Editing Brownian motion (section)

==Statistical mechanics theories==

=== Einstein's theory ===
There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the [[mean squared displacement]] of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities.<ref name="Einstein-1956">{{cite book | last = Einstein | first = Albert | year = 1956 | orig-year = 1926 | title = Investigations on the Theory of the Brownian Movement | url = http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf | archive-url = https://ghostarchive.org/archive/20221009/http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf | archive-date = 2022-10-09 | url-status = live | publisher=Dover Publications | access-date=2013-12-25 }}</ref> In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the [[molecular weight]] in grams, of a gas.<ref>{{cite book | editor-last = Stachel | editor-first = J. | year = 1989 | chapter = Einstein's Dissertation on the Determination of Molecular Dimensions | chapter-url = http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf | archive-url = https://ghostarchive.org/archive/20221009/http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf | archive-date = 2022-10-09 | url-status = live | title=The Collected Papers of Albert Einstein, Volume 2 | publisher=Princeton University Press }}</ref> In accordance to [[Avogadro's law]], this volume is the same for all ideal gases, which is 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the [[Avogadro number]], and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the [[molar mass]] of the gas by the [[Avogadro constant]].

[[File:Diffusion of Brownian particles.svg|thumb|upright=1.2|The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a [[Dirac delta function]], indicating that all the particles are located at the origin at time ''t'' = 0. As ''t'' increases, the distribution flattens (though remains bell-shaped), and ultimately becomes uniform in the limit that time goes to infinity.]]

The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval.<ref name="Einstein-1905"/> Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 10<sup>14</sup> collisions per second.<ref name="Feynman-1964"/>

He regarded the increment of particle positions in time <math>\tau</math> in a one-dimensional (''x'') space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a [[random variable]] (<math>q</math>) with some [[probability density function]] <math>\varphi(q)</math> (i.e., <math>\varphi(q) </math> is the probability density for a jump of magnitude <math>q</math>, i.e., the probability density of the particle incrementing its position from <math>x</math> to <math>x + q</math> in the time interval <math>\tau</math>). Further, assuming conservation of particle number, he expanded the [[number density]] <math>\rho(x,t+\tau)</math> (number of particles per unit volume around <math>x</math>) at time <math>t + \tau</math> in a [[Taylor series]],
<math display="block">\begin{align}
\rho(x, t+\tau)
={}& \rho(x,t) + \tau \frac{\partial\rho(x,t)}{\partial t} + \cdots
\\[2ex]
={}& \int_{-\infty}^{\infty} \rho(x - q, t) \, \varphi(q) \, dq
= \mathbb{E}_q{\left[\rho(x - q, t)\right]}
\\[1ex]
={}&
\rho(x, t) \, \int_{-\infty}^{\infty} \varphi(q) \, dq
- \frac{\partial\rho}{\partial x} \, \int_{-\infty}^{\infty} q \, \varphi(q) \, dq
+ \frac{\partial^2 \rho}{\partial x^2} \, \int_{-\infty}^{\infty} \frac{q^2}{2} \varphi(q) \, dq + \cdots
\\[1ex]
={}& \rho(x, t) \cdot 1 - 0 + \cfrac{\partial^2 \rho}{\partial x^2} \, \int_{-\infty}^{\infty} \frac{q^2}{2} \varphi(q) \, dq + \cdots
\end{align}</math>
where the second equality is by definition of <math>\varphi</math>. The [[integral]] in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd [[moment (mathematics)|moments]]) vanish because of space symmetry. What is left gives rise to the following relation:
<math display="block">\frac{\partial\rho}{\partial t}
= \frac{\partial^2 \rho}{\partial x^2} \cdot \int_{-\infty}^{\infty} \frac{q^2}{2 \tau} \varphi(q) \, dq + \text{higher-order even moments.}</math>
Where the coefficient after the [[Laplacian]], the second moment of probability of displacement <math>q</math>, is interpreted as [[mass diffusivity]] ''D'':
<math display="block">D = \int_{-\infty}^{\infty} \frac{q^2}{2 \tau} \varphi(q) \, dq.</math>
Then the density of Brownian particles {{mvar|ρ}} at point {{mvar|x}} at time {{mvar|t}} satisfies the [[diffusion equation]]:
<math display="block">\frac{\partial\rho}{\partial t} = D\cdot \frac{\partial^2\rho}{\partial x^2},</math>

Assuming that ''N'' particles start from the origin at the initial time ''t'' = 0, the diffusion equation has the solution
<math display="block">\rho(x,t) = \frac{N}{\sqrt{4\pi Dt}} \exp{\left(-\frac{x^2}{4Dt}\right)}.</math>
This expression (which is a [[normal distribution]] with the mean <math> \mu = 0</math> and variance <math> \sigma^2 = 2Dt</math> usually called Brownian motion <math> B_t</math>) allowed Einstein to calculate the [[moment (mathematics)|moments]] directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by
<math display="block">\mathbb{E}{\left[x^2\right]} = 2 D t.</math>
This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root.<ref name="Einstein-1956"/> His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.<ref>{{cite book | last = Lavenda | first = Bernard H. | year = 1985 | title = Nonequilibrium Statistical Thermodynamics | url = https://archive.org/details/nonequilibriumst00lave | url-access = limited | page = [https://archive.org/details/nonequilibriumst00lave/page/n29 20] | publisher = John Wiley & Sons | isbn = 978-0-471-90670-4 }}</ref>

The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium.

In his original treatment, Einstein considered an [[osmotic pressure]] experiment, but the same conclusion can be reached in other ways.

Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of {{math|1=''v'' = ''μmg''}}, where {{mvar|m}} is the mass of the particle, {{mvar|g}} is the acceleration due to gravity, and {{mvar|μ}} is the particle's [[Einstein relation (kinetic theory)|mobility]] in the fluid. [[Sir George Stokes, 1st Baronet|George Stokes]] had shown that the mobility for a spherical particle with radius {{mvar|r}} is <math>\mu = \tfrac{1}{6\pi\eta r}</math>, where {{mvar|η}} is the [[dynamic viscosity]] of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the [[Barometric formula|barometric distribution]]
<math display="block">\rho = \rho_o \, \exp\left({-\frac{m g h}{k_\text{B} T}}\right),</math>
where {{math|''ρ'' − ''ρ''<sub>o</sub>}} is the difference in density of particles separated by a height difference, of <math>h = z - z_o</math>, {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]] (the ratio of the [[universal gas constant]], {{math|''R''}}, to the Avogadro constant, {{math|''N''{{sub|A}}}}), and {{math|''T''}} is the [[Thermodynamic temperature|absolute temperature]].

[[File:Brownian motion gamboge.jpg|thumb|[[Jean Baptiste Perrin|Perrin]] examined the equilibrium ([[Barometric formula|barometric distribution]]) of granules (0.6 [[micron]]s) of [[gamboge]], a viscous substance, under the microscope. The granules move against gravity to regions of lower concentration. The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air.]]

[[Dynamic equilibrium]] is established because the more that particles are pulled down by [[gravity]], the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by [[Fick's laws of diffusion|Fick's law]],
<math display="block">J = -D \frac{d\rho}{dh},</math>
where {{math|1=''J'' = ''ρv''}}. Introducing the formula for {{mvar|ρ}}, we find that
<math display="block">v = \frac{D m g}{k_\text{B} T}.</math>

In a state of dynamical equilibrium, this speed must also be equal to {{math|1=''v'' = ''μmg''}}. Both expressions for {{mvar|v}} are proportional to {{math|''mg''}}, reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical [[charged particle]]s of charge {{mvar|q}} in a uniform [[electric field]] of magnitude {{mvar|E}}, where {{math|''mg''}} is replaced with the [[electrostatic force]] {{math|''qE''}}. Equating these two expressions yields the [[Einstein relation (kinetic theory)|Einstein relation]] for the diffusivity, independent of {{math|''mg''}} or {{math|''qE''}} or other such forces:
<math display="block"> \frac{\mathbb{E}{\left[x^2\right]}}{2t} = D = \mu k_\text{B} T
= \frac{\mu R T}{N_\text{A}} = \frac{RT}{6\pi\eta r N_\text{A}}.</math>
Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the [[Boltzmann constant]] as {{math|1=''k''<sub>B</sub> = ''R'' / ''N''{{sub|A}}}}, and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant {{math|''R''}}, the temperature {{mvar|T}}, the viscosity {{mvar|η}}, and the particle radius {{mvar|r}}, the Avogadro constant {{math|''N''{{sub|A}}}} can be determined.

The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by [[J. J. Thomson]]<ref name="Thomson-1904">{{cite book | last = Thomson | first = J. J. | year = 1904 | title =Electricity and Matter | url = https://archive.org/details/electricitymatte00thomuoft | pages = [https://archive.org/details/electricitymatte00thomuoft/page/80 80]–83 | publisher = Yale University Press }}</ref> in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a [[concentration gradient]] given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other".<ref name="Thomson-1904"/>

An identical expression to Einstein's formula for the diffusion coefficient was also found by [[Walther Nernst]] in 1888<ref>{{cite journal | last = Nernst | first = Walther | year = 1888 | title = Zur Kinetik der in Lösung befindlichen Körper | language = de | journal = [[Zeitschrift für Physikalische Chemie]] | volume = 9 | pages = 613–637 }}</ref> in which he expressed the diffusion coefficient as the ratio of the [[osmotic pressure]] to the ratio of the [[Friction|frictional force]] and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by [[Stokes's law]]. He writes <math>k' = p_o/k</math> for the diffusion coefficient {{mvar|k&prime;}}, where <math>p_o</math> is the osmotic pressure and {{mvar|k}} is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the [[ideal gas law]] per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's.<ref>{{cite book | last = Leveugle | first = J. | year = 2004 | title = La Relativité, Poincaré et Einstein, Planck, Hilbert | page = 181 | publisher = Harmattan }}</ref> The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the [[mean free path]].<ref>{{cite book | last = Townsend | first = J.E.S. | year = 1915 | title = Electricity in Gases | url = https://archive.org/details/electricityinga00towngoog | page = [https://archive.org/details/electricityinga00towngoog/page/n282 254] | publisher = Clarendon Press }}</ref>

Confirming Einstein's formula experimentally proved difficult.
Initial attempts by [[Theodor Svedberg]] in 1906 and 1907 were critiqued by Einstein and by Perrin as not measuring a quantity directly comparable to the formula. [[Victor Henri]] in 1908 took cinematographic shots through a microscope and found quantitative disagreement with the formula but again the analysis was uncertain.<ref>{{Cite journal |last=Maiocchi |first=Roberto |date=September 1990 |title=The case of Brownian motion |url=https://www.cambridge.org/core/journals/british-journal-for-the-history-of-science/article/abs/case-of-brownian-motion/7E6FCB8188956D072CC83581B5645099 |journal=The British Journal for the History of Science |language=en |volume=23 |issue=3 |pages=257–283 |doi=10.1017/S0007087400043983 |issn=1474-001X}}</ref> Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909.<ref>{{Cite journal |last=Haw |first=M D |date=2002-08-26 |title=Colloidal suspensions, Brownian motion, molecular reality: a short history |url=https://iopscience.iop.org/article/10.1088/0953-8984/14/33/315 |journal=Journal of Physics: Condensed Matter |volume=14 |issue=33 |pages=7769–7779 |doi=10.1088/0953-8984/14/33/315}}</ref><ref>{{Cite journal |last=Brush |first=Stephen G. |date=1968 |title=A History of Random Processes: I. Brownian Movement from Brown to Perrin |url=https://www.jstor.org/stable/41133279 |journal=Archive for History of Exact Sciences |volume=5 |issue=1 |pages=1–36 |issn=0003-9519}}</ref> The confirmation of Einstein's theory constituted empirical progress for the [[Kinetic theory of gases|kinetic theory of heat]]. In essence, Einstein showed that the motion can be predicted directly from the kinetic model of [[thermal equilibrium]]. The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the [[second law of thermodynamics]] as being an essentially statistical law.<ref>See P. Clark 1976, p. 97</ref>

[[File:Brownian Motion.ogv|thumb|320px|Brownian motion model of the trajectory of a particle of dye in water.]]

===Smoluchowski model===
{{Primary sources|date=January 2025}}
[[Marian Smoluchowski|Smoluchowski]]'s theory of Brownian motion<ref>{{cite journal | last = Smoluchowski | first = M. M. | year = 1906 | title = Sur le chemin moyen parcouru par les molécules d'un gaz et sur son rapport avec la théorie de la diffusion | language = fr | trans-title = On the average path taken by gas molecules and its relation with the theory of diffusion | url = https://archive.org/stream/bulletininternat1906pols#page/202/mode/2up | journal = [[Bulletin International de l'Académie des Sciences de Cracovie]] | page = 202 }}</ref> starts from the same premise as that of Einstein and derives the same probability distribution {{math|''ρ''(''x'', ''t'')}} for the displacement of a Brownian particle along the {{mvar|x}} in time {{mvar|t}}. He therefore gets the same expression for the mean squared displacement: {{nowrap|<math>\mathbb{E}{\left[(\Delta x)^2\right]}</math>.}} However, when he relates it to a particle of mass {{mvar|m}} moving at a velocity {{mvar|u}} which is the result of a frictional force governed by Stokes's law, he finds
<math display="block">\mathbb{E}{\left[(\Delta x)^2\right]} = 2Dt
= t \frac{32}{81} \frac{mu^2}{\pi \mu a}
= t\frac{64}{27} \frac{\frac{1}{2}mu^2}{3 \pi \mu a},</math>
where {{mvar|μ}} is the viscosity coefficient, and {{mvar|a}} is the radius of the particle. Associating the kinetic energy <math>mu^2/2</math> with the thermal energy {{math|''RT''/''N''}}, the expression for the mean squared displacement is {{math|64/27}} times that found by Einstein. The fraction 27/64 was commented on by [[Arnold Sommerfeld]] in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."<ref>See p. 535 in {{cite journal | last = Sommerfeld | first = A. | year = 1917 | title = Zum Andenken an Marian von Smoluchowski |language = de | trans-title = In Memory of Marian von Smoluchowski | journal = [[Physikalische Zeitschrift]] | volume = 18 | issue = 22 | pages = 533–539 }}</ref>

Smoluchowski<ref>{{cite journal | last = Smoluchowski | first = M. M. | year = 1906 | title = Essai d'une théorie cinétique du mouvement Brownien et des milieux troubles | url = https://archive.org/stream/bulletininternat1906pols#page/577/mode/2up | language = fr | trans-title = Test of a kinetic theory of Brownian motion and turbid media | journal = [[Bulletin International de l'Académie des Sciences de Cracovie]] | page = 577 }}</ref> attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal.
If the probability of {{mvar|m}} gains and {{math|''n'' − ''m''}} losses follows a [[binomial distribution]],
<math display="block">P_{m,n} = \binom{n}{m} 2^{-n},</math>
with equal {{em|a priori}} probabilities of 1/2, the mean total gain is
<math display="block">\mathbb{E}{\left[2m-n\right]} = \sum_{m=\frac{n}{2}}^n (2m-n)P_{m,n}=\frac{n n!}{2^{n+1} \left [ \left (\frac{n}{2} \right )! \right ]^2}.</math>

If {{mvar|n}} is large enough so that Stirling's approximation can be used in the form
<math display="block">n!\approx\left(\frac{n}{e}\right)^n\sqrt{2\pi n},</math>
then the expected total gain will be{{citation needed|date=July 2012}}
<math display="block">\mathbb{E}{\left[2m - n\right]} \approx \sqrt{\frac{n}{2\pi}},</math>
showing that it increases as the square root of the total population.

Suppose that a Brownian particle of mass {{mvar|M}} is surrounded by lighter particles of mass {{mvar|m}} which are traveling at a speed {{mvar|u}}. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be {{math|''mu''/''M''}}. This ratio is of the order of {{val|e=-7|u=cm/s}}. But we also have to take into consideration that in a gas there will be more than 10<sup>16</sup> collisions in a second, and even greater in a liquid where we expect that there will be 10<sup>20</sup> collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 10<sup>8</sup> to 10<sup>10</sup> collisions in one second, then velocity of the Brownian particle may be anywhere between {{val|10|-|1000|u=cm/s}}. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.

These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, {{mvar|U}}, which depends on the collisions that tend to accelerate and decelerate it. The larger {{mvar|U}} is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of energy applies, the kinetic energy of the Brownian particle, {{nowrap|<math>MU^2/2</math>,}} will be equal, on the average, to the kinetic energy of the surrounding fluid particle, {{nowrap|<math>mu^2/2</math>.}}

In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.<ref>{{cite journal | last=von Smoluchowski | first = M. | year = 1906 | title = Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen | language = de | journal = [[Annalen der Physik]] | volume = 326 | issue = 14 | pages = 756–780 | bibcode = 1906AnP...326..756V | doi = 10.1002/andp.19063261405 | url = https://zenodo.org/record/1424073 }}</ref> The model assumes collisions with {{math|''M'' ≫ ''m''}} where {{mvar|M}} is the test particle's mass and {{mvar|m}} the mass of one of the individual particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision always imparts the same magnitude of {{nowrap|Δ''V''}}. If {{math|''N''<sub>R</sub>}} is the number of collisions from the right and {{math|''N''<sub>L</sub>}} the number of collisions from the left then after {{mvar|N}} collisions the particle's velocity will have changed by {{math|Δ''V''(2''N''<sub>R</sub> − ''N'')}}. The [[multiplicity (mathematics)|multiplicity]] is then simply given by:
<math display="block"> \binom{N}{N_\text{R}} = \frac{N!}{N_\text{R}!(N - N_\text{R})!}</math>
and the total number of possible states is given by {{math|2<sup>''N''</sup>}}. Therefore, the probability of the particle being hit from the right {{math|''N''<sub>R</sub>}} times is:
<math display="block">P_N(N_\text{R}) = \frac{N!}{2^N N_\text{R}!(N-N_\text{R})!}</math>

As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions don't apply. For example, the assumption that on average occurs an equal number of collisions from the right as from the left falls apart once the particle is in motion. Also, there would be a distribution of different possible {{math|Δ''V''}}s instead of always just one in a realistic situation.

===Langevin equation===
{{main|Langevin equation}}
The [[diffusion equation]] yields an approximation of the time evolution of the [[probability density function]] associated with the position of the particle going under a Brownian movement under the physical definition. The approximation becomes valid on timescales much larger than the timescale of individual atomic collisions, since it does not include a term to describe the acceleration of particles during collision. The time evolution of the position of the Brownian particle over all time scales described using the [[Langevin equation]], an equation that involves a random force field representing the effect of the [[thermal fluctuations]] of the solvent on the particle.<ref name=Velocity-2010/> At longer times scales, where acceleration is negligible, individual particle dynamics can be approximated using [[Brownian dynamics]] in place of [[Langevin dynamics]].

===Astrophysics: star motion within galaxies===

In [[stellar dynamics]], a massive body (star, [[black hole]], etc.) can experience Brownian motion as it responds to [[gravitational|gravitational forces]] from surrounding stars.<ref name="Merritt-2013">{{cite book | last = Merritt | first = David | year = 2013 | title = Dynamics and Evolution of Galactic Nuclei | page = 575 | publisher = Princeton University Press | isbn = 9781400846122 | ol = 16802359W }}</ref> The rms velocity {{mvar|V}} of the massive object, of mass {{mvar|M}}, is related to the rms velocity <math>v_\star</math> of the background stars by
<math display="block"> MV^2 \approx m v_\star^2 </math>
where <math>m \ll M</math> is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both <math>v_\star</math> and {{mvar|V}}.<ref name="Merritt-2013"/> The Brownian velocity of [[Sagittarius A*|Sgr A*]], the [[supermassive black hole]] at the center of the [[Milky Way galaxy]], is predicted from this formula to be less than 1&nbsp;km&nbsp;s<sup>−1</sup>.<ref name="Reid-2004">{{cite journal | last1 = Reid | first1 = M. J. | last2 = Brunthaler | first2 = A. | year = 2004 | title = The Proper Motion of Sagittarius A*. II. The Mass of Sagittarius A* | journal = [[The Astrophysical Journal]] | volume = 616 | issue = 2 | pages = 872–884 | arxiv = astro-ph/0408107 | bibcode = 2004ApJ...616..872R | doi = 10.1086/424960 | s2cid = 16568545 }}</ref>