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43-433: Ram air refers to the principle of using the airflow created by a moving object to increase ambient pressure, known as ram pressure . Often, the purpose of a ram air system is to increase an engine's power. The term "ram air" may also refer to: Ram pressure where ρ {\displaystyle \rho } is the density of the fluid; P ram {\displaystyle P_{\text{ram}}}

86-423: A galaxy cluster moving through a hot intracluster medium would experience a pressure of where P r {\displaystyle P_{r}} is the ram pressure, ρ e {\displaystyle \rho _{e}} the intracluster gas density, and v {\displaystyle v} the speed of the galaxy relative to the medium. This pressure can strip gas out of

129-525: A square matrix A j , the trace is the sum of the diagonal elements, hence the sum over a common index A i . The outer product of the column vector u by the row vector v j yields an m  ×  n matrix A : A i j = u i v j = ( u v ) i j {\displaystyle {A^{i}}_{j}=u^{i}v_{j}={(uv)^{i}}_{j}} Since i and j represent two different indices, there

172-401: A force blowing it apart over a nearly instantaneous span of time. In essence, the meteoroid is ripped apart by its own speed. This occurs when fine tendrils of superheated air force their way into cracks and faults in the leading face's surface. Once this high pressure plasma gains entry to the meteoroid's interior it exerts tremendous force on the body's internal structure. This occurs because

215-536: A more gradual few billion years. Recent radio observation of carbon monoxide (CO) emission from three galaxies ( NGC 4330 , NGC 4402 , and NGC 4522 ) in the Virgo cluster point to the molecular gas not being stripped but instead being compressed by the ram pressure. Increased Hα emission, a sign of star formation, corresponds to the compressed CO region, suggesting that star formation may be accelerated, at least temporarily, while ram pressure stripping of neutral hydrogen

258-413: A smaller space than formerly. Ram pressure and the very high temperatures it causes are the reasons few meteors make it all the way to the ground and most simply burn up or are ablated into tiny fragments . Larger or more solid meteorites may explode instead in a meteor airburst . The use of the term explosion is somewhat loose in this context, and can be confusing. This confusion is exacerbated by

301-692: A term. When dealing with covariant and contravariant vectors, where the position of an index indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; see § Superscripts and subscripts versus only subscripts below. In terms of covariance and contravariance of vectors , They transform contravariantly or covariantly, respectively, with respect to change of basis . In recognition of this fact,

344-442: A typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation . An index that is summed over is a summation index , in this case " i   ". It is also called a dummy index since any symbol can replace " i   " without changing the meaning of the expression (provided that it does not collide with other index symbols in

387-400: A volume V {\displaystyle V} bounded by a surface S {\displaystyle S} is and the momentum per second it carries into the body is equal to the ram pressure term. This discussion can be extended to 'drag' forces; if all matter incident upon a surface transfers all its momentum to the volume, this is equivalent (in terms of momentum transfer) to

430-609: Is a special case of matrix multiplication. The matrix product of two matrices A ij and B jk is: C i k = ( A B ) i k = ∑ j = 1 N A i j B j k {\displaystyle \mathbf {C} _{ik}=(\mathbf {A} \mathbf {B} )_{ik}=\sum _{j=1}^{N}A_{ij}B_{jk}} equivalent to C i k = A i j B j k {\displaystyle {C^{i}}_{k}={A^{i}}_{j}{B^{j}}_{k}} For

473-477: Is an attractive mechanism to explain not only the presence of isolated dwarf galaxies away from galaxy clusters with particularly low hydrogen abundance to stellar mass ratio, but also the compression of gas in the centre of a dwarf galaxy and the subsequent reignition of star formation . Meteoroids enter Earth's atmosphere from outer space traveling at hypersonic speeds of at least 11 km/s (7 mi/s) and often much faster. Despite moving through

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516-417: Is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see § Application below). Typically, ( x x x ) would be equivalent to the traditional ( x y z ) . In general relativity , a common convention is that In general, indices can range over any indexing set , including an infinite set . This should not be confused with

559-475: Is equivalent to using the product rule and the Kronecker delta δ i j {\displaystyle \delta _{ij}} . The first term in the brackets is the isotropic thermal pressure, and the second is the ram pressure. In this context, ram pressure is momentum transfer by advection (flow of matter carrying momentum across a surface into a body). The mass per unit second flowing into

602-472: Is equivalent to the equation v i = ∑ j ( a i b j x j ) {\textstyle v_{i}=\sum _{j}(a_{i}b_{j}x^{j})} . Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within

645-419: Is fluid velocity, ρ {\displaystyle \rho } the fluid density, and g → {\displaystyle {\vec {g}}} the gravitational acceleration. The Eulerian rate of change of momentum in direction i {\displaystyle i} at a point is thus (using Einstein notation ): Substituting the conservation of mass, expressed as this

688-449: Is ongoing. More recently, it has been shown that ram pressure can also lead to the removal of gas in isolated dwarf galaxies that plunge through the cosmic web (the so-called cosmic web stripping process). Although the typical overdensity within the cosmic web is significantly lower than that found in the environment of galaxy clusters , the high relative speed between a dwarf and the cosmic web renders ram pressure efficient. This

731-434: Is simplified by the convention to: y = x i e i {\displaystyle y=x^{i}e_{i}} The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors . That is, in this context x should be understood as the second component of x rather than the square of x (this can occasionally lead to ambiguity). The upper index position in x

774-408: Is that it represents the invariant quantities with a simple notation. In physics, a scalar is invariant under transformations of basis. In particular, a Lorentz scalar is invariant under a Lorentz transformation . The individual terms in the sum are not. When the basis is changed, the components of a vector change by a linear transformation described by a matrix. This led Einstein to propose

817-830: Is the Levi-Civita symbol . Since the basis is orthonormal, raising the index i {\displaystyle i} does not alter the value of ε i j k {\displaystyle \varepsilon _{ijk}} , when treated as a tensor. The product of a matrix A ij with a column vector v j is: u i = ( A v ) i = ∑ j = 1 N A i j v j {\displaystyle \mathbf {u} _{i}=(\mathbf {A} \mathbf {v} )_{i}=\sum _{j=1}^{N}A_{ij}v_{j}} equivalent to u i = A i j v j {\displaystyle u^{i}={A^{i}}_{j}v^{j}} This

860-450: Is the momentum flux per second in the i {\displaystyle i} direction through a surface with normal in the j {\displaystyle j} direction. u i , u j {\displaystyle u_{i},u_{j}} are the components of the fluid velocity in these directions. The total Cauchy stress tensor σ i j {\displaystyle \sigma _{ij}}

903-517: Is the sum of this ram pressure and the isotropic thermal pressure (in the absence of viscosity ). In the simple case when the relative velocity is normal to the surface, and momentum is fully transferred to the object, the ram pressure becomes The Eulerian form of the Cauchy momentum equation for a fluid is for isotropic pressure p {\displaystyle p} , where u → {\displaystyle {\vec {u}}}

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946-429: Is the vector and v are its components (not the i th covector v ), w is the covector and w i are its components. The basis vector elements e i {\displaystyle e_{i}} are each column vectors, and the covector basis elements e i {\displaystyle e^{i}} are each row covectors. (See also § Abstract description ; duality , below and

989-528: Is thought to have profound effects on the evolution of galaxies. As galaxies fall toward the center of a cluster, more and more of their gas is stripped out, including the cool, denser gas that is the source of continued star formation . Spiral galaxies that have fallen at least to the core of both the Virgo and Coma clusters have had their gas (neutral hydrogen) depleted in this way and simulations suggest that this process can happen relatively quickly, with 100% depletion occurring in 100 million years to

1032-729: The cross product of two vectors with respect to a positively oriented orthonormal basis, meaning that e 1 × e 2 = e 3 {\displaystyle \mathbf {e} _{1}\times \mathbf {e} _{2}=\mathbf {e} _{3}} , can be expressed as: u × v = ε j k i u j v k e i {\displaystyle \mathbf {u} \times \mathbf {v} =\varepsilon _{\,jk}^{i}u^{j}v^{k}\mathbf {e} _{i}} Here, ε j k i = ε i j k {\displaystyle \varepsilon _{\,jk}^{i}=\varepsilon _{ijk}}

1075-413: The examples ) In the presence of a non-degenerate form (an isomorphism V → V , for instance a Riemannian metric or Minkowski metric ), one can raise and lower indices . A basis gives such a form (via the dual basis ), hence when working on R with a Euclidean metric and a fixed orthonormal basis , one has the option to work with only subscripts. However, if one changes coordinates,

1118-461: The case of an orthonormal basis , we have u j = u j {\displaystyle u^{j}=u_{j}} , and the expression simplifies to: ⟨ u , v ⟩ = ∑ j u j v j = u j v j {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\sum _{j}u^{j}v^{j}=u_{j}v^{j}} In three dimensions,

1161-511: The compression-heated air. In other words, kinetic energy is converted into heated air via ram pressure, and that heated air is quickly moved away from object surface with minimal physical interaction, and hence minimal heating of the body. This was counter-intuitive at the time, when sharp, streamlined profiles were assumed to be better. This blunt-body concept was used in Apollo -era capsules. Einstein notation In mathematics , especially

1204-494: The convention that repeated indices imply the summation is to be done. As for covectors, they change by the inverse matrix . This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is. The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality . For example, V ⊗  V ,

1247-410: The explosion, and it causes the meteoroid to disintegrate with hypersonic velocity , a speed comparable to that of explosive detonation . Harry Julian Allen and Alfred J. Eggers of NACA used an insight about ram pressure to propose the blunt-body concept : a large, blunt body entering the atmosphere creates a boundary layer of compressed air which serves as a buffer between the body surface and

1290-1047: The following notation uses the same symbol both for a vector or covector and its components , as in: v = v i e i = [ e 1 e 2 ⋯ e n ] [ v 1 v 2 ⋮ v n ] w = w i e i = [ w 1 w 2 ⋯ w n ] [ e 1 e 2 ⋮ e n ] {\displaystyle {\begin{aligned}v=v^{i}e_{i}={\begin{bmatrix}e_{1}&e_{2}&\cdots &e_{n}\end{bmatrix}}{\begin{bmatrix}v^{1}\\v^{2}\\\vdots \\v^{n}\end{bmatrix}}\\w=w_{i}e^{i}={\begin{bmatrix}w_{1}&w_{2}&\cdots &w_{n}\end{bmatrix}}{\begin{bmatrix}e^{1}\\e^{2}\\\vdots \\e^{n}\end{bmatrix}}\end{aligned}}} where v

1333-594: The following operations in Einstein notation as follows. The inner product of two vectors is the sum of the products of their corresponding components, with the indices of one vector lowered (see #Raising and lowering indices ): ⟨ u , v ⟩ = ⟨ e i , e j ⟩ u i v j = u j v j {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\langle \mathbf {e} _{i},\mathbf {e} _{j}\rangle u^{i}v^{j}=u_{j}v^{j}} In

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1376-418: The galaxy where, essentially, the gas is gravitationally bound to the galaxy less strongly than the force from the intracluster medium 'wind' due to the ram pressure. Evidence of this ram pressure stripping can be seen in the image of NGC 4402 . These ram pressure stripped galaxies will often have a large trailing tail and because of this they are commonly called "Jellyfish galaxies." Ram pressure stripping

1419-514: The matter entering the volume (the context above). On the other hand, if only velocity perpendicular to the surface is transferred, there are no shear forces, and the effective pressure on that surface increases by where u n {\displaystyle u_{n}} is the velocity component perpendicular to the surface. What is the sea level ram air pressure at 100 mph ? Within astronomy and astrophysics, James E. Gunn and J. Richard Gott first suggested that galaxies in

1462-435: The rarified upper reaches of Earth's atmosphere the immense speed at which a meteor travels nevertheless rapidly compresses the air in its path, creating a shock wave . The meteoroid then experiences what is known as ram pressure. As the air in front of the meteoroid is compressed its temperature quickly rises. This is not due to friction , rather it is simply a consequence of many molecules and atoms being made to occupy

1505-510: The row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product. In Einstein notation, the usual element reference A m n {\displaystyle A_{mn}} for the m {\displaystyle m} -th row and n {\displaystyle n} -th column of matrix A {\displaystyle A} becomes A m n {\displaystyle {A^{m}}_{n}} . We can then write

1548-403: The rule e i ( e j ) = δ j i . {\displaystyle \mathbf {e} ^{i}(\mathbf {e} _{j})=\delta _{j}^{i}.} where δ is the Kronecker delta . As Hom ⁡ ( V , W ) = V ∗ ⊗ W {\displaystyle \operatorname {Hom} (V,W)=V^{*}\otimes W}

1591-409: The same term). An index that is not summed over is a free index and should appear only once per term. If such an index does appear, it usually also appears in every other term in an equation. An example of a free index is the " i   " in the equation v i = a i b j x j {\displaystyle v_{i}=a_{i}b_{j}x^{j}} , which

1634-399: The superheated air now exerts its force over a much larger surface area, as when the wind suddenly fills a sail . This sudden rise in the force exerted on the meteoroid overwhelms the body's structural integrity and it begins to break up. The breakup of the meteoroid yields an even larger total surface area for the superheated air to act upon and a cycle of amplification rapidly occurs. This is

1677-466: The tendency for airburst energies to be expressed in terms of nuclear weapon yields, as when the Tunguska airburst is given a rating in megatons of TNT . Large meteoroids do not explode in the sense of chemical or nuclear explosives. Rather, at a critical moment in its atmospheric entry the enormous ram pressure experienced by the leading face of the meteoroid converts the body's immense momentum into

1720-417: The tensor product of V with itself, has a basis consisting of tensors of the form e ij = e i ⊗ e j . Any tensor T in V ⊗  V can be written as: T = T i j e i j . {\displaystyle \mathbf {T} =T^{ij}\mathbf {e} _{ij}.} V  * , the dual of V , has a basis e , e , ..., e which obeys

1763-546: The usage of linear algebra in mathematical physics and differential geometry , Einstein notation (also known as the Einstein summation convention or Einstein summation notation ) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus ; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces . It

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1806-408: The way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see Covariance and contravariance of vectors . In the above example, vectors are represented as n × 1 matrices (column vectors), while covectors are represented as 1 × n matrices (row covectors). When using the column vector convention: The virtue of Einstein notation

1849-639: Was introduced to physics by Albert Einstein in 1916. According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables ), it implies summation of that term over all the values of the index. So where the indices can range over the set {1, 2, 3} , y = ∑ i = 1 3 x i e i = x 1 e 1 + x 2 e 2 + x 3 e 3 {\displaystyle y=\sum _{i=1}^{3}x^{i}e_{i}=x^{1}e_{1}+x^{2}e_{2}+x^{3}e_{3}}

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