Sunrise (or sunup ) is the moment when the upper rim of the Sun appears on the horizon in the morning , at the start of the Sun path . The term can also refer to the entire process of the solar disk crossing the horizon.
81-523: Although the Sun appears to "rise" from the horizon, it is actually the Earth's motion that causes the Sun to appear. The illusion of a moving Sun results from Earth observers being in a rotating reference frame ; this apparent motion caused many cultures to have mythologies and religions built around the geocentric model , which prevailed until astronomer Nicolaus Copernicus formulated his heliocentric model in
162-403: A r = d e f ( d 2 r d t 2 ) r {\displaystyle \mathbf {a} _{\mathrm {r} }\ {\stackrel {\mathrm {def} }{=}}\ \left({\tfrac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}\right)_{\mathrm {r} }} is the apparent acceleration in
243-452: A is the Euler acceleration and m is the mass of the body. The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. It begins with the relation between a particle's coordinates in a rotating frame and its coordinates in an inertial (stationary) frame. Then, by taking time derivatives, formulas are derived that relate the velocity of
324-620: A cloud seed . More and more water accumulates on the seed until a visible cloud is formed. In the case of ship tracks, the cloud seeds are stretched over a long narrow path where the wind has blown the ship's exhaust, so the resulting clouds resemble long strings over the ocean. The warming caused by human-produced greenhouse gases has been somewhat offset by the cooling effect of human-produced aerosols. In 2020, regulations on fuel significantly cut sulfur dioxide emissions from international shipping by approximately 80%, leading to an unexpected global geoengineering termination shock. Aerosols in
405-717: A cloud seed . More and more water accumulates on the seed until a visible cloud is formed. In the case of ship tracks, the cloud seeds are stretched over a long narrow path where the wind has blown the ship's exhaust, so the resulting clouds resemble long strings over the ocean. The warming caused by human-produced greenhouse gases has been somewhat offset by the cooling effect of human-produced aerosols. In 2020, regulations on fuel significantly cut sulfur dioxide emissions from international shipping by approximately 80%, leading to an unexpected global geoengineering termination shock. The liquid or solid particles in an aerosol have diameters typically less than 1 μm . Larger particles with
486-409: A histogram with the area of each bar representing the proportion of particles in that size bin, usually normalised by dividing the number of particles in a bin by the width of the interval so that the area of each bar is proportionate to the number of particles in the size range that it represents. If the width of the bins tends to zero , the frequency function is: where Therefore, the area under
567-446: A rotation matrix . Introduce the unit vectors ı ^ , ȷ ^ , k ^ {\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}} representing standard unit basis vectors in the rotating frame. The time-derivatives of these unit vectors are found next. Suppose
648-472: A non-rotating planet, winds and currents tend to flow to the right of this direction north of the equator , and to the left of this direction south of the equator. This effect is responsible for the rotation of large cyclones (see Coriolis effects in meteorology ). In classical mechanics , the Euler acceleration (named for Leonhard Euler ), also known as azimuthal acceleration or transverse acceleration
729-506: A significant effect on Earth's climate: volcanic, desert dust, sea-salt, that originating from biogenic sources and human-made. Volcanic aerosol forms in the stratosphere after an eruption as droplets of sulfuric acid that can prevail for up to two years, and reflect sunlight, lowering temperature. Desert dust, mineral particles blown to high altitudes, absorb heat and may be responsible for inhibiting storm cloud formation. Human-made sulfate aerosols , primarily from burning oil and coal, affect
810-506: A significant effect on Earth's climate: volcanic, desert dust, sea-salt, that originating from biogenic sources and human-made. Volcanic aerosol forms in the stratosphere after an eruption as droplets of sulfuric acid that can prevail for up to two years, and reflect sunlight, lowering temperature. Desert dust, mineral particles blown to high altitudes, absorb heat and may be responsible for inhibiting storm cloud formation. Human-made sulfate aerosols , primarily from burning oil and coal, affect
891-429: A significant settling speed make the mixture a suspension , but the distinction is not clear. In everyday language, aerosol often refers to a dispensing system that delivers a consumer product from a spray can . Diseases can spread by means of small droplets in the breath , sometimes called bioaerosols . Aerosol is defined as a suspension system of solid or liquid particles in a gas. An aerosol includes both
SECTION 10
#1732787180438972-498: Is 16 arcminutes. These two angles combine to define sunrise to occur when the Sun's center is 50 arcminutes below the horizon, or 90.83° from the zenith . The timing of sunrise varies throughout the year and is also affected by the viewer's latitude and longitude , altitude , and time zone . These changes are driven by the axial tilt of Earth, daily rotation of the Earth, the planet's movement in its annual elliptical orbit around
1053-528: Is a vector function that is written as f ( t ) = f 1 ( t ) ı ^ + f 2 ( t ) ȷ ^ + f 3 ( t ) k ^ , {\displaystyle {\boldsymbol {f}}(t)=f_{1}(t){\hat {\boldsymbol {\imath }}}+f_{2}(t){\hat {\boldsymbol {\jmath }}}+f_{3}(t){\hat {\boldsymbol {k}}}\ ,} and we want to examine its first derivative then (using
1134-436: Is also known as the transport theorem in analytical dynamics and is also sometimes referred to as the basic kinematic equation . A velocity of an object is the time-derivative of the object's position, so The time derivative of a position r ( t ) {\displaystyle {\boldsymbol {r}}(t)} in a rotating reference frame has two components, one from the explicit time dependence due to motion of
1215-409: Is an acceleration that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the angular velocity of the reference frame 's axis. This article is restricted to a frame of reference that rotates about a fixed axis. The Euler force is a fictitious force on a body that is related to the Euler acceleration by F = m a , where
1296-574: Is either ı ^ {\displaystyle {\hat {\boldsymbol {\imath }}}} or ȷ ^ . {\displaystyle {\hat {\boldsymbol {\jmath }}}.} Introduce unit vectors ı ^ , ȷ ^ , k ^ {\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},\ {\hat {\boldsymbol {k}}}} , now representing standard unit basis vectors in
1377-422: Is taken as the sum of the real and fictitious forces. This equation has exactly the form of Newton's second law, except that in addition to F , the sum of all forces identified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton's second law in the noninertial frame provided we agree that in the noninertial frame we must add an extra force-like term, often called
1458-456: Is the Coriolis acceleration . The last term, − d Ω d t × r {\displaystyle -{\tfrac {\mathrm {d} {\boldsymbol {\Omega }}}{\mathrm {d} t}}\times \mathbf {r} } , is the Euler acceleration and is zero in uniformly rotating frames. When the expression for acceleration is multiplied by the mass of
1539-564: Is the mass of the object being acted upon by these fictitious forces . Notice that all three forces vanish when the frame is not rotating, that is, when Ω = 0 . {\displaystyle {\boldsymbol {\Omega }}=0\ .} For completeness, the inertial acceleration a i {\displaystyle \mathbf {a} _{\mathrm {i} }} due to impressed external forces F i m p {\displaystyle \mathbf {F} _{\mathrm {imp} }} can be determined from
1620-768: Is the same as found using a vector cross product with the rotation vector Ω {\displaystyle {\boldsymbol {\Omega }}} pointed along the z-axis of rotation Ω = ( 0 , 0 , Ω ) , {\displaystyle {\boldsymbol {\Omega }}=(0,\ 0,\ \Omega ),} namely, d d t u ^ = Ω × u ^ , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega \times }}{\hat {\boldsymbol {u}}}\ ,} where u ^ {\displaystyle {\hat {\boldsymbol {u}}}}
1701-459: The ( x , y ) {\displaystyle (x,y)} components are expressed in the stationary frame. Likewise, ȷ ^ ( t ) = ( − sin θ ( t ) , cos θ ( t ) ) . {\displaystyle {\hat {\boldsymbol {\jmath }}}(t)=(-\sin \theta (t),\ \cos \theta (t))\ .} Thus
SECTION 20
#17327871804381782-511: The mass concentration ( M ), defined as the mass of particulate matter per unit volume, in units such as μg/m . Also commonly used is the number concentration ( N ), the number of particles per unit volume, in units such as number per m or number per cm . Particle size has a major influence on particle properties, and the aerosol particle radius or diameter ( d p ) is a key property used to characterise aerosols. Aerosols vary in their dispersity . A monodisperse aerosol, producible in
1863-454: The Earth . (This article considers only frames rotating about a fixed axis. For more general rotations, see Euler angles .) All non-inertial reference frames exhibit fictitious forces ; rotating reference frames are characterized by three: and, for non-uniformly rotating reference frames, Scientists in a rotating box can measure the rotation speed and axis of rotation by measuring these fictitious forces. For example, Léon Foucault
1944-448: The azimuths of sunrise on other dates are complex, but they can be estimated with reasonable accuracy by using the analemma . The figure on the right is calculated using the solar geometry routine in Ref. as follows: An interesting feature in the figure on the right is apparent hemispheric symmetry in regions where daily sunrise and sunset actually occur. This symmetry becomes clear if
2025-510: The inertial force . It is convenient to consider magnetic resonance in a frame that rotates at the Larmor frequency of the spins. This is illustrated in the animation below. The rotating wave approximation may also be used. Aerosol An aerosol is a suspension of fine solid particles or liquid droplets in air or another gas . Aerosols can be generated from natural or human causes . The term aerosol commonly refers to
2106-522: The power function distribution , occasionally applied to atmospheric aerosols; the exponential distribution , applied to powdered materials; and for cloud droplets, the Khrgian–Mazin distribution. For low values of the Reynolds number (<1), true for most aerosol motion, Stokes' law describes the force of resistance on a solid spherical particle in a fluid. However, Stokes' law is only valid when
2187-2667: The product rule of differentiation): d d t f = d f 1 d t ı ^ + d ı ^ d t f 1 + d f 2 d t ȷ ^ + d ȷ ^ d t f 2 + d f 3 d t k ^ + d k ^ d t f 3 = d f 1 d t ı ^ + d f 2 d t ȷ ^ + d f 3 d t k ^ + [ Ω × ( f 1 ı ^ + f 2 ȷ ^ + f 3 k ^ ) ] = ( d f d t ) r + Ω × f {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}&={\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\imath }}}}{\mathrm {d} t}}f_{1}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} {\hat {\boldsymbol {\jmath }}}}{\mathrm {d} t}}f_{2}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+{\frac {\mathrm {d} {\hat {\boldsymbol {k}}}}{\mathrm {d} t}}f_{3}\\&={\frac {\mathrm {d} f_{1}}{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}+{\frac {\mathrm {d} f_{2}}{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}+{\frac {\mathrm {d} f_{3}}{\mathrm {d} t}}{\hat {\boldsymbol {k}}}+\left[{\boldsymbol {\Omega }}\times \left(f_{1}{\hat {\boldsymbol {\imath }}}+f_{2}{\hat {\boldsymbol {\jmath }}}+f_{3}{\hat {\boldsymbol {k}}}\right)\right]\\&=\left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times {\boldsymbol {f}}\end{aligned}}} where ( d f d t ) r {\displaystyle \left({\frac {\mathrm {d} {\boldsymbol {f}}}{\mathrm {d} t}}\right)_{\mathrm {r} }} denotes
2268-427: The skewness associated with a long tail of larger particles. Also for a quantity that varies over a large range, as many aerosol sizes do, the width of the distribution implies negative particles sizes, which is not physically realistic. However, the normal distribution can be suitable for some aerosols, such as test aerosols, certain pollen grains and spores . A more widely chosen log-normal distribution gives
2349-514: The terminal velocity of a particle undergoing gravitational settling in still air. Neglecting buoyancy effects, we find: where The terminal velocity can also be derived for other kinds of forces. If Stokes' law holds, then the resistance to motion is directly proportional to speed. The constant of proportionality is the mechanical mobility ( B ) of a particle: A particle traveling at any reasonable initial velocity approaches its terminal velocity exponentially with an e -folding time equal to
2430-488: The troposphere , tends to mute sunset and sunrise colors, while volcanic ejecta that is instead lofted into the stratosphere (as thin clouds of tiny sulfuric acid droplets), can yield beautiful post-sunset colors called afterglows and pre-sunrise glows. A number of eruptions, including those of Mount Pinatubo in 1991 and Krakatoa in 1883 , have produced sufficiently high stratospheric sulfuric acid clouds to yield remarkable sunset afterglows (and pre-sunrise glows) around
2511-400: The winter solstice , also varying by latitude. The offset between the dates of the solstice and the earliest or latest sunrise time is caused by the eccentricity of Earth's orbit and the tilt of its axis, and is described by the analemma, which can be used to predict the dates. Variations in atmospheric refraction can alter the time of sunrise by changing its apparent position. Near the poles,
Sunrise - Misplaced Pages Continue
2592-504: The 16th century. Architect Buckminster Fuller proposed the terms "sunsight" and "sunclipse" to better represent the heliocentric model, though the terms have not entered into common language. Astronomically, sunrise occurs for only an instant, namely the moment at which the upper limb of the Sun appears tangent to the horizon. However, the term sunrise commonly refers to periods of time both before and after this point: The stage of sunrise known as false sunrise actually occurs before
2673-490: The 20 μm range show a particularly long persistence time in air conditioned rooms due to their "jet rider" behaviour (move with air jets, gravitationally fall out in slowly moving air); as this aerosol size is most effectively adsorbed in the human nose, the primordial infection site in COVID-19 , such aerosols may contribute to the pandemic. Aerosol particles with an effective diameter smaller than 10 μm can enter
2754-530: The 20th century, the term Coriolis force began to be used in connection with meteorology . Perhaps the most commonly encountered rotating reference frame is the Earth . Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the northern hemisphere , and to the left in the southern . Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on
2835-403: The Earth's atmosphere can influence its climate, as well as human health. Volcanic eruptions release large amounts of sulphuric acid , hydrogen sulfide and hydrochloric acid into the atmosphere. These gases represent aerosols and eventually return to earth as acid rain , having a number of adverse effects on the environment and human life. When aerosols absorb pollutants, it facilitates
2916-454: The Sun , and the Earth and Moon's paired revolutions around each other . The analemma can be used to make approximate predictions of the time of sunrise. In late winter and spring, sunrise as seen from temperate latitudes occurs earlier each day, reaching its earliest time shortly before the summer solstice ; although the exact date varies by latitude. After this point, the time of sunrise gets later each day, reaching its latest shortly after
2997-408: The Sun truly reaches the horizon because Earth's atmosphere refracts the Sun's image. At the horizon, the average amount of refraction is 34 arcminutes , though this amount varies based on atmospheric conditions. Also, unlike most other solar measurements, sunrise occurs when the Sun's upper limb , rather than its center, appears to cross the horizon. The apparent radius of the Sun at the horizon
3078-918: The angle in the x − y {\displaystyle x-y} -plane formed at time t {\displaystyle t} by ( x ′ , y ′ ) {\displaystyle \left(x',y'\right)} and the x {\displaystyle x} -axis), and if the two reference frames coincide at time t = 0 {\displaystyle t=0} (meaning ( x ′ , y ′ , z ′ ) = ( x , y , z ) {\displaystyle \left(x',y',z'\right)=(x,y,z)} when t = 0 , {\displaystyle t=0,} so take θ 0 = 0 {\displaystyle \theta _{0}=0} or some other integer multiple of 2 π {\displaystyle 2\pi } ),
3159-451: The behavior of clouds. Although all hydrometeors , solid and liquid, can be described as aerosols, a distinction is commonly made between such dispersions (i.e. clouds) containing activated drops and crystals, and aerosol particles. The atmosphere of Earth contains aerosols of various types and concentrations, including quantities of: Aerosols can be found in urban ecosystems in various forms, for example: The presence of aerosols in
3240-431: The behavior of clouds. When aerosols absorb pollutants, it facilitates the deposition of pollutants to the surface of the earth as well as to bodies of water. This has the potential to be damaging to both the environment and human health. Ship tracks are clouds that form around the exhaust released by ships into the still ocean air. Water molecules collect around the tiny particles ( aerosols ) from exhaust to form
3321-439: The bronchi, while the ones with an effective diameter smaller than 2.5 μm can enter as far as the gas exchange region in the lungs, which can be hazardous to human health. For a monodisperse aerosol, a single number—the particle diameter—suffices to describe the size of the particles. However, more complicated particle-size distributions describe the sizes of the particles in a polydisperse aerosol. This distribution defines
Sunrise - Misplaced Pages Continue
3402-424: The colors are scattered out of the beam by air molecules and airborne particles , changing the final color of the beam the viewer sees. Because the shorter wavelength components, such as blue and green, scatter more strongly, these colors are preferentially removed from the beam. At sunrise and sunset, when the path through the atmosphere is longer, the blue and green components are removed almost completely, leaving
3483-449: The deposition of pollutants to the surface of the earth as well as to bodies of water. This has the potential to be damaging to both the environment and human health. Aerosols interact with the Earth's energy budget in two ways, directly and indirectly. Ship tracks are clouds that form around the exhaust released by ships into the still ocean air. Water molecules collect around the tiny particles ( aerosols ) from exhaust to form
3564-444: The diameter of the spherical particle with a density of 1000 kg/m and the same settling velocity as the irregular particle. Neglecting the slip correction, the particle settles at the terminal velocity proportional to the square of the aerodynamic diameter, d a : where This equation gives the aerodynamic diameter: One can apply the aerodynamic diameter to particulate pollutants or to inhaled drugs to predict where in
3645-472: The environment of the particle upon which they act. Instead, centrifugal force originates in the rotation of the frame of reference within which observations are made. The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist Gaspard-Gustave Coriolis in connection with hydrodynamics , and also in the tidal equations of Pierre-Simon Laplace in 1778. Early in
3726-442: The first time derivatives of Ω {\displaystyle {\boldsymbol {\Omega }}} inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide. Carrying out the differentiations and re-arranging some terms yields the acceleration relative to the rotating reference frame, a r {\displaystyle \mathbf {a} _{\mathrm {r} }} where
3807-578: The frames are aligned at t = 0 {\displaystyle t=0} and the z {\displaystyle z} -axis is the axis of rotation. Then for a counterclockwise rotation through angle Ω t {\displaystyle \Omega t} : ı ^ ( t ) = ( cos θ ( t ) , sin θ ( t ) ) {\displaystyle {\hat {\boldsymbol {\imath }}}(t)=(\cos \theta (t),\ \sin \theta (t))} where
3888-558: The frequency curve between two sizes a and b represents the total fraction of the particles in that size range: It can also be formulated in terms of the total number density N : Assuming spherical aerosol particles, the aerosol surface area per unit volume ( S ) is given by the second moment : And the third moment gives the total volume concentration ( V ) of the particles: The particle size distribution can be approximated. The normal distribution usually does not suitably describe particle size distributions in aerosols because of
3969-469: The general rotating frame. As they rotate they will remain normalized and perpendicular to each other. If they rotate at the speed of Ω ( t ) {\displaystyle \Omega (t)} about an axis along the rotation vector Ω ( t ) {\displaystyle {\boldsymbol {\Omega }}(t)} then each unit vector u ^ {\displaystyle {\hat {\boldsymbol {u}}}} of
4050-399: The hemispheric relation in to the sunrise equation is applied to the x- and y-components of the solar vector presented in Ref. Air molecules and airborne particles scatter white sunlight as it passes through the Earth's atmosphere. This is done by a combination of Rayleigh scattering and Mie scattering . As a ray of white sunlight travels through the atmosphere to an observer, some of
4131-406: The inertial frame of reference, and r {\displaystyle \mathrm {r} } means the rotating frame of reference. Acceleration is the second time derivative of position, or the first time derivative of velocity where subscript i {\displaystyle \mathrm {i} } means the inertial frame of reference, r {\displaystyle \mathrm {r} }
SECTION 50
#17327871804384212-403: The laboratory, contains particles of uniform size. Most aerosols, however, as polydisperse colloidal systems, exhibit a range of particle sizes. Liquid droplets are almost always nearly spherical, but scientists use an equivalent diameter to characterize the properties of various shapes of solid particles, some very irregular. The equivalent diameter is the diameter of a spherical particle with
4293-403: The laws of motion in a rotating reference frame: Treat the fictitious forces like real forces, and pretend you are in an inertial frame. Obviously, a rotating frame of reference is a case of a non-inertial frame. Thus the particle in addition to the real force is acted upon by a fictitious force...The particle will move according to Newton's second law of motion if the total force acting on it
4374-507: The longer-wavelength orange and red hues seen at those times. The remaining reddened sunlight can then be scattered by cloud droplets and other relatively large particles to light up the horizon red and orange. The removal of the shorter wavelengths of light is due to Rayleigh scattering by air molecules and particles much smaller than the wavelength of visible light (less than 50 nm in diameter). The scattering by cloud droplets and other particles with diameters comparable to or larger than
4455-437: The mixture of particulates in air, and not to the particulate matter alone. Examples of natural aerosols are fog , mist or dust . Examples of human caused aerosols include particulate air pollutants , mist from the discharge at hydroelectric dams , irrigation mist, perfume from atomizers , smoke , dust , sprayed pesticides , and medical treatments for respiratory illnesses. Several types of atmospheric aerosol have
4536-670: The non-zero angle subtended by the solar disc. Neglecting the effects of refraction and the Sun's non-zero size, whenever sunrise occurs, in temperate regions it is always in the northeast quadrant from the March equinox to the September equinox and in the southeast quadrant from the September equinox to the March equinox. Sunrises occur approximately due east on the March and September equinoxes for all viewers on Earth. Exact calculations of
4617-542: The number frequency as: where: The log-normal distribution has no negative values, can cover a wide range of values, and fits many observed size distributions reasonably well. Other distributions sometimes used to characterise particle size include: the Rosin-Rammler distribution , applied to coarsely dispersed dusts and sprays; the Nukiyama–Tanasawa distribution, for sprays of extremely broad size ranges;
4698-409: The object itself in the rotating reference frame, and another from the frame's own rotation. Applying the result of the previous subsection to the displacement r ( t ) , {\displaystyle {\boldsymbol {r}}(t),} the velocities in the two reference frames are related by the equation where subscript i {\displaystyle \mathrm {i} } means
4779-475: The particle as seen in the two frames, and the acceleration relative to each frame. Using these accelerations, the fictitious forces are identified by comparing Newton's second law as formulated in the two different frames. To derive these fictitious forces, it's helpful to be able to convert between the coordinates ( x ′ , y ′ , z ′ ) {\displaystyle \left(x',y',z'\right)} of
4860-466: The particle, the three extra terms on the right-hand side result in fictitious forces in the rotating reference frame, that is, apparent forces that result from being in a non-inertial reference frame , rather than from any physical interaction between bodies. Using Newton's second law of motion F = m a , {\displaystyle \mathbf {F} =m\mathbf {a} ,} we obtain: where m {\displaystyle m}
4941-519: The particles and the suspending gas, which is usually air. Meteorologists and climatologists often refer to them as particle matter, while the classification in sizes ranges like PM2.5 or PM10, is useful in the field of atmospheric pollution as these size range play a role in ascertain the harmful effects in human health. Frederick G. Donnan presumably first used the term aerosol during World War I to describe an aero- solution , clouds of microscopic particles in air. This term developed analogously to
SECTION 60
#17327871804385022-615: The rate of change of f {\displaystyle {\boldsymbol {f}}} as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as: d d t f = [ ( d d t ) r + Ω × ] f . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\boldsymbol {f}}=\left[\left({\frac {\mathrm {d} }{\mathrm {d} t}}\right)_{\mathrm {r} }+{\boldsymbol {\Omega }}\times \right]{\boldsymbol {f}}\ .} This result
5103-436: The relative amounts of particles, sorted according to size. One approach to defining the particle size distribution uses a list of the sizes of every particle in a sample. However, this approach proves tedious to ascertain in aerosols with millions of particles and awkward to use. Another approach splits the size range into intervals and finds the number (or proportion) of particles in each interval. These data can be presented in
5184-401: The relaxation time: where: To account for the effect of the shape of non-spherical particles, a correction factor known as the dynamic shape factor is applied to Stokes' law. It is defined as the ratio of the resistive force of the irregular particle to that of a spherical particle with the same volume and velocity: where: The aerodynamic diameter of an irregular particle is defined as
5265-461: The respiratory tract such particles deposit. Pharmaceutical companies typically use aerodynamic diameter, not geometric diameter, to characterize particles in inhalable drugs. The previous discussion focused on single aerosol particles. In contrast, aerosol dynamics explains the evolution of complete aerosol populations. The concentrations of particles will change over time as a result of many processes. External processes that move particles outside
5346-614: The reverse transformation is x ′ = x cos ( − θ ( t ) ) − y sin ( − θ ( t ) ) {\displaystyle x'=x\cos(-\theta (t))-y\sin(-\theta (t))} y ′ = x sin ( − θ ( t ) ) + y cos ( − θ ( t ) ) . {\displaystyle y'=x\sin(-\theta (t))+y\cos(-\theta (t))\ .} This result can be obtained from
5427-752: The rotating coordinate system (such as ı ^ , ȷ ^ , {\displaystyle {\hat {\boldsymbol {\imath }}},\ {\hat {\boldsymbol {\jmath }}},} or k ^ {\displaystyle {\hat {\boldsymbol {k}}}} ) abides by the following equation: d d t u ^ = Ω × u ^ . {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {u}}}={\boldsymbol {\Omega }}\times {\boldsymbol {\hat {u}}}\ .} So if R ( t ) {\displaystyle R(t)} denotes
5508-473: The rotating frame of reference, and where the expression, again, Ω × {\displaystyle {\boldsymbol {\Omega }}\times } in the bracketed expression on the left is to be interpreted as an operator working onto the bracketed expression on the right. As Ω × Ω = 0 {\displaystyle {\boldsymbol {\Omega }}\times {\boldsymbol {\Omega }}={\boldsymbol {0}}} ,
5589-981: The rotating reference frame and the coordinates ( x , y , z ) {\displaystyle (x,y,z)} of an inertial reference frame with the same origin. If the rotation is about the z {\displaystyle z} axis with a constant angular velocity Ω {\displaystyle \Omega } (so z ′ = z {\displaystyle z'=z} and d θ d t ≡ Ω , {\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t}}\equiv \Omega ,} which implies θ ( t ) = Ω t + θ 0 {\displaystyle \theta (t)=\Omega t+\theta _{0}} for some constant θ 0 {\displaystyle \theta _{0}} where θ ( t ) {\displaystyle \theta (t)} denotes
5670-462: The rotating reference frame, the term − Ω × ( Ω × r ) {\displaystyle -{\boldsymbol {\Omega }}\times ({\boldsymbol {\Omega }}\times \mathbf {r} )} represents centrifugal acceleration , and the term − 2 Ω × v r {\displaystyle -2{\boldsymbol {\Omega }}\times \mathbf {v} _{\mathrm {r} }}
5751-421: The same value of some physical property as the irregular particle. The equivalent volume diameter ( d e ) is defined as the diameter of a sphere of the same volume as that of the irregular particle. Also commonly used is the aerodynamic diameter , d a . People generate aerosols for various purposes, including: Some devices for generating aerosols are: Several types of atmospheric aerosol have
5832-463: The sunlight's wavelengths (more than 600 nm) is due to Mie scattering and is not strongly wavelength-dependent. Mie scattering is responsible for the light scattered by clouds, and also for the daytime halo of white light around the Sun ( forward scattering of white light). Sunset colors are typically more brilliant than sunrise colors, because the evening air contains more particles than morning air. Ash from volcanic eruptions , trapped within
5913-625: The term hydrosol , a colloid system with water as the dispersed medium. Primary aerosols contain particles introduced directly into the gas; secondary aerosols form through gas-to-particle conversion. Key aerosol groups include sulfates, organic carbon, black carbon, nitrates, mineral dust, and sea salt, they usually clump together to form a complex mixture. Various types of aerosol, classified according to physical form and how they were generated, include dust, fume, mist, smoke and fog. There are several measures of aerosol concentration. Environmental science and environmental health often use
5994-1334: The time derivative of these vectors, which rotate without changing magnitude, is d d t ı ^ ( t ) = Ω ( − sin θ ( t ) , cos θ ( t ) ) = Ω ȷ ^ ; {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\imath }}}(t)=\Omega (-\sin \theta (t),\ \cos \theta (t))=\Omega {\hat {\boldsymbol {\jmath }}}\ ;} d d t ȷ ^ ( t ) = Ω ( − cos θ ( t ) , − sin θ ( t ) ) = − Ω ı ^ , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\hat {\boldsymbol {\jmath }}}(t)=\Omega (-\cos \theta (t),\ -\sin \theta (t))=-\Omega {\hat {\boldsymbol {\imath }}}\ ,} where Ω ≡ d d t θ ( t ) . {\displaystyle \Omega \equiv {\frac {\mathrm {d} }{\mathrm {d} t}}\theta (t).} This result
6075-444: The time-of-day variation is extreme, since the Sun crosses the horizon at a very shallow angle and thus rises more slowly. Accounting for atmospheric refraction and measuring from the leading edge slightly increases the average duration of day relative to night . The sunrise equation , however, which is used to derive the time of sunrise and sunset, uses the Sun's physical center for calculation, neglecting atmospheric refraction and
6156-438: The total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as electromagnetic forces ) using Newton's second law in the inertial frame: F i m p = m a i {\displaystyle \mathbf {F} _{\mathrm {imp} }=m\mathbf {a} _{\mathrm {i} }} Newton's law in the rotating frame then becomes In other words, to handle
6237-601: The transformation from rotating coordinates to inertial coordinates can be written x = x ′ cos ( θ ( t ) ) − y ′ sin ( θ ( t ) ) {\displaystyle x=x'\cos(\theta (t))-y'\sin(\theta (t))} y = x ′ sin ( θ ( t ) ) + y ′ cos ( θ ( t ) ) {\displaystyle y=x'\sin(\theta (t))+y'\cos(\theta (t))} whereas
6318-493: The transformation taking basis vectors of the inertial- to the rotating frame, with matrix columns equal to the basis vectors of the rotating frame, then the cross product multiplication by the rotation vector is given by Ω × = R ′ ( t ) ⋅ R ( t ) T {\displaystyle {\boldsymbol {\Omega }}\times =R'(t)\cdot R(t)^{T}} . If f {\displaystyle {\boldsymbol {f}}}
6399-469: The velocity of the gas at the surface of the particle is zero. For small particles (< 1 μm) that characterize aerosols, however, this assumption fails. To account for this failure, one can introduce the Cunningham correction factor , always greater than 1. Including this factor, one finds the relation between the resisting force on a particle and its velocity: where This allows us to calculate
6480-399: The world. The high altitude clouds serve to reflect strongly reddened sunlight still striking the stratosphere after sunset, down to the surface. Rotating reference frame A rotating frame of reference is a special case of a non-inertial reference frame that is rotating relative to an inertial reference frame . An everyday example of a rotating reference frame is the surface of
6561-593: Was able to show the Coriolis force that results from Earth's rotation using the Foucault pendulum . If Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning carousel . In classical mechanics , centrifugal force is an outward force associated with rotation . Centrifugal force is one of several so-called pseudo-forces (also known as inertial forces ), so named because, unlike real forces , they do not originate in interactions with other bodies situated in
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