circle: radius

Radius

http://upload.wikimedia.org/wikipedia/commons/thumb/1/1d/CIRCLE_1.svg/220px-CIRCLE_1.svg.png

Circle illustration

In classical geometry, the radius of a circle or sphere is any line segment from its center to its perimeter the radius of a circle or sphere is the length of any such segment. By extension, the diameter is defined as twice the radius:^[1]

therefore $r = \frac{d}{2}.$

If the object does not have an obvious center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity.

For regular polygons, the radius is the same as its circumradius.^[2] The inradius of a regular polygon is also called apothem. In graph theory, the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.^[3]

The name comes from Latin radius, meaning "ray" but also the spoke of a chariot wheel.^[4] The plural of radius can be either radii or the conventional English plural radiuses.^[5]

The radius of the circle with perimeter (circumference) C is

$r = \frac{C}{2\pi}= \frac{C}{\tau}.$

Contents

3.1 Radius from side

4 Formulas for hypercubes

4.1 Radius from side

Radius from area

The radius of a circle with area A is

$r = \sqrt{\frac{A}{\pi}} = \sqrt{\frac{2A}{\tau}}.$

The radius is half the diameter.

Radius from three points

To compute the radius of a circle going through three points P₁, P₂, P₃, the following formula can be used:

$r=\frac{|P_1-P_3|}{2\sin\theta}$

where θ is the angle $\angle P_1 P_2 P_3.$

This formula uses the Sine Rule.

Formulas for regular polygons

These formulas assume a regular polygon with n sides.

Radius from side

The radius can be computed from the side s by:

$r = R_n\, s$ where $R_n = \frac{1}{2 \sin \frac{\pi}{n}} \quad\quad \begin{array}{r|ccr|c} n & R_n & & n & R_n\\ \hline 2 & 0.50000000 & & 10 & 1.6180340- \\ 3 & 0.5773503- & & 11 & 1.7747328- \\ 4 & 0.7071068- & & 12 & 1.9318517- \\ 5 & 0.8506508+ & & 13 & 2.0892907+ \\ 6 & 1.00000000 & & 14 & 2.2469796+ \\ 7 & 1.1523824+ & & 15 & 2.4048672- \\ 8 & 1.3065630- & & 16 & 2.5629154+ \\ 9 & 1.4619022+ & & 17 & 2.7210956- \end{array}$

Formulas for hypercubes

Radius from side

The radius of a d-dimensional hypercube with side s is

$r = \frac{s}{2}\sqrt{d}.$

Atomic radius

Types of radii

http://upload.wikimedia.org/wikipedia/commons/thumb/2/23/Helium_atom_QM.svg/180px-Helium_atom_QM.svg.png

Diagram of a helium atom, showing the electron probability density as shades of gray.

The atomic radius of a chemical element is a measure of the size of its atoms, usually the mean or typical distance from the nucleus to the boundary of the surrounding cloud of electrons. Since the boundary is not a well-defined physical entity, there are various non-equivalent definitions of atomic radius.

Depending on the definition, the term may apply only to isolated atoms, or also to atoms in condensed matter, covalently bound in molecules, or in ionized and excited states; and its value may be obtained through experimental measurements, or computed from theoretical models. Under some definitions, the value of the radius may depend on the atom's state and context.^[1]

The concept is difficult to define because the electrons do not have definite orbits, or sharply defined ranges. Rather, their positions must be described as probability distributions that taper off gradually as one moves away from the nucleus, without a sharp cutoff. Moreover, in condensed matter and molecules, the electron clouds of the atoms usually overlap to some extent, and some of the electrons may roam over a large region encompassing two or more atoms.

Despite these conceptual difficulties, under most definitions the radii of isolated neutral atoms range between 30 and 300 pm (trillionths of a meter), or between 0.3 and 3 angstroms. Therefore, the radius of an atom is more than 10,000 times the radius of its nucleus (1–10 fm),^[2] and less than 1/1000 of the wavelength of visible light (400–700 nm).

http://upload.wikimedia.org/wikipedia/commons/thumb/0/00/Ethanol-3D-vdW.png/180px-Ethanol-3D-vdW.png

The approximate shape of a molecule of ethanol, CH₃CH₂OH. Each atom is modeled by a sphere with the element's Van der Waals radius.

For many purposes, atoms can be modeled as spheres. This is only a crude approximation, but it can provide quantitative explanations and predictions for many phenomena, such as the density of liquids and solids, the diffusion of fluids through molecular sieves, the arrangement of atoms and ions in crystals, and the size and shape of molecules.^[^{citation needed}^]

Atomic radii vary in a predictable and explicable manner across the periodic table. For instance, the radii generally decrease along each period (row) of the table, from the alkali metals to the noble gases; and increase down each group (column). The radius increases sharply between the noble gas at the end of each period and the alkali metal at the beginning of the next period. These trends of the atomic radii (and of various other chemical and physical properties of the elements) can be explained by the electron shell theory of the atom; they provided important evidence for the development and confirmation of quantum theory.

Contents

History

In 1920, shortly after it had become possible to determine the sizes of atoms using X-ray crystallography, it was suggested that all atoms of the same element have the same radii.^[3] However, in 1923, when more crystal data had become available, it was found that the approximation of an atom as a sphere does not necessarily hold when comparing the same atom in different crystal structures.^[4]

Definitions

Widely used definitions of atomic radius include:

Van der Waals radius: in principle, half the minimum distance between the nuclei of two atoms of the element that are not bound to the same molecule.^[5]
Ionic radius: the nominal radius of the ions of an element in a specific ionization state, deduced from the spacing of atomic nuclei in crystalline salts that include that ion. In principle, the spacing between two adjacent oppositely charged ions (the length of the ionic bond between them) should equal the sum of their ionic radii.^[5]
Covalent radius: the nominal radius of the atoms of an element when covalently bound to other atoms, as deduced the separation between the atomic nuclei in molecules. In principle, the distance between two atoms that are bound to each other in a molecule (the length of that covalent bond) should equal the sum of their covalent radii.^[5]
Metallic radius: the nominal radius of atoms of an element when joined to other atoms by metallic bonds.^[^{citation needed}^]
Bohr radius: the radius of the lowest-energy electron orbit predicted by Bohr model of the atom (1913).^[6]^[7] It is only applicable to atoms and ions with a single electron, such as hydrogen, singly ionized helium, and positronium. Although the model itself is now obsolete, the Bohr radius for the hydrogen atom is still regarded as an important physical constant.

Empirically measured atomic radii

The following table shows empirically measured covalent radii for the elements, as published by J. C. Slater in 1964.^[8] The values are in picometers (pm), with an accuracy of about 5 pm. The shade of the box ranges from red to yellow as the radius increases; gray indicates lack of data.

Group (vertical)

Period (horizontal)

H
25

Li
145

Be
105

B
85

C
70

N
65

O
60

F
50

Na
180

Mg
150

Al
125

Si
110

P
100

S
100

Cl
100

K
220

Ca
180

Sc
160

Ti
140

V
135

Cr
140

Mn
140

Fe
140

Co
135

Ni
135

Cu
135

Zn
135

Ga
130

Ge
125

As
115

Se
115

Br
115

Rb
235

Sr
200

Y
180

Zr
155

Nb
145

Mo
145

Tc
135

Ru
130

Rh
135

Pd
140

Ag
160

Cd
155

In
155

Sn
145

Sb
145

Te
140

I
140

Cs
260

Ba
215

Hf
155

Ta
145

W
135

Re
135

Os
130

Ir
135

Pt
135

Au
135

Hg
150

Tl
190

Pb
180

Bi
160

Po
190

Ra
215

Uut

Uup

Uus

Uuo

Lanthanides

La
195

Ce
185

Pr
185

Nd
185

Pm
185

Sm
185

Eu
185

Gd
180

Tb
175

Dy
175

Ho
175

Er
175

Tm
175

Yb
175

Lu
175

Actinides

Ac
195

Th
180

Pa
180

U
175

Np
175

Pu
175

Am
175

Explanation of the general trends

http://upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Atomic_number_to_radius_graph.png/220px-Atomic_number_to_radius_graph.png

A graph comparing the calculated atomic radius of elements with atomic numbers 1-100. Accuracy of ±5 pm.

The way the atomic radius varies with increasing atomic number can be explained by the arrangement of electrons in shells of fixed capacity. The shells are generally filled in order of increasing radius, since the negatively charged electrons are attracted by the positively charged protons in the nucleus. As the atomic number increases along each row of the periodic table, the additional electrons go into the same outermost shell; whose radius gradually contracts, due to the increasing nuclear charge. In a noble gas, the outermost shell is completely filled; therefore, the additional electron of next alkali metal will go into the next outer shell, accounting for the sudden increase in the atomic radius.

The increasing nuclear charge is partly counterbalanced by the increasing number of electrons, a phenomenon that is known as shielding; which explains why the size of atoms usually increases down each column. However, there is one notable exception, known as the lanthanide contraction: the 5d block of elements are much smaller than one would expect, due to the shielding caused by the 4f electrons.

The following table summarizes the main phenomena that influence the atomic radius of an element:

factor	principle	increase with...	*tend to*	effect on radius
electron shells	quantum mechanics	principal and azimuthal quantum numbers	increase atomic radius	increases down each column
nuclear charge	attractive force acting on electrons by protons in nucleus	atomic number	decrease atomic radius	decreases along each period
shielding	repulsive force acting on outermost shell electrons by inner electrons	number of electron shells	increase atomic radius	reduces the effect of the 2nd factor

Lanthanide contraction

Main article: Lanthanide contraction

The electrons in the 4f-subshell, which is progressively filled from cerium (Z = 58) to lutetium (Z = 71), are not particularly effective at shielding the increasing nuclear charge from the sub-shells further out. The elements immediately following the lanthanides have atomic radii which are smaller than would be expected and which are almost identical to the atomic radii of the elements immediately above them.^[9] Hence hafnium has virtually the same atomic radius (and chemistry) as zirconium, and tantalum has an atomic radius similar to niobium, and so forth. The effect of the lanthanide contraction is noticeable up to platinum (Z = 78), after which it is masked by a relativistic effect known as the inert pair effect.

Due to lanthanide contraction, the 5 following observations can be drawn:

The size of Ln³⁺ ions regularly decreases with atomic number. According to Fajans' rules, decrease in size of Ln³⁺ ions increases the covalent character and decreases the basic character between Ln³⁺ and OH⁻ ions in Ln(OH)₃. Hence the order of size of Ln³⁺ is given:
La³⁺ > Ce³⁺ > ..., ... > Lu³⁺.
There is a regular decrease in their ionic radii.
There is a regular decrease in their tendency to act as a reducing agent, with increase in atomic number.
The second and third rows of d-block transition elements are quite close in properties.
Consequently, these elements occur together in natural minerals and are difficult to separate.

d-Block contraction

Main article: d-block contraction

The d-block contraction is less pronounced than the lanthanide contraction but arises from a similar cause. In this case, it is the poor shielding capacity of the 3d-electrons which affects the atomic radii and chemistries of the elements immediately following the first row of the transition metals, from gallium (Z = 31) to bromine (Z = 35).^[9]

Calculated atomic radii

The following table shows atomic radii computed from theoretical models, as published by Enrico Clementi and others in 1967.^[10] The values are in picometres (pm).

Group (vertical)

Period (horizontal)

H
53

He
31

Li
167

Be
112

B
87

C
67

N
56

O
48

F
42

Ne
38

Na
190

Mg
145

Al
118

Si
111

P
98

S
88

Cl
79

Ar
71

K
243

Ca
194

Sc
184

Ti
176

V
171

Cr
166

Mn
161

Fe
156

Co
152

Ni
149

Cu
145

Zn
142

Ga
136

Ge
125

As
114

Se
103

Br
94

Kr
88

Rb
265

Sr
219

Y
212

Zr
206

Nb
198

Mo
190

Tc
183

Ru
178

Rh
173

Pd
169

Ag
165

Cd
161

In
156

Sn
145

Sb
133

Te
123

I
115

Xe
108

Cs
298

Ba
253

Hf
208

Ta
200

W
193

Re
188

Os
185

Ir
180

Pt
177

Au
174

Hg
171

Tl
156

Pb
154

Bi
143

Po
135

Rn
120

Uut

Uup

Uus

Uuo

Lanthanides

Pr
247

Nd
206

Pm
205

Sm
238

Eu
231

Gd
233

Tb
225

Dy
228

Er
226

Tm
222

Yb
222

Lu
217

Actinides

Bend radius

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Bend radius

Bend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, sheet, cable or hose without kinking it, damaging it, or shortening its life. The smaller the bend radius, the greater is the material flexibility (as the radius of curvature decreases, the curvature increases). The diagram below illustrates a cable with a seven-centimeter bend radius.

The minimum bend radius is the radius below which an object such as a cable should not be bent.

Fiber optics

The minimum bend radius is of particular importance in the handling of fiber-optic cables, which are often used in telecommunications. The minimum bending radius will vary with different cable designs. The manufacturer should specify the minimum radius to which the cable may safely be bent during installation, and for the long term. The former is somewhat shorter than the latter. The minimum bend radius is in general also a function of tensile stresses, e.g., during installation, while being bent around a sheave while the fiber or cable is under tension. If no minimum bend radius is specified, one is usually safe in assuming a minimum long-term low-stress radius not less than 15 times the cable diameter.

Beside mechanical destruction, another reason why one should avoid excessive bending of fiber-optic cables is to minimize microbending and macrobending losses. Microbending causes light attenuation induced by deformation of the fiber while macrobending causes the leakage of light through the fiber cladding and this is more likely to happen where the fiber is excessively bent.

Bohr radius

Bohr radius
Symbol:	a₀
Named after:	Niels Bohr
Value in meters:	≈ 5.29×10⁻¹¹m
Value in picometers:	≈ 52.9 pm
Value in angstroms:	≈ 0.529 Å

The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom. The precise definition of the Bohr radius is:

$a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_{\mathrm{e}} e^2} = \frac{\hbar}{m_{\mathrm{e}}\,c\,\alpha}$

where:

$\varepsilon_0 \$ is the permittivity of free space

$\hbar \$ is the reduced Planck's constant

$m_{\mathrm{e}} \$ is the electron rest mass

$e \$ is the elementary charge

$c \$ is the speed of light in vacuum

$\alpha \$ is the fine structure constant.

Or, in Gaussian units the Bohr radius is simply

$a_0=\frac{\hbar^2}{m_e e^2}$

According to 2010 CODATA the Bohr radius has a value of 5.2917721092(17)×10⁻¹¹ m (i.e., approximately 53 pm or 0.53 angstroms).^[1]^[2]

In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. The model says that the electrons orbit only at certain distances from the nucleus, depending on their energy. In the simplest atom, hydrogen, a single electron orbits the nucleus and its smallest possible orbit, with lowest energy, has an orbital radius almost equal to the Bohr radius. (It is not exactly the Bohr radius due to the reduced mass effect. They differ by about 0.1%.)

Although the Bohr model is no longer in use, the Bohr radius remains very useful in atomic physics calculations, due in part to its simple relationship with other fundamental constants. (This is why it is defined using the true electron mass rather than the reduced mass, as mentioned above.) For example, it is the unit of length in atomic units.

According to the modern, quantum-mechanical understanding of the hydrogen atom, the average distance −its expectation value− between electron and proton is ≈1.5a₀,^[3] somewhat different than the value in the Bohr model (≈a₀), but certainly the same order of magnitude.

The Bohr radius of the electron is one of a trio of related units of length, the other two being the Compton wavelength of the electron $\lambda_{\mathrm{e}} \$ and the classical electron radius $r_{\mathrm{e}} \$ . The Bohr radius is built from the electron mass $m_{\mathrm{e}}$ , Planck's constant $\hbar \$ and the electron charge $e \$ . The Compton wavelength is built from $m_{\mathrm{e}} \$ , $\hbar \$ and the speed of light $c \$ . The classical electron radius is built from $m_{\mathrm{e}} \$ , $c \$ and $e \$ . Any one of these three lengths can be written in terms of any other using the fine structure constant $\alpha \$ :

$r_{\mathrm{e}} = \frac{\alpha \lambda_{\mathrm{e}}}{2\pi} = \alpha^2 a_0.$

The Compton wavelength is about 20 times smaller than the Bohr radius, and the classical electron radius is about 1000 times smaller than the Compton wavelength.

Contents

Reduced Bohr radius

The Bohr radius including the effect of reduced mass in the hydrogen atom can be given by the following equation:

$\ a_0^* \ = \frac{\lambda_{\mathrm{p}} + \lambda_{\mathrm{e}}}{2\pi\alpha},$

where

$\lambda_{\mathrm{p}} \$ is the Compton wavelength of the proton.

$\lambda_{\mathrm{e}} \$ is the Compton wavelength of the electron.

$\alpha \$ is the fine structure constant.

In the above equation, the effect of the reduced mass is achieved by using the increased Compton wavelength, which is just the Compton wavelengths of the electron and the proton added together.

Radius of gyration

Radius of gyration or gyradius is the name of several related measures of the size of an object, a surface, or an ensemble of points. It is calculated as the root mean square distance of the objects' parts from either its center of gravity or a given axis.

Contents

3.1 Derivation of identity

Applications in structural engineering

In structural engineering, the two-dimensional radius of gyration is used to describe the distribution of cross sectional area in a column around its centroidal axis. The radius of gyration is given by the following formula

$R_{\mathrm{g}}^{2} = \frac{I}{A},$

$R_{\mathrm{g}} = \sqrt{ \frac {I} {A} },$

where I is the second moment of area and A is the total cross-sectional area. The gyration radius is useful in estimating the stiffness of a column. However, if the principal moments of the two-dimensional gyration tensor are not equal, the column will tend to buckle around the axis with the smaller principal moment. For example, a column with an elliptical cross-section will tend to buckle in the direction of the smaller semiaxis.

It also can be referred to as the radial distance from a given axis at which the mass of a body could be concentrated without altering the rotational inertia of the body about that axis.

In engineering, where people deal with continuous bodies of matter, the radius of gyration is usually calculated as an integral.

Applications in mechanics

The radius of gyration (r) about a given axis can be computed in terms of the mass moment of inertia I around that axis, and the total mass m;

$r_{\mathrm{g}}^{2} = \frac{I}{m},$

$r_{\mathrm{g}} = \sqrt{ \frac{I}{m}},$

I is a scalar, and is not the moment of inertia tensor. ^[1]

Molecular applications

In polymer physics, the radius of gyration is used to describe the dimensions of a polymer chain. The radius of gyration of a particular molecule at a given time is defined as:

$R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2},$

where $\mathbf{r}_{\mathrm{mean}}$ is the mean position of the monomers. As detailed below, the radius of gyration is also proportional to the root mean square distance between the monomers:

$R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{2N^{2}} \sum_{i,j} \left( \mathbf{r}_{i} - \mathbf{r}_{j} \right)^{2}.$

As a third method, the radius of gyration can also be computed by summing the principal moments of the gyration tensor.

Since the chain conformations of a polymer sample are quasi infinite in number and constantly change over time, the "radius of gyration" discussed in polymer physics must usually be understood as a mean over all polymer molecules of the sample and over time. That is, the radius of gyration which is measured is an average over time or ensemble:

$R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N} \langle \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2} \rangle,$

where the angular brackets $\langle \ldots \rangle$ denote the ensemble average.

An entropically governed polymer chain (i.e. in so called theta conditions) follows a random walk in three dimensions. The radius of gyration for this case is given by

$R_{\mathrm{g}} = \frac{1}{ \sqrt 6\ } \ \sqrt N\ a.$

Note that although

represents the contour length of the polymer,

is strongly dependent of polymer stiffness and can vary over orders of magnitude.

is reduced accordingly.

One reason that the radius of gyration is an interesting property is that it can be determined experimentally with static light scattering as well as with small angle neutron- and x-ray scattering. This allows theoretical polymer physicists to check their models against reality. The hydrodynamic radius is numerically similar, and can be measured with Dynamic Light Scattering (DLS).

Derivation of identity

To show that the two definitions of $R_{\mathrm{g}}^{2}$ are identical, we first multiply out the summand in the first definition:

$R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ \frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} - \mathbf{r}_{\mathrm{mean}} \right)^{2} = \frac{1}{N} \sum_{k=1}^{N} \left[ \mathbf{r}_{k} \cdot \mathbf{r}_{k} + \mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}} - 2 \mathbf{r}_{k} \cdot \mathbf{r}_{\mathrm{mean}} \right].$

Carrying out the summation over the last two terms and using the definition of $\mathbf{r}_{\mathrm{mean}}$ gives the formula

$R_{\mathrm{g}}^{2} \ \stackrel{\mathrm{def}}{=}\ -\mathbf{r}_{\mathrm{mean}} \cdot \mathbf{r}_{\mathrm{mean}} + \frac{1}{N} \sum_{k=1}^{N} \left( \mathbf{r}_{k} \cdot \mathbf{r}_{k} \right).$

Filling radius

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In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating Systolic geometry in its modern form.

The filling radius of a simple loop C in the plane is defined as the largest radius, R>0, of a circle that fits inside C:

$\mathrm{FillRad}(C\subset \mathbb{R}^2) = R.$

Contents

Dual definition via neighborhoods

There is a kind of a dual point of view that allows one to generalize this notion in an extremely fruitful way, as shown by Gromov. Namely, we consider the $\epsilon$ -neighborhoods of the loop C, denoted

$U_\epsilon C \subset \mathbb{R}^2.$

As $\epsilon>0$ increases, the $\epsilon$ -neighborhood $U_\epsilon C$ swallows up more and more of the interior of the loop. The last point to be swallowed up is precisely the center of a largest inscribed circle. Therefore we can reformulate the above definition by defining $\mathrm{FillRad}(C\subset \mathbb{R}^2)$ to be the infimum of $\epsilon > 0$ such that the loop C contracts to a point in $U_\epsilon C$ .

Given a compact manifold X imbedded in, say, Euclidean space E, we could define the filling radius relative to the imbedding, by minimizing the size of the neighborhood $U_\epsilon X\subset E$ in which X could be homotoped to something smaller dimensional, e.g., to a lower dimensional polyhedron. Technically it is more convenient to work with a homological definition.

Homological definition

Denote by A the coefficient ring $\mathbb{Z}$ or $\mathbb{Z}_2$ , depending on whether or not X is orientable. Then the fundamental class, denoted [X], of a compact n-dimensional manifold X, is a generator of the homology group $H_n(X;A)\simeq A$ , and we set

$\mathrm{FillRad}(X\subset E) = \inf \left\{ \epsilon > 0 \left| \;\iota_\epsilon([X])=0\in H_n(U_\epsilon X) \right. \right\},$

where $\iota_\epsilon$ is the inclusion homomorphism.

To define an absolute filling radius in a situation where X is equipped with a Riemannian metric g, Gromov proceeds as follows. One exploits an imbedding due to Kazimierz Kuratowski (the first name is sometimes spelled with a "C"). One imbeds X in the Banach space $L^\infty(X)$ of bounded Borel functions on X, equipped with the sup norm $\|\;\|$ . Namely, we map a point $x\in X$ to the function $f_x\in L^\infty(X)$ defined by the formula

for all $y\in X$ , where d is the distance function defined by the metric. By the triangle inequality we have $d(x,y) = \| f_x - f_y \|,$ and therefore the imbedding is strongly isometric, in the precise sense that internal distance and ambient distance coincide. Such a strongly isometric imbedding is impossible if the ambient space is a Hilbert space, even when X is the Riemannian circle (the distance between opposite points must be π, not 2!). We then set $E= L^\infty(X)$ in the formula above, and define

$\mathrm{FillRad}(X)=\mathrm{FillRad} \left( X\subset L^{\infty}(X) \right).$

Relation to diameter and systole

The exact value of the filling radius is known in very few cases. A general inequality relating the filling radius and the Riemannian diameter of X was proved in (Katz, 1983): the filling radius is at most a third of the diameter. In some cases, this yields the precise value of the filling radius. Thus, the filling radius of the Riemannian circle of length 2π, i.e. the unit circle with the induced Riemannian distance function, equals π/3, i.e. a sixth of its length. This follows by combing the diameter upper bound mentioned above with Gromov's lower bound in terms of the systole (Gromov, 1983). More generally, the filling radius of real projective space with a metric of constant curvature is a third of its Riemannian diameter, see (Katz, 1983). Equivalently, the filling radius is a sixth of the systole in these cases. The precise value is also known for the n-spheres (Katz, 1983).

The filling radius is linearly related to the systole of an essential manifold M. Namely, the systole of such an M is at most six times its filling radius, see (Gromov, 1983). The inequality is optimal in the sense that the boundary case of equality is attained by the real projective spaces as above

Schwarzschild radius

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http://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/MassProperties.svg/320px-MassProperties.svg.png

The relation between properties of mass and their associated physical constants. Every massive object is believed to exhibit all five properties. However, due to extremely large or extremely small constants, it is generally impossible to verify more than two or three properties for any object.

The Schwarzschild radius (r_s) represents the ability of mass to cause curvature in space and time.
The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.
Inertial mass (m) represents the Newtonian response of mass to forces.
Rest energy (E₀) represents the ability of mass to be converted into other forms of energy.
The Compton wavelength (λ) represents the quantum response of mass to local geometry.

The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is the radius of a sphere such that, if all the mass of an object is compressed within that sphere, the escape speed from the surface of the sphere would equal the speed of light. An example of an object smaller than its Schwarzschild radius is a black hole. Once a stellar remnant collapses within this radius, light cannot escape and the object is no longer visible.^[1] It is a characteristic radius associated with every quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild who calculated this exact solution for the theory of general relativity in 1915.

Contents

5 Other uses for the Schwarzschild radius

History

In 1915, Karl Schwarzschild obtained an exact solution^[2]^[3] to Einstein's field equations for the gravitational field outside a non-rotating, spherically symmetric body (see Schwarzschild metric). Using the definition $M=\frac {Gm} {c^2}$ , the solution contained a term of the form $\frac {1} {2M-r}$ ; where the value of

making this term singular has come to be known as the Schwarzschild radius. The physical significance of this singularity, and whether this singularity could ever occur in nature, was debated for many decades; a general acceptance of the possibility of a black hole did not occur until the second half of the 20th century.

Parameters

The Schwarzschild radius of an object is proportional to the mass. Accordingly, the Sun has a Schwarzschild radius of approximately 3.0 km (1.9 mi) while the Earth's is only about 9.0 mm, the size of a peanut. The observable universe's mass has a Schwarzschild radius of approximately 10 billion light years.

	(m)	(g/cm³)
Universe	4.46×10^25[^{citation needed}^] (~10B ly)	8×10^−29[^{citation needed}^] (9.9×10⁻³⁰^[4])
Milky Way	2.08×10¹⁵ (~0.2 ly)	3.72×10⁻⁸
Sun	2.95×10³	1.84×10¹⁶
Earth	8.87×10⁻³	2.04×10²⁷

An object whose radius is smaller than its Schwarzschild radius is called a black hole. The surface at the Schwarzschild radius acts as an event horizon in a non-rotating body (a rotating black hole operates slightly differently). Neither light nor particles can escape through this surface from the region inside, hence the name "black hole". The Schwarzschild radius of the (currently hypothesized) supermassive black hole at our Galactic Center would be approximately 13.3 million kilometres.

Formula for the Schwarzschild radius

The Schwarzschild radius is proportional to the mass with a proportionality constant involving the gravitational constant and the speed of light:

$r_\mathrm{s} = \frac{2Gm}{c^2},$

where:

$r_s\!$ is the Schwarzschild radius;

$G\!$ is the gravitational constant;

$m\!$ is the mass of the object;

$c\!$ is the speed of light in vacuum.

The proportionality constant, 2G/c², is approximately 1.48×10⁻²⁷ m/kg, or 2.95 km/solar mass.

An object of any density can be large enough to fall within its own Schwarzschild radius,

$V_s \propto \rho^{-3/2},$ ^[^{citation needed}^]

where:

$V_s\!$ is the volume of the object;

$\rho\!$ is its density.

Classification by Schwarzschild radius

Supermassive black hole

Assuming constant density, the Schwarzschild radius of a body is proportional to its mass, but the radius is proportional to the cube root of the volume and hence the mass. Therefore, as one accumulates matter at normal density (10³ kg/m³, for example, the density of water), its Schwarzschild radius increases more quickly than its radius. At around 150 million(1.5 x 10⁸) times the mass of the Sun, such an accumulation will fall inside its own Schwarzschild radius and thus it would be a supermassive black hole of 150 million solar masses. (Supermassive black holes up to 18 billion(1.8 x 10¹⁰) solar masses have been observed.^[5]) The supermassive black hole in the center of our galaxy (4.5 ± 0.4 million solar masses) constitutes observationally the most convincing evidence for the existence of black holes in general. It is thought that large black holes like these don't form directly in one collapse of a cluster of stars. Instead they may start as a stellar-sized black hole and grow larger by the accretion of matter and other black holes. An empirical correlation between the size of supermassive black holes and the stellar velocity dispersion $\sigma$ of a galaxy bulge ^[6] is called the M-sigma relation.

Stellar black hole

If one accumulates matter at nuclear density (the density of the nucleus of an atom, about 10¹⁸ kg/m³; neutron stars also reach this density), such an accumulation would fall within its own Schwarzschild radius at about 3 solar masses and thus would be a stellar black hole.

Primordial black hole

Conversely, a small mass has an extremely small Schwarzschild radius. A mass similar to Mount Everest has a Schwarzschild radius smaller than a nanometre. Its average density at that size would be so high that no known mechanism could form such extremely compact objects. Such black holes might possibly be formed in an early stage of the evolution of the universe, just after the Big Bang, when densities were extremely high. Therefore these hypothetical miniature black holes are called primordial black holes.

Other uses for the Schwarzschild radius

The Schwarzschild radius in gravitational time dilation

Gravitational time dilation near a large, slowly rotating, nearly spherical body, such as the earth or sun can be reasonably approximated using the Schwarzschild radius as follows:

$\frac{t_r}{t} = \sqrt{1 - \frac{r_s}{r}}$

where:

$t_r\!$ is the elapsed time for an observer at radial coordinate "r" within the gravitational field;

$t\!$ is the elapsed time for an observer distant from the massive object (and therefore outside of the gravitational field);

$r\!$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object);

$r_s\!$ is the Schwarzschild radius.

The results of the Pound, Rebka experiment in 1959 were found to be consistent with predictions made by general relativity. By measuring Earth’s gravitational time dilation, this experiment indirectly measured Earth’s Schwarzschild radius.

The Schwarzschild radius in Newtonian gravitational fields

The Newtonian gravitational field near a large, slowly rotating, nearly spherical body can be reasonably approximated using the Schwarzschild radius as follows:

$\frac{g}{r_s} \left( \frac{r}{c} \right)^2 = \frac{1}{2}$

where:

$g\!$ is the gravitational acceleration at radial coordinate "r";

$r_s\!$ is the Schwarzschild radius of the gravitating central body;

$r\!$ is the radial coordinate;

$c\!$ is the speed of light in vacuum.

On the surface of the Earth:

$\frac{9.80665 m/s^2}{8.870056 mm} \left( \frac{6375416 m}{299792458 m/s} \right)^2 = \left(1105.59 s^{-2} \right) \left(0.0212661 s\right)^2 = \frac{1}{2}.$

The Schwarzschild radius in Keplerian orbits

For all circular orbits around a given central body:

$\frac{r}{r_s} \left( \frac{v}{c} \right)^2 = \frac{1}{2}$

where:

$r\!$ is the orbit radius;

$r_s\!$ is the Schwarzschild radius of the gravitating central body;

$v\!$ is the orbital speed;

$c\!$ is the speed of light in vacuum.

This equality can be generalized to elliptic orbits as follows:

$\frac{a}{r_s} \left( \frac{2 \pi a}{c T} \right)^2 = \frac{1}{2}$

where:

$a\!$ is the semi-major axis;

$T\!$ is the orbital period.

For the Earth orbiting the Sun:

$\frac{1 \,\mathrm{AU}}{2953.25\,\mathrm m} \left( \frac{2 \pi \,\mathrm{AU}}{\mathrm{light\,year}} \right)^2 = \left(50 655 379.7 \right) \left(9.8714403 \times 10^{-9} \right)= \frac{1}{2}.$

Relativistic circular orbits and the photon sphere

The Keplerian equation for circular orbits can be generalized to the relativistic equation for circular orbits by accounting for time dilation in the velocity term:

$\frac{r}{r_s} \left( \frac{v}{c} \sqrt{1 - \frac{r_s}{r}} \right)^2 = \frac{1}{2}$

$\frac{r}{r_s} \left( \frac{v}{c} \right)^2 \left(1 - \frac{r_s}{r} \right) = \frac{1}{2}$

$\left( \frac{v}{c} \right)^2 \left( \frac{r}{r_s} - 1 \right) = \frac{1}{2}.$

This final equation indicates that an object orbiting at the speed of light would have an orbital radius of 1.5 times the Schwarzschild radius. This is a special orbit known as the photon sphere.

Radius of convergence

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In mathematics, the radius of convergence of a power series is the radius of the largest disk in which the series converges. It is either a non-negative real number or ∞. When it is positive, the power series converges absolutely and uniformly on compact sets inside the open disk of radius equal to the radius of convergence, and it is the Taylor series of the analytic function to which it converges.

Contents

3 Radius of convergence in complex analysis

Definition

For a power series ƒ defined as:

$f(z) = \sum_{n=0}^\infty c_n (z-a)^n,$

where

a is a complex constant, the center of the disk of convergence,

c_n is the n^th complex coefficient, and

z is a complex variable.

The radius of convergence r is a nonnegative real number or ∞ such that the series converges if

$|z-a| < r\,$

and diverges if

$|z-a| > r.\,$

In other words, the series converges if z is close enough to the center and diverges if it is too far away. The radius of convergence specifies how close is close enough. On the boundary, that is, where |z − a| = r, the behavior of the power series may be complicated, and the series may converge for some values of z and diverge for others. The radius of convergence is infinite if the series converges for all complex numbers z.

Finding the radius of convergence

Two cases arise. The first case is theoretical: when you know all the coefficients

then you take certain limits and find the precise radius of convergence. The second case is practical: when you construct a power series solution of a difficult problems you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms. In this second case, extrapolating a plot estimates the radius of convergence.

Theoretical radius

The radius of convergence can be found by applying the root test to the terms of the series. The root test uses the number

$C = \limsup_{n\rightarrow\infty}\sqrt[n]{|c_n(z-a)^n|} = \limsup_{n\rightarrow\infty}\sqrt[n]{|c_n|}|z-a|$

"lim sup" denotes the limit superior. The root test states that the series converges if C < 1 and diverges if C > 1. It follows that the power series converges if the distance from z to the center a is less than

$r = \frac{1}{\limsup_{n\rightarrow\infty}\sqrt[n]{|c_n|}}$

and diverges if the distance exceeds that number; this statement is the Cauchy–Hadamard theorem. Note that r = 1/0 is interpreted as an infinite radius, meaning that ƒ is an entire function.

The limit involved in the ratio test is usually easier to compute, and when that limit exists, it shows that the radius of convergence is finite.

$r = \lim_{n\rightarrow\infty} \left| \frac{c_n}{c_{n+1}} \right|.$

This is shown as follows. The ratio test says the series converges if

$\lim_{n\to\infty} \frac{|c_{n+1}(z-a)^{n+1}|}{|c_n(z-a)^n|} < 1.$

That is equivalent to

$|z - a| < \frac{1}{\lim_{n\to\infty} \frac{|c_{n+1}|}{|c_n|}} = \lim_{n\to\infty} \left|\frac{c_n}{c_{n+1}}\right|.$

Practical estimation of radius

http://upload.wikimedia.org/wikipedia/commons/thumb/9/96/Domb_Sykes_plot_Hinch.svg/400px-Domb_Sykes_plot_Hinch.svg.png

Domb–Sykes plot of the function $f(\varepsilon)=\varepsilon\,(1+\varepsilon^3)\,/\,\sqrt{(1+2\varepsilon)}.$ ^[1] On the left (a) is a straightforward plot of the ratio of the power-series coefficients $c_{n-1}/c_{n}$ as a function of index

; on the right, (b) is the Domb–Sykes plot of $c_{n}/c_{n-1}$ as a function of

. The solid green line is the straight-line asymptote in the Domb–Sykes plot, and intercepts the vertical axis at −2 and has a slope +1. So there is a singularity at $\varepsilon=-\tfrac12$ and the radius of convergence is $r=\tfrac12.$

Suppose you only know a finite number of coefficients

, say ten to a hundred. Typically, as

increases, these coefficients settle into a regular behavior determined by the nearest radius-limiting singularity.

When the behavior of the coefficients is one of constant sign or alternating sign, Domb and Sykes^[2] proposed plotting $c_n/c_{n-1}$ against

, fitting a straight line extrapolation, and taking the intercept of this line as an estimate the reciprocal

of the radius of convergence. Negative

means the convergence-limiting singularity is on the negative axis. This procedure is called a Domb–Sykes plot.

When the coefficients settle into having a periodic pattern of signs then use a test proposed by Mercer and Roberts.^[3] Compute

from $b_n^2=(c_{n+1}c_{n-1}-c_n^2)/(c_nc_{n-2}-c_{n-1}^2)$ and plot

versus

. Extrapolate to

to again estimate the reciprocal

of the radius of convergence.

You may also estimate two subsidiary quantities. Estimate the exponent

of the convergence limiting singularity because the slope of the straight line extrapolation is

. Estimate the angle $\theta$ , from the real axis, of the convergence limiting singularities by plotting $(c_{n-1}b_n/c_n+c_{n+1}/c_n/b_n)/2$ versus

. Then extrapolating to

estimates $\cos\theta$ .

Radius of convergence in complex analysis

A power series with a positive radius of convergence can be made into a holomorphic function by taking its argument to be a complex variable. The radius of convergence can be characterized by the following theorem:

The radius of convergence of a power series f centered on a point a is equal to the distance from a to the nearest point where f cannot be defined in a way that makes it holomorphic.

The set of all points whose distance to a is strictly less than the radius of convergence is called the disk of convergence.

http://upload.wikimedia.org/wikipedia/commons/thumb/0/0f/TaylorComplexConv.png/300px-TaylorComplexConv.png

A graph of the functions explained in the text: Approximations in blue, circle of convergence in white

The nearest point means the nearest point in the complex plane, not necessarily on the real line, even if the center and all coefficients are real. For example, the function

$f(z)=\frac{1}{1+z^2}$

has no singularities on the real line, since

has no real roots. Its Taylor series about 0 is given by

$\sum_{n=0}^\infty (-1)^n z^{2n}.$

The root test shows that its radius of convergence is 1. In accordance with this, the function ƒ(z) has singularities at ±i, which are at a distance 1 from 0.

For a proof of this theorem, see analyticity of holomorphic functions.

A simple example

The arctangent function of trigonometry can be expanded in a power series familiar to calculus students:

$\arctan(z)=z-\frac{z^3}{3}+\frac{z^5}{5}-\frac{z^7}{7}+\cdots .$

It is easy to apply the root test in this case to find that the radius of convergence is 1.

A more complicated example

Consider this power series:

$\frac{z}{e^z-1}=\sum_{n=0}^\infty \frac{B_n}{n!} z^n$

where the rational numbers B_n are the Bernoulli numbers. It may be cumbersome to try to apply the ratio test to find the radius of convergence of this series. But the theorem of complex analysis stated above quickly solves the problem. At z = 0, there is in effect no singularity since the singularity is removable. The only non-removable singularities are therefore located at the other points where the denominator is zero. We solve

$e^z-1=0\,$

by recalling that if z = x + iy and e^iy = cos(y) + i sin(y) then

$e^z = e^x e^{iy} = e^x(\cos(y)+i\sin(y)),\,$

and then take x and y to be real. Since y is real, the absolute value of cos(y) + i sin(y) is necessarily 1. Therefore, the absolute value of e^z can be 1 only if e^x is 1; since x is real, that happens only if x = 0. Therefore z is pure imaginary and cos(y) + i sin(y) = 1. Since y is real, that happens only if cos(y) = 1 and sin(y) = 0, so that y is an integral multiple of 2π. Consequently the singular points of this function occur at

z = a nonzero integer multiple of 2πi.

The singularities nearest 0, which is the center of the power series expansion, are at ±2πi. The distance from the center to either of those points is 2π, so the radius of convergence is 2π.

Convergence on the boundary

If the power series is expanded around the point a and the radius of convergence is r, then the set of all points z such that |z − a| = r is a circle called the boundary of the disk of convergence. A power series may diverge at every point on the boundary, or diverge on some points and converge at other points, or converge at all the points on the boundary. Furthermore, even if the series converges everywhere on the boundary (even uniformly), it does not necessarily converge absolutely.

Example 1: The power series for the function ƒ(z) = 1/(1 − z), expanded around z = 0, which is simply

$\sum_{n=0}^\infty z^n,$

has radius of convergence 1, and diverges at every point on the boundary.

Example 2: The power series for g(z) = −ln(1 − z), expanded around z = 0, which is

$\sum_{n=1}^\infty \frac{1}{n} z^n,$

has radius of convergence 1, and diverges for z = 1 but converges for all other points on the boundary. The function ƒ(z) of Example 1 is the derivative of g(z).

Example 3: The power series

$\sum_{n=1}^\infty \frac{1}{n^2} z^n$

has radius of convergence 1 and converges everywhere on the boundary absolutely. If h is the function represented by this series on the unit disk, then the derivative of h(z) is equal to g(z)/z with g of Example 2. It turns out that h(z) is the dilogarithm function.

Example 4: The power series

$\sum_{i=1}^\infty a_i z^i \text{ where } a_i = \frac{(-1)^{n-1}}{2^nn}\text{ for }n=\lfloor\log_2(i)\rfloor+1\text{, the unique integer with }2^{n-1}\le i < 2^n,$

has radius of convergence 1 and converges uniformly on the entire boundary {|z| = 1}, but does not converge absolutely on the boundary.^[4]

Comments on rate of convergence

If we expand the function

$f(x)=\sin x = \sum^{\infin}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\text{ for all } x$

around the point x = 0, we find out that the radius of convergence of this series is $\scriptstyle\infty$ meaning that this series converges for all complex numbers. However, in applications, one is often interested in the precision of a numerical answer. Both the number of terms and the value at which the series is to be evaluated affect the accuracy of the answer. For example, if we want to calculate ƒ(0.1) = sin(0.1) accurate up to five decimal places, we only need the first two terms of the series. However, if we want the same precision for x = 1, we must evaluate and sum the first five terms of the series. For ƒ(10), one requires the first 18 terms of the series, and for ƒ(100), we need to evaluate the first 141 terms.

So the fastest convergence of a power series expansion is at the center, and as one moves away from the center of convergence, the rate of convergence slows down until you reach the boundary (if it exists) and cross over, in which case the series will diverge.

A graphical example

Consider the function 1/(z² + 1).

This function has poles at z = ±i.

As seen in the first example, the radius of convergence of this function's series in powers of (z − 0) is 1, as the distance from 0 to each of those poles is 1.

Then the Taylor series of this function around z = 0 will only converge if |z| < 1, as depicted on the example on the right.

Abscissa of convergence of a Dirichlet series

An analogous concept is the abscissa of convergence of a Dirichlet series

$\sum_{n=1}^\infty {a_n \over n^s}.$

Such a series converges if the real part of s is greater than a particular number depending on the coefficients a_n: the abscissa of convergence

circle

Wednesday, January 23, 2013

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