the vibrating membrane.

1) Solution to the Static Case
First we will recall that the governing equation in (7.2.2) is,

Our governing PDE can be rewritten as the Weak Form shown here, Next we will integrate by parts, rearrange terms, apply Gauss' Theorem and arrive at our final formulation.

It is customary to represent the final expression in (7.2.8) as a more concise formulation shown here,
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$$ \displaystyle \begin{align} \tilde{m}(w,u) + \tilde{k}(w,u) &= \tilde{f}(w) \quad \forall \ w=0 \ on \ \Gamma_g \\ \tilde{m}(w,u) &= \int\limits_\Omega {w\rho \frac} d\Omega =0 \\ \tilde{k}(w,u) &= \int\limits_\Omega {T\nabla w\nabla u} d\Omega \\ \tilde{f}(w) &= - \int_{\Gamma h} {whd{\Gamma _h}}  + \int\limits_\Omega  {wf} d\Omega \\ \end{align} $$     (7.2.9)
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2) Solution to Eigenvalue Problem: Separation of Variables
The governing PDE is a function of $$ \displaystyle x, y,$$ and $$ \displaystyle t $$ we can rewrite $$ \displaystyle u(x,y,t)$$ as a product, where we seek a solutions of the form,

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$$ \displaystyle u (x,y,t) = {X}(x) {Y}(y) {T}(t) $$     (7.2.12)
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Our governing PDE is for the static membrane is shown here,
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$$ \displaystyle T\left( {\frac + \frac} \right) = \rho \frac $$     (7.2.13)
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Upon rearrangement we obtain the following form which is more suitable for applying Separation of Variables,
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$$ \displaystyle \left( {\frac + \frac} \right) = \frac{\rho }{T}\frac $$     (7.2.14)
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Substituting the appropriate derivatives of (7.2.12) into (7.2.14) gives,
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$$ \displaystyle \left( {XYT + YXT} \right) = \frac{\rho }{T}T''XY $$ (7.2.15)
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Dividing the LHS and RHS by the product $$ \displaystyle X(x)Y(y)T(t) $$ and setting the equation equal to a constant $$ \displaystyle -\omega^2 $$, (In this case the constant is set up to work with the natural frequency calculation):
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$$ \displaystyle \frac{X} + \frac{Y} = \frac{\rho }{T}\frac{T} = - {\omega ^2} $$     (7.2.16)
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Letting $${X} $$ and $$ {Y} $$ be equal to the arbitrary constants $$\displaystyle k_1 $$ and $$\displaystyle k_2 $$ we get the following ODEs shown in (7.2.17) and (7.2.18). One can consider these ODEs as being constrained by equation (7.2.19).
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$$ \displaystyle {\frac{X}} = - {k_1}^2 $$     (7.2.17)
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$$ \displaystyle {\frac{Y}} = - {k_2}^2 $$    (7.2.18)
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$$ \displaystyle {\omega ^2} = {k_1}^2 + {k_2}^2 $$    (7.2.19)
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Upon rearrangement of (7.2.17) and (7.2.18) $$ \displaystyle X(x) $$ and $$ \displaystyle Y(y) $$ are represented in the two second order homogeneous ODEs.
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$$ \displaystyle \begin{align} {X''} + {k_1}^2{X} &= 0 \\ {Y''} + {k_2}^2{Y} &= 0 \\ \end{align} $$    (7.2.20)
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$$ \displaystyle X(x) $$ and $$ \displaystyle Y(y) $$ have the same general solutions,
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$$ \displaystyle {X}=Acos({k_1}x)+Bsin({k_2}x) $$     (7.2.21)
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For the simplicity of imposing the BCs let us assume the domain is $$ \displaystyle {x,y}\in [0,L] $$. When we impose the BC by evaluating at 0, the sin term drops out therefore making $$ \displaystyle A=0 $$. Next observe at L, $$\displaystyle k_{1}L = n\pi $$ and $$\displaystyle k_{2}L = m\pi $$.

Similarly, the second order homogeneous ODE for $$ \displaystyle T(t) $$ is,
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$$ \displaystyle {T''} + \frac{T}{\rho }{\omega ^2}{T} = 0 $$     (7.2.22)
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Which has the following general solution,
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$$ \displaystyle {T} = C\cos \left( {\omega\sqrt {\frac{T}{\rho }} t} \right) + D\sin \left( {\omega\sqrt {\frac{T}{\rho }} t} \right) $$     (7.2.23)
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Therefore, the natural frequency is $$ \displaystyle \omega\sqrt {\frac{T}{\rho }} $$. Expressing the natural frequeny in terms of equation (7.2.19) gives us,
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$$ \displaystyle \omega_{m,n}=\sqrt {\frac{T}{\rho }}\sqrt {k_1^2+k_2^2}=\sqrt {\frac{T}{\rho }\left[ {{{\left( {\frac{{n\pi }}{L}} \right)}^2} + {{\left( {\frac{{m\pi }}{L}} \right)}^2}} \right]} $$     (7.2.24)
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Substituting the numerical values for density, temperature, and BC's from the problem statement and letting $$ \displaystyle m=n=1 $$, $$ \displaystyle m=2 $$ and $$ \displaystyle n=1 $$, and $$ \displaystyle m=n=2 $$,

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$$ \displaystyle \begin{align} {\omega _{1,1}} &= 5.13 \\ {\omega _{2,1}} &= 8.11 \\ {\omega _{2,2}} &= 10.26 \\ \end{align} $$   (7.2.25)
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=Problem 7.3: 2-D Static Vibrating Membrane Analysis Using Quadratic and Rectangular Meshes=

Given: The 2-D Vibrating Membrane Data Set
Given the data set provided below.

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$$ \begin{align} \displaystyle T \cdot div(\nabla \mathbf{u}) + \mathbf{f} &= \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} \\ T &:= \mathbf{Tension} \left( \frac{force}{length} \right) \\ \mathbf{u}(\mathbf{x},t) &:= \mathbf{Transverse \quad Displacement} \\ \mathbf{f}(\mathbf{x},t) &:= \mathbf{Distribute \quad Transverse \quad Force} \\ \rho(\mathbf{x}) &:= \mathbf{Mass \quad Density}\left( \frac{mass}{area}\right) \\ \Omega &:= \mathbf{Membrane \quad Boundary}\\ d\Omega &:= \mathbf{Displacement \quad on \quad the \quad Membrane \quad Boundary}\\ \end{align} $$     (7.3.1)
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1) Static (Steady State):

 * Let $$ \displaystyle \Omega = \bigcirc $$ be a circle of unit radians.
 * Find the solution for the static vibrating membrane $$ \displaystyle \mathbf{u}_s^{h} $$ to $$ 10^{-6} $$ accuracy at the center for the problem formulated below.

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$$ \begin{align} \displaystyle T \cdot div(\nabla \mathbf{u}) + \mathbf{f} &= 0 \\ T &= 4 \\ \mathbf{f}(\mathbf{x},t) &= 1\quad in \quad \Omega= \bigcirc \\ g &= 0\quad on \quad \Gamma_g=d\Omega \implies \frac{d^{2}\mathbf{u}}{dt^{2}}= 0 \\ \Omega &= \bigcirc \\ \end{align} $$     (7.3.2)
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Solution: Matlab
After modifying code to no avail, I was fortunate enough to use the abilities of wikiversity to find a similar code and then modified it. The code used to solve this problem was referenced from Deshpande. I made modifications to the code to allow for the use of a greater number of elements. Here I used 16 elements instead of the 9 used by team 6. In modifying the code I also used code from Fish and Belytschko, A First Course in Finite Elements. The code was found through their website and was in chapter 9 [|F&B]. The code used is as follows

The graphs produced by the code are

=Problem 7.4=

Given: The Governing Equation and Boundary Conditions
Governing equation was derived in meeting 32-4.
 * $$\displaystyle Q_{1}$$:Heat flow into w throgh $$\displaystyle \partial w$$
 * $$\displaystyle Q_{2}$$:Heat generated in w by heat source.
 * $$\displaystyle Q_{3}$$:Heat in w due to change in term u.

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$$  \displaystyle Q_{1}=-\int_{\partial w}\mathbf{q}\mathbf{n}d(\partial w)=\int_{w}div\mathbf{q}dw $$     (7.4.1)
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$$  \displaystyle Q_{2}=\int_{w}f(\mathbf{x},t)dw $$     (7.4.2)
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$$  \displaystyle Q_{3}=\int_{w}\rho c\frac{\partial u}{\partial t}dw $$     (7.4.3)
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$$  \displaystyle Q_{1}+Q_{2}=Q_{3} $$     (7.4.4)
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$$  \displaystyle \mathbf{q}=-\mathbf{K}grad(u) $$     (7.4.5)
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$$  \displaystyle div(\mathbf{K}grad(u))+f=\rho c\frac{\partial u}{\partial t} $$ (7.4.6)
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Boundry Conditions

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$$  \displaystyle u(\mathbf{x},t)]_{\Gamma _{g}}=g(\mathbf{x},t) $$     (7.4.7)
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$$  \displaystyle (q.n)]_{\Gamma _{h}}=h(\mathbf{x},t) $$     (7.4.8)
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$$  \displaystyle (q.n)]_{\Gamma _{H}}=H(u-u_{\infty }) $$     (7.4.9)
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WRF

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$$  \displaystyle \int_{\Omega }w[div(Kgrad(u))+f-\rho c\frac{\partial u}{\partial t}]d\Omega =0 $$     (7.4.10)
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$$  \displaystyle\begin{align} K=  & \int_{\Omega }wdiv(Kgrad(u))d\Omega =\frac{\partial }{\partial x_{i}}(K_{ij}\frac{\partial u}{\partial x_{j}})\\ & \int_{\Omega }n_{i}wK_{ij}\frac{\partial u}{\partial x_{j}}d(\partial\Omega)-\int_{\Omega }\frac{\partial w}{\partial x_{i}}K_{ij}\frac{\partial u}{\partial x_{j}}d(\Omega) \end{align} $$     (7.4.11)
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$$  \displaystyle K=-\int_{\Gamma _{g}}wnqd\Gamma _{g}-\int_{\Gamma _{h}}wnqd\Gamma _{h}-\int_{\Gamma _{H}}wnqd\Gamma _{H}+\int_{\Omega }grad(w).K.grad(u)d\Omega $$     (7.4.12)
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We deliberately chose w i.e. i vanishes on essential boundry. So the first term vanishes too. In the first term we have unknown flux that we can not deal with right now. But other fluxes are defined.We can write our continuous form of components as:

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$$  \displaystyle \tilde{m}(w,u)=\int_{\Omega}w\rho c\frac{\partial u}{\partial t}d\Omega $$     (7.4.13)
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$$  \displaystyle \tilde{k} (w,u)=\int_{\Omega}grad(w)Kdiv(u)d\Omega+\int_{\Gamma_{H}}wHud\Gamma_{H} $$     (7.4.14)
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$$  \displaystyle \tilde{f}(w,u)=\int_{\Omega}wfd\Omega-\int_{\Gamma_{h}}whd\Gamma_{h}+\int_{\Gamma_{H}}wHu_{\infty }d\Gamma_{H} $$     (7.4.15)
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!Note this time f depeneds on both w and u.

Discrete WF

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$$  \displaystyle w^{app}=\sum_{I=1}^{n}N(x,y)_{I}c_{I} $$     (7.4.16)
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$$  \displaystyle u^{app}=\sum_{J=1}^{n}N(x,y)_{J}d_{J} $$     (7.4.17)
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$$  \displaystyle grad(u)=grad(N(x,y)_{I})d_{I}=B(x,y)_{I}d_{I} $$     (7.4.18)
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$$  \displaystyle \tilde{m}(w,u)+\tilde{k}(w,u)=\tilde{f}(w,u) $$     (7.4.19) Now lets substitute everything in the equation above.
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$$  \displaystyle \sum_{I=1}^{n}c^{T}\left [\sum_{J=1}^{n}\left \{ \int_{\Omega }\rho cN_{I}N_{J}d_{I}^{s}d\Omega+\int_{\Omega}B_{I}KB_{J}d_{I}d\Omega -\int_{\Omega}N_{I}fd\Omega+\int_{\Gamma_{h}}N_{I}hd\Gamma_{h}+\int_{\Omega}N_{I}N_{J}(\Gamma_{H})H(d(\Gamma_{H})-d_{\infty })\right \} \right ]=0 $$     (7.4.20)
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$$  \displaystyle \tilde{\mathbf{d}}=\begin{bmatrix} \bar{de}\\ df
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\end{bmatrix} $$     (7.4.21)
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$$  \displaystyle \tilde{\mathbf{w}}=\begin{bmatrix} 0\\ wf
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\end{bmatrix} $$     (7.4.22)
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$$  \displaystyle \tilde{\mathbf{K}}=\begin{bmatrix} K_{E} &K_{EF} \\ K_{EF}^{T} & K_{F} \end{bmatrix} $$     (7.4.23)
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Example of calculation of elements of K & M matrices when only one node lie on essential boundary.
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$$  \displaystyle\begin{align} & K_{E}=\int_{\Omega }B_{1}.K.B_{1}d\Omega+\int\limits_ {{N_1}H{N_1}d{\Gamma _H}} &\\ & K_{EF}=\int_{\Omega }B_{1}.K.B_{J}d\Omega+\int\limits_ {{N_1}H{N_j}d{\Gamma _H}} & J=2,n\\ & K_{FF}=\int_{\Omega }B_{I}.K.B_{J}d\Omega+\int\limits_ {{N_i}H{N_j}d{\Gamma _H}} & I,J=2,n
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\end{align} $$     (7.4.24)
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$$  \displaystyle \tilde{\mathbf{M}}=\begin{bmatrix} M_{E} &M_{EF} \\ M_{EF}^{T} & M_{F} \end{bmatrix} $$     (7.4.25)
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$$  \displaystyle\begin{align} & M_{E}=\int_{\Omega }N_{1}\rho cN_{1}d\Omega&\\ & M_{EF}=\int_{\Omega }N_{1}\rho cN_{J}d\Omega& j=2,n\\ & M_{FF}=\int_{\Omega }N_{I}\rho cN_{J}d\Omega& i,j=2,n
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\end{align} $$     (7.4.26)
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$$ \underline {\tilde c} = \left[ {\begin{array}{ccccccccccccccc} {\underline } \\ \hline \end{array}} \right] $$     (7.4.27)
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$$  \displaystyle \underline {\tilde c} = \left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right] $$     (7.4.28)
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Structure of d matrix
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$$  \displaystyle \underline {\tilde d} = \left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right] $$     (7.4.29)
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Structure of K matrix for this type of problem
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$$  \displaystyle {\left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right]_{_{1 \times \tilde n}}}^T{\left[ {\begin{array}{ccccccccccccccc} &\vline & && \\ \hline &\vline & && \\ &\vline & && \\ &\vline & && \end{array}} \right]_{\tilde n \times \tilde n}}{\left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right]_{_{\tilde n \times 1}}}+{\left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right]_{_{1 \times \tilde n}}}^T{\left[ {\begin{array}{ccccccccccccccc} {}&\vline & {}&{}&{} \\ \hline {}&\vline & {}&{}&{} \\ {}&\vline & {}&{\underline K _{HH}^H}&{} \\ {}&\vline & {}&{}&{} \end{array}} \right]_{\tilde n \times \tilde n}}{\left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right]_{_{\tilde n \times 1}}} $$     (7.4.30)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

Components of force matrix
 * {| style="width:100%" border="0"

$$  \displaystyle {\left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right]_{_{1 \times \tilde n}}}^T\left[ {\begin{array}{ccccccccccccccc} {\underline F _E^f} \\ \hline {\underline F _h^f} \\ {\underline F _H^f} \\ {\underline F _R^f} \end{array}} \right] + {\left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right]_{_{1 \times \tilde n}}}^T\left[ {\begin{array}{ccccccccccccccc} {} \\ \hline {\underline F _h^h} \\ {} \\  {} \end{array}} \right] + {\left[ {\begin{array}{ccccccccccccccc} \\ \hline \\   \\ \end{array}} \right]_{_{1 \times \tilde n}}}^T\left[ {\begin{array}{ccccccccccccccc} {} \\ \hline {} \\  {\underline F _H^H} \\ {} \end{array}} \right] $$     (7.4.31)
 * style="width:95%" |
 * style="width:95%" |
 *

 * }

 * {| style="width:100%" border="0"

$$  \displaystyle \begin{matrix} Temperature & at & global & nodes \end{matrix} $$
 * style="width:95%" |
 * style="width:95%" |
 * }

 * {| style="width:100%" border="0"

$$  \displaystyle \begin{matrix} Temperature & Contours \end{matrix} $$
 * style="width:95%" |
 * style="width:95%" |
 * }

=Contributing Members & Referenced Lecture=

title: User:Eml5526.s11.team2.greatsemesterHW7


title: File:Fe1.s11.mtg41.djvu


text:
Talk:Historical geology

I was categorising some uncategorised categories (as one does), and I noticed that all the links under the Geological Time Scale heading are topics, but in the category namespace. In fact, Historical Geology is the only listed page in each of the categories, so it's all the wrong way around! I'm gonna change this around, so a few dangerously large-looking edits will probably show up on recent changes. Gaidheal1 22:13, 13 April 2011 (UTC)

title: Talk:Historical geology


title: File:Nm1.s11.mtg41.djvu


text:
Precambrian/Archean eon

The Archean Eon does not have an initial time boundary, however, by evidence, it is a popular belief that the Archean Eon began around 3.8 billion years ago and terminates at 2.5 billion years ago.The Archean Eon contains about 45% of earth's history. The oldest fossils are found in the Archean Eon. The age of the oldest preserved rocks on Earth's surface has been formally proposed as a definition for the base of the Archaean.

Some examples of these old preserved rocks are Apex Chert (3.465 billion years old) and stromatolites (3.45 billion years old) from Australia, and the Swaziland microfossils from Africa (also about 3.45 billion years old)

During the Archean Eon the earth acquired its basic configuration with the mantel and crust surrounding the core. This general period, from about 3.0 to 2.5 billion years ago, was the period of maximum continent formation. 70% of continental landmasses date from this period (Thus, most of the continents are extremely ancient). Modern Earth sciences recognize that the present continents are built around cores of extremely ancient rock, called "shields". A large part of Australia is a "shield", as is much of Canada, India, Siberia, and Scandinavia.

None of the Archean eon eras have been further subdivided into periods.

Eoarchean (era) The Eoarechean era is bound at 3.8 to 3.6 billion years ago.It is best known through the Isua Greenstone Belt, which is the oldest known rock formation (3.8 - 3.7 billion years old). It is located in Greenland. The Greenstone belt contains metamorphosed volcanic (mafic) and sedimentary rocks.Crust formation continued throughout this era.

Paleoarchean (era) The Paleoarchean era is bound at 3.6 to 3.2 billion years ago. Both archaea and eubacteria evolved during the paleoarchean, implying that the last universal common ancestor (LUCA) of all life of earth existed during this era. The oldest stromatolites date back to about 3.5 billion years, within the Paleoarchean.Vast numbers of cyanobacteria were instrumental in changing the geology of earth and the evolution of life through the production of oxygen. The carbon isotopes in rocks point to photosynthesis. Continent formation continued throughout the Paleoarchean.

Mesoarchean (era) The Mesoarchean era is bound at 3.2 to 2.8 billion years ago. Plate tectonics forced the separation of the Kaapvaal and Pilbara cratons, and the separation of these ancient parts of South Africa and Australia was complete by the end of the Mesoarchean, around 2.8 billion years ago. Another super continent that may have originated during the mesoarchean was Ur. The first incontrovertible fossils appear from this time. All life from this period was consequently anaerobic. The oldest banded iron formations (BIFs) are dated to this period.

Neoarchean (era) The Neoarchean era is bound at 2.5 to 2.8 billion years ago. Cyanobacteria started producing significant amounts of oxygen in this period.Lead to the Oxygen Catastrophe during the early proterozoic, in which rising levels of oxygen poisoned much of the life that existed at the time. Large continents first appeared on earth, with modern plate tectonics (with subduction zones, continental plates sliding over each other and the upwelling of lava to produce new crust where continental plates tore apart). This is when the first large continent shields were formed.

title: Precambrian/Archean eon


title: Literature/1973


text:
Precambrian/Proterozoic eon

The Proterozoic Eon spanned from 2.5 Billion Years Ago to 542 Million Years Ago and comprises 42% of Earth's entire History. The Eon is divided into the following three Eras:
 * Paleo-Proterozoic Era - 2.6 Billion Years Ago to 1.6 Billion Years Ago
 * Meso-Proterozoic Era - 1.6 Billion Years Ago to 1.0 Billion Years Ago
 * Neo-Proterozoic Era - 1.0 Billion Years Ago to 542 Million Years Ago

During the Proterozoic Eon, the Earth started to exhibit modern characteristics of plate tectonics, sedimentation and a global climate. Major orogenies occurred throughout the Proterozoic Eon. During the Paleo-Proterozoic Era, the first large continent Laurentia was formed and the Earth had its first ice age. The supercontinet Rodinia was formed during the Meso-Proterozoic ERA and broke apart during the Neo-Proterozoic. The Neo-Proterozoic Era also saw the creation of the modern day Pacific Ocean, and increased glaciation; as well as the proliferation of prokaryotic and eukaryotic organisms and the evolution of multi-cellular organisms.

Laurentia
The first major event of the Paleo-Proterozoic Era was the formation of of Laurentia. Laurentia was the Earth's first large continent, and it formed when many of the separated Precambrian provinces came together. Plate tectonics moved the provinces together and sutured them along orogens as the continental crust collided. Orogens are linear tracts of deformed rocks created by compressional forces and metamorphism that accompany mountain building. The Wopmay orogen in Canada's Northwest Territory is one example of an orogeny and shows a sequence of events called a Wilson Cycle. A Wilson cycle has three steps: The rifting, plate tectonics and deposition of sediment that occur in the Wilson Cycle are a Proterozoic example of modern geological processes that occurred during the Proterozoic Eon.
 * 1) The opening of an ocean basin
 * 2) Sedimentation occurs along the margins of the rifting and spreading continents
 * 3) Closing of the ocean basin through plate tectonics

Earth's First Ice age
During the Paleo-proterozoic Era, the Earth experienced global glaciation. Evidence of this eustatic glaciation lies North of Lake Huron in the form of alternating layers of varved mudstones and tillites that were deposited between 2.6 billion years ago and 2.1 billion years ago and show that glaciers were once present. A varve is a thin sedimentary layer or pair of layers that represent the depositional record of a single year and usually form in glacial meltwater lakes. Tillites are unsorted glacial drift that has been converted into solid rock. The Paleo-proterozic glaciation was most likely the Earth's first ice age.

Banded Iron Formations(BIF)
Though they started appearing in the Archean Eon, Banded Iron Formations (or BIFs) became prevalent in the Paleo-Proterozoic. BIFs are cherts that show alternating bands of dark red and light gray. The dark red bands are made up of hematite and magnetite (iron oxide minerals). The light gray bands are chemically precipitated quartz. The formations formed on shallow sea floors as underwater hydrothermal vents dissolved iron and other elements. The iron reacted with the oxygen to form compounds that became part of the sediment on the sea floor. The alternating iron oxide bands correlate to the oscillating levels of oxygen in the Proterozoic atmosphere. When atmospheric oxygen levels were high, free oxygen was able to combine with the iron and make the rust-red color. As oxygen levels dropped, there was no free oxygen to combine with the iron and the gray bands were made. A banded iron sequence near Lake Superior contains an interesting rock formation called the Gunflint Chert. The 1.9 billion year old Gunflint Chert contains prokaryotic organisms and cyanobacteria. Cyanobacteria are prokaryotic, photosynthetic microorganisms that possess chlorophyll and produce oxygen.

Rodinia
The Meso-Proterozoic Era brought about the creation of the new supercontinent Rodinia. Rodinia was formed when Laurentia collided with other smaller land masses. Its size rivaled that of the supercontinent Pangea. The continental collisions that occurred during the formation of Rodinia created numerous orogenies, including the Greenville Orogeny in present day North America. Greenville rocks originally started out as carbonates and sandstones, but the orogenic forces placed upon them as land masses collided turned them into metamorphic rocks that contained many igneous intrusions. The figure to the right shows how the Earth’s land masses assembled to form Rodinia. Continents during this time were oriented in a far different manner than they are today. For instance, the east coast of North America was next to Antarctica and Australia. Rodinia was fully assembled by the end of the Meso-Proterozoic.

Rodinia Breaks Up
About 750 million years ago, during the Neo-Proterozoic Era, Rodinia began to break apart. The proto-Pacific Ocean (also called the Panthalassa) was created as rifting occurred, land masses separated and ocean water filled rift valleys. The aptly named proto-Pacific ocean eventually became the modern day Pacific Ocean.

Ice Age
During the Neo-Proterozoic, comprehensive global glaciation occurred again. Evidence of the Varangian glaciation is seen in tillites, dropstones and striations found in Neo-Proterozoic rocks on virtually all of Earth's landmasses. There are two prevalent hypotheses as to the cause of the glaciation. One is that highly reflective land masses reflected the Sun's energy away from the Earth's surface. As the ice caps grew, the new ice reflected even more energy and the Earth's temperature continued to drop and glaciers formed. The second hypothesis is that carbon dioxide levels in the Earth's atmosphere dropped and decreased the greenhouse effect. The decrease of carbon dioxide and the increase of oxygen were further perpetuated by the proliferation of photosynthetic organisms. The lack of the greenhouse gasses resulted in cooling the Earth and formation of glaciers.

Proterozoic Life
While Prokaryotes appeared on the Earth during the Archean, they flourished during the Proterozoic Eon. Prokaryotes are any cellular organisms that have no nuclear membrane, no organelles in the cytoplasm except ribosomes, and have genetic material in the form of single continuous strands forming coils or loops. Stromatolites, which still exist today in Sharks Bay Australia are one example of a Prokaryotic organism. Prokaryotes became more structurally complex and evolved into Eukaryotes. Eukaryotes are organisms that have as its fundamental structural unit a cell, that contains specialized organelles in the cytoplasm, a membrane-bound nucleus enclosing genetic material organized into chromosomes, and an elaborate system of division by mitosis or meiosis. Some examples are stromatolites, Animeka, Kakabekia, Eosphaera, and Acritarchs. Evidence of multi-cellular animals (Metazoans) from approximately 570 million years ago was discovered in South China and Australia. Some examples of Proterozoic Metazoans are Cyclomedusa, Tribrachidium, Cloudina, and Charniodiscus. Eukaryotes are incredible important because they are the first cells in Earth history to reproduce sexually. Acritarchs are one example of a Proterozoic Eukaryote. Just before the end of the Proterozoic Eon, there was one last major jump in evolutionary diversification. Eukaryotes evolved into multi-cellular animals, or metazoans. Metazoans are multi-cellular animals that possess more than one kind of cell and have their cells organized into tissues and organs. Kimberella was a jellyfish like Proterozoic metazoan of special note because it was the first organism to contain a coelum, or body cavity in which the digestive tract and internal organs were suspended. Metazoans were a major building block in the evolution of much of life on earth today.

title: Precambrian/Proterozoic eon


title: Category:1973


title: Literature/1972


title: Category:1972


title: Literature/1971


title: Category:1971


title: Literature/1970


title: Category:1970


title: Literature/1969


title: Category:1969


text:
Phanerozoic/Triassic period

251 to 200 million years ago

The Triassic period is the oldest of the three periods comprising the Mesozoic (Middle Life) era on the Geologic Time Scale. Named by the German geologist Friedrich August Von Alberti (1795-1878),the Triassic takes its name from the German Trias Supergroup, comprised of "threefold" lithostratigraphic units deposited during and now associated with the Triassic period. These sequences of rock strata are evident in western and central Europe and consist of both marine and continental facies. From space, Triassic Earth would look much different than what we know today, with its land masses contained in one supercontinent, Pangea, centered along the equator extending from the north to south poles.

Beneath the Triassic Earth surface, incredible tectonic and volcanic processes were happening in preparation for the breakup of Pangea. Tensional forces began to build and the supercontinent started to slowly unzip, a process that would span the entire 51 million years that demarcate the Triassic. Eventually, Pangea would break apart to form into the continents we know today. The Triassic began at the end of the Great Dying, a mass extinction event occurring at the end of the Permian period. This mass extinction of most early life on Earth opened the way for a remarkable change of all her inhabitants, and so begins the Mesozoic era of "middle life." Plant life was also remarkably different from the types of diversity we see today. The Triassic closes with the "end-Triassic extinction," another mass extinction event that profoundly changed life on Earth once again.

Continents and Tectonics
Earth appeared remarkably different entering the Triassic period 251 million years ago. All the continental land masses were joined together as one supercontinent we call Pangea, which in Greek means “Entire Earth.” Pangea was centered along the equator extending from the north to south poles and remained relatively unchanged throughout the 51 million year Triassic period. But beneath the vast surface of Triassic-Earth, incredible tectonic processes were actively preparing to break Pangea apart into the continents we are familiar with today.

The Triassic Earth experienced crustal rifting along normal fault lines as the crust initiates the process of separating and fragmenting Pangea into smaller continental units, eventually giving way to basaltic lava flows through the resulting fissures . Extensional tectonics will eventually unzip Pangea and eventually form what is now Europe and Africa, separating Mexico from South America. The Tethys Sea, shaped like a huge tongue protruding into Pangea from the east, eventually becomes the Mediterranean Sea. Seafloor spreading at divergent plate boundaries commenced the formation of today’s Atlantic mid-oceanic ridge.

Subduction along what is now the western margin of North America produces vast mountain ranges. Accretionary tectonics continues to merge displaced terranes of old island arcs and microcontinents onto the west coast of North America, increasing the land mass of Pangea. Triassic continental land masses were positioned high above sea level, exposing the mountain ranges to relentless erosion. The sedimentary record resulting from these processes on the west coast of North America is represented by the Moenkopi, Shinarump and Navajo formations .

Along the East coast of what is now the United States, fault-bounded rift basins are developing as the Allegheny orogenies are being eroded. The basins become filled with these eroded sediments to become the Newark Supergroup of the Late Triassic. The implication of this formation is that Pangea is unzipping and separating. Sedimentary formations are intruded by igneous dikes and sills indicating significant igneous activity extending from present day Massachusetts down to North Carolina. The Palisades Sill igneous intrusion (see figure) illustrates the immense igneous processes along this area during the Triassic. Such extreme rifting was accompanied by massive basaltic lava flows from the open fissures and exposed magma as the continents broke apart, likely contributing to the Late Triassic mass extinction through increased global warming.

Animal Life

 * Invertebrates - The Triassic period emerged from the Great Dying of the Permian, an extinction event that decimated life on Earth, providing opportunity for the surviving life forms to adapt and for new forms to evolve. Bivalves, ammonoids and brachiopods survived the Permian-Trassic extinction and dominated invertebrate life of the Triassic, becoming extremely complex and diverse throughout the Triassic and into the Jurassic. New life forms also begin to appear in the fossil record including the squid like coloeids as well as echinoderms, resembling modern day starfish and sea urchins. Small coral and sponge reefs appear in the shallow warm seas along the equator in the Tethys Sea but there is no evidence that they existed in mass as we see today.  Also missing from the fossil record are phytoplankton which we know are so important to sea life, but this missing record is likely from a lack of fossilized preservation of their soft-bodies.  Arthropods were likely abundant during the Triassic because they are found in both the Permian period preceding the Triassic as well as the later periods, the Jurassic and Cretaceous, of the Mesozoic. The lobster and crab crustaceans also make their first appearance during the Triassic.

 * Vertebrates - As the Triassic begins and the Permian-Triassic extinction comes to an end, 70% of land species present in the Permian have disappeared. Earth is now ready for a new line of vertebrates to emerge in place of the departed and that’s exactly what happened.  Life on land attained new levels of modernization and diversity.  Reptilian populations increase in number and variety existing both on land and in the sea.  The fish-eating plesiosaurs and the dolphin-like giant reptilian ichthyosaurs are just two examples of many sea dwellers.   Pterosuars were the first group of reptiles to take to the air in flight.  Many diminutive diapsids, lizard-like reptiles, also populated land and sea, and tortoises also made their debut in the Triassic. The therapsids, endothermic mammal-like reptilian synapsids, greatly diversified from their Permian ancestors and although prominent in the beginning of the Triassic, they eventually lose their stronghold to the ectothermic archosaurs, a large group of diapsids notable for the single openings in front of their eyes on each side of their skull. These predominantly carnivorous creatures lived both on land and in semi-aquatic environments. Some bipedal varieties walked upright while other quadruped types walked agilely on all fours. Archosaurs later develop into the first dinosaurs, getting their start in the late Triassic. Archosaurs are the ancient ancestors of our modern day birds and crocodiles. It’s important to recognize that throughout the Triassic, these creatures are constantly adapting, evolving and diversifying into new and fantastic life-forms.  The first small shrew-like mammals appeared toward the end of the Triassic.  These tiny creatures occupied a tiny niche in the hostile continents of the Triassic that would eventually allow them to populate the Earth through to present day. Throughout the Triassic, mammals remain relatively insignificant until the end the Mesozoic in the Cretaceous.

Plant Life

During the Triassic, forestation of conifers, ginkgos, cycads, and bennettitaleans dominated the lands. The northern hemisphere was consisted primarily of conifers and seedless ferns, and the southern hemisphere eventually was taken over by seed ferns. The arid interior of Pangea sustained the gymnosperms, like the evergreen conifer, which relied on the wind for pollination and exposed its seeds to more easily reproduce. The conifers and cycads remain throughout the entire Triassic period. Coal-rich deposits suggest accumulations of plant matter and amphibian fossils indicate swamplands existed. Abundant ferns point to tropical forests and as Pangea separated later in the Mesozoic, the vegetation likely thinned out. Some smaller lycopod varieties survived from the Permian, but their spores didn't thrive in the dry Triassic climate. The ferns, however, flourished and subjugated the Triassic landscape.

Climate

The unique climate of the Triassic Period resulted from a combination of the conditions and processes happening at that time. The supercontinent Pangea remained centered on the equator, extending from the north to the south pole. One massive ocean, Panthallasa, encompasses the globe and surrounds Pangea. Volcanism in the sea and across Earth's continental crust begins the slow process of unzipping the supercontinent as fault zones create rift valleys across the lands.

Together, these conditions created a warm and dry climate. Pangea sat relatively high above sea level and because the continent was so massive, the interior became increasingly dry and arid because of its increased distance from the sea. Volcanism released enormous volumes of carbon dioxide into the air, further warming the atmosphere, melting any remaining polar ice and ultimately causing global sea levels to rise. The supercontinents vast size disrupts ocean water circulation which further contributes to global warming conditions. These processes took place over tens of millions of years and Triassic Earth continued on a steady path of global warming throughout its history.

Extinction Events

Two Extinction Events and their impact on life on earth.

The Triassic-Jurassic extinction occurred 205 million years ago and this event is believed to be second greatest mass extinction in Earth’s history, second only to the Permian-Triassic extinction. It is estimated to have wiped out 50% of all species, greatly affecting the larger species including many therapsids, all large crurotarsans and most large amphibians, which until then, dominated the terrain. The fossil evidence indicates this extinction occurred in a geologic instant of about 10,000 years. Its cause is unknown, but the usual suspects have been suggested, including extra-terrestrial impact, volcanic traps or massive sustained volcanic eruptions over the course of a million years, eventually resulting in catastrophic atmospheric changes. Unfortunately, enough concrete evidence just doesn’t exist at this time to allow scientists to achieve general consensus.
 * Triassic–Jurassic Extinction 205 million years ago

The beginnings of the Triassic period and the Mesozoic era are marked by the end of an extinction event that destroyed over 90% of marine and 70% of land species. This Permian-Triassic extinction is notably referred to as "the Great Dying" because the desolation to life far exceeds that inferred by any of Earth’s other extinction events. It is believed that a chain of related events brought about the Great Dying. At that time, Earth’s land masses sat high above sea level and were concentrated into one supercontinent, Pangea. This supercontinent would have significantly influenced Earth’s processes and Earth’s atmosphere. The interior of early Pangea would likely have been arid, dry and possibly cold. Earth’s poles were extremely cold and the huge continent would have prevented equatorial circulation, significantly restricting marine fauna habitation, not to mention the reduction of coastline habitat that Pangea now offered to marine life. As the shallow epicontinental seas receded from the outer parts of the continental mass, shallow warm sea environments were lost, along with the diverse life forms they supported. Dynamic volcanic outgassing activity likely disrupted atmospheric conditions plunging Earth on a path of global warming, melting Earth's ice and releasing frozen methane gas into the atmosphere. Finally, isotopic evidence found in Permian-Triassic sediments supports the possibility of an extraterrestrial impact that would have resulted in tsunamis, volcanism and further atmospheric changes. This extinction event was significant to the Triassic period because it left Earth full of rich and vacant environments hungry to be filled with the new and diversified life forms known to have developed during the Triassic.
 * Permian-Triassic Extinction 251 million years ago

title: Phanerozoic/Triassic period


title: Literature/1968


title: Category:1968


title: Literature/1958


title: Category:1958


text:
Phanerozoic/Cambrian period

Kenneth Sobieski This page is under construction

General
The Cambrian Period, named for a sparsly fossiliferous sequence of rocks in Wales(Latin name: Cambria) occurred between 542 Million Years Ago (MYA) and 488 MYA. The system was named by Adam Sedgwick, a Geology professor at Cambridge University, in the 1830s.

Climate
The Cambrian world was bracketed between two ice ages, one during the late Late Proterozoic and the other during the Ordovician. During these ice ages, the decrease in global temperature led to mass extinctions. Cooler conditions eliminated many warm water species, and glaciation lowered global sea level. However, during the Cambrian there was no significant ice formation. None of the continents were located at the poles, and so land temperatures remained mild. In fact, global climate was probably warmer and more unifrom than it is today. With the beginning of the Cambrian at the retreat of Proterozoic ice, the sea level rose significantly. Lowland areas such as Baltica were flooded and much of the world was covered by epeiric seas. This event opened up new habitats where marine invertebrates, such as the trilobites, radiated and flourished.

Land Mass Configuration
Breakup of Rodinia into:
 * Gondwana (today's southern continents)
 * Laurentia (most of North America)
 * Siberia
 * Baltica

Intra-Cambrian Periods
Russian-Kazakhian Stages
 * Batyrbayan 		(491.5 - 488)
 * Aksayan 		(493 - 491.5)
 * Sakian 			(494.5 - 493)
 * Ayusokkanian 		(501 - 494.5)
 * Mayan 			(502-501)
 * Amgan 			(513 - 502)
 * Toyonian		(518.5 - 513)
 * Botomian 		(524 - 518.5)
 * Atdabanian 		(530 - 524)
 * Tommotian 		(534 - 530)
 * Nemakit-Daldynian 	(542 - 534)

Regional Stages
 * Dolgellian 		(492.5 - 488)
 * Festiniogian 		(496.8 - 492.5)
 * Maentwrogian 		(501 - 496.8)
 * Lenian			(524 - 513)

North American Stages
 * Sunwaptan 		(493MYA To 491MYA)
 * Steptoan		(494.5MYA To 493MYA)
 * Marjuman 		(504 To 494.5 Ma)
 * Delamaran 		(512MYA To 504MYA)
 * Dyeran 			(524.5MYA To 512MYA)
 * Montezuman 		(529.5MYA To 524.5MYA)

Australian Stages
 * Payntonian  		(491 - 488)
 * Idamean 		(497 - 494)
 * Mindyallan 		(501-497)
 * Boomerangian 		(504-501)
 * Undillian 		(506-504)
 * Florian 		(508-506)
 * Templetonian 		(510-508)
 * Ordian     		(520 - 510)

Chinese Stages
 * Zhungxian      	(505 - 501)
 * Zuzhuangian 		(509-505)
 * Maozhangian 		(513 - 509)
 * Longwangmioan 		(518-513)
 * Changlangpuan 		(523-518)
 * Qungzusian     	(532 - 523)
 * Meishuchuan 		(542 +/- 0.3 To 532 Ma)

Stratigraphy
On of the most famous stratigraphic formations is the Burgess Shale in what is now British Columbia's Yoho National Park, in an area of the Canadian Rocky Mountains known as the Burgess Pass.

Flora and Fauna: The Cambrian Explosion
With the breakup of Rodinia, and the melting of glaciers, the world gained much more warm, shallow, oxygen-rich coastlines which life could exploit. The favorable climate lead to what is popularly known as "The Cambrian Explosion." The Cambrian Explosion seems, in fact, to have been an "arms race" between the predators and the prey, which may also help to explain the diversity. Land Animals Sea Animals Land Plants Sea Plants
 * NONE
 * Invertibrates
 * Aysheaia pedunculata
 * Hallucigenia
 * Anomalocaris
 * Opabinia
 * Wiwaxia
 * Arthropods
 * trilobites
 * most abundant and diverse animals of Cambrian time
 * long antennae, compound eyes, many jointed legs, and a hard exoskeleton
 * ranged from a few millimeters (1 mm = 0.25 inches) to 45 centimeters (18 inches).
 * archaeocyathids
 * hyoliths
 * monoplacophorans
 * inarticulate brachiopods
 * primitive echinoderms
 * articulate brachiopods
 * stony and lacy bryozoans
 * stromatoporoids
 * cephalopods
 * crinoids and blastoids
 * starfish
 * graptolites
 * bivalves
 * gastropods
 * vertebrates
 * echinoids
 * crustaceans
 * gymnolaemate bryozoans
 * Vertibrates
 * Haikouichthys
 * NONE
 * Single-Celled
 * Algaes
 * Fungi (molds, mushrooms, and yeasts)
 * lichens (cooperation between fungi and algae to form a single-organism partnership)
 * mosses
 * Multi-Celled
 * None

title: Phanerozoic/Cambrian period


title: Category:1956


text:
Phanerozoic/Cretaceous period

The Cretaceous period is the third and final period in the Mesozoic Era. It began 145.5 million years ago after the Jurassic Period and ended 65.5 million years ago, before the  Paleogene Period of the Cenozoic Era. The word Cretaceous comes from Latin word for "chalk bearing" which is named for the white chalk cliffs of Dover England.

Epochs
The Cretaceous period is composed of the Early Cretaceous Epoch and the Late Cretaceous Epoch.

Early Cretaceous
145.5 MA to 99.6 MA

The Cretaceous period started where the Jurassic period left off. Before the start of the Jurassic period, Gondwana and Laurasia started breaking up. During the Early Cretaceous period, the two continents continued to break up.

In terms of life, ichthyosaurs started to decline in this Epoch. Ichthyosaurs, Greek for "fish lizard", were fish-like marine reptiles. They flourished in the Mesozoic Era. Where the ichthyosaurs started declining, psittacosaurs, spinosaurs and coelurosaurs started appearing.

Late Cretaceous
99.6 MA to 65.5 MA

The chalky sediments - as nowadays exposed in the white cliffs of Dover - formed during this Epoch.

At the end of the Cretaceous, the climate started to cool and a mass extinction occurred.

Climate
The climate was hot and humid during most of the Cretaceous Period. Fossils from the Cretaceous indicate the presence of tropical and subtropical plants in locations far away from the equator. Some plants existed in Antarctica and what is now Alaska. As the continents separated, water was able to circulate in the equatorial and mid-latitudes. The circulation of seawater distributed the warmth between the separating continents and around the globe. There were no glaciers, which contributed to a much higher sea level with shallow seas covering extensive parts of the continents.

The climate started to cool at the end of the Cretaceous Period.

Continental Configuration
During the Cretaceous Period the lithospheric plates started to transform into the present configuration. Laurasia and Gondwanaland continued to break up, widening the Atlantic Ocean. South America and Africa completely separated by the Late Cretaceous. Australia remained connected to Antarctica. India moved northward toward Asia.

There was continued mountain building in the western North American continent. The Sevier orogeny started during the Nevadan but continued to the Cretaceous. Associated with the Sevier orogeny were large bodies of Magma created from subduction. These bodies cooled to form the Siera Nevada Batholiths. Another mountain-building event was the Laramide Orogeny which resulted from the collision and subduction of the Farallon plate. The angle of subduction lowered, pushing mountain-building farther inland. The Rocky Mountains were created from the Laramide Orogeny. The Laramide Orogeny continued into the Paleogene Period.

Most recognizable continents were covered in a shallow sea. The North American continent developed a depression when the Farallon continent had previously collided with it. This was a foreland basin. The consequence was a shallow epicontinental sea along the middle of North America, stretching North and South. This area of water is known today as the Western Interior Seaway. There are facies that show a pattern of fining upward, followed by a coarsening upward, reflecting a transgression followed by a regression. In many parts of the world, due to the high sea level, marine limestones are abundant. Because of the high sea level, chalk deposited on the continents. One example of this is the Dover cliffs of southern England.

Fauna

Coccolithophores became abundant during the Late Cretaceous. Coccolithophores are single-celled marine plants that live in large numbers throughout the upper layers of the ocean. They are characterized by a plate or shell called coccoliths that are made of calcium carbonate. The accumulation of coccoliths in this period formed large deposits of chalk in places such as the White Cliffs of Dover. This chalk probably developed when the seas were high, and because of the amount of calcium carbonate in the coccoliths. The name "Cretaceous" comes from this deposition of chalk.

Fossils indicate the presence of many dinosaurs, fish, reptiles and amphibians. New species of pterosaurs developed. Pterosaurs were flying reptiles that had wings made of a membrane of skins stretched between fingers. Many predatory fish also emerged from this period, including teleost fish and sharks. Teleost fish were ray-finned fish that preyed on ammonoids. Reptiles were also marine predators. One example of this is the mosasaur, which were marine reptiles that were probably ancestors to modern snakes. Pleisosaurs also existed in this period and were long-necked marine reptiles with fins.

There was a decline in epifaunal sea-floor animals such as brachiopods and crinoids, probably because of an increase in predators in the Cretaceous seas. This brought about an increase in infaunal organisms (burrowing and swimming molluscs, echinoderms).

Flora
Flowering plants appeared, also known as angiosperms. They replaced gymnosperms and were spurned by the diversification of insects and bees.

Also in this period, conifers became dominant gymnosperms.

Extinction Event

At the end of the Cretaceous Period, a mass extinction event occured. Previously called the KT mass extinction, now known as the K-Pg mass extiction, this event was observed in the geologic record by a thin band - the K-Pg boundary. Many plants and animal species died out in this short amount of time. Non-flying dinosaur fossils are found below the K-Pg boundary, meaning many dinosaurs died out during this extinction. Also affected were ammonoids, marine reptiles (plesiosaurs and mosasaurs), rudists, belemnites and more. There was a drastic reductions in coccolithophores, planktonic foraminifera and radiolarians.

Several theories try to explain this event. The most prevalent theory being a single or multiple meteorites hitting the earth, and/or increased volcanic activity. Plants that depended on solar energy for photosynthesis died out. As a result, herbivores that depended on those plants died out and predators that depended on the herbivores died out. Some scientists suggest that a drastic change in climate or sea level caused the mass extinction.

title: Phanerozoic/Cretaceous period


title: Literature/1956


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Phanerozoic/Jurassic period

Overview of Jurassic Period
The Jurassic Period takes place after the Triassic Period and before the Cretaceous Period. This period is well known for the reign of the dinosaurs of its time and the global tropical landscape. The name Jurassic comes from the Jura Mountains located in Europe, near the border of France and Switzerland.

The Jurassic Period can be divided into three epochs:
 * Lias (200-180 million years ago)
 * Dogger (180-154 million years ago)
 * Malm (154-145 million years ago)

title: Phanerozoic/Jurassic period


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Phanerozoic/Mississippian Period

Rogelio Paz This page is under construction

The Mississippian Period takes place about 360 to 325 million years ago. A part of the Carboniferous Period, named so because of the unconformity and the rocks marking the unconformity are different then those in Europe and found mainly in the American state of Mississippi.

Rocks
In the most part, coal was largely deposited during this time period. The rocks of the Mississippian Period are mostly limestones. Much of this was deposited as limy mud in the shallow seas.

Mountain Building
The most notable occurrence of orogeny is the formation of the Appalachian Mountains, which were formed when continents of Laurasia crashed with Gondwanaland. What is now North America and Africa collided to form the southern Appalachian Mountains. The Appalachians formed due to three major orogenic events: the Taconic orogeny, Acadian orogeny, and Allegheny orogeny.

Animal Life
This time period saw a reduction of trilobites and an increase in marine life such as corals and mollusks. The large amounts of Mississippian limestone found demonstrate the great diversity in crinoids. Sharks, acanthodians, and sarcopterygian fishes were diverse, while placoderms began to die out.

Fish had evolved into tetrapods such as ichthyostegas, during the Devonian period. These animals, although still fish, were one step closer to becoming land animals through the development of "arms and feet" like appendages. This led to amphibians which were making the transition from oceans to land. Labyrinthodonts were mainly swamp based animals who fed on insects, fish, and even each other. These animals lived mainly on solid land, returning to the water only to reproduce. The oldest fossils of amniotic animals can be found dated back the the Late Mississippian. This marked a significant achievement in the evolution of animals as they could finally reproduce without the need to return to water. These animals were reptiles such as the Petrolacosaurus.

title: Phanerozoic/Mississippian Period