Oort Constants
Encyclopedia
The Oort constants and are empirically derived parameters that characterize the local rotational properties of our galaxy, the Milky Way
Milky Way
The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky...

, in the following manner:


where and are the rotational velocity and distance to the Galactic center
Galactic Center
The Galactic Center is the rotational center of the Milky Way galaxy. It is located at a distance of 8.33±0.35 kpc from the Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius where the Milky Way appears brightest...

, respectively, measured at the position of the Sun
Sun
The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

. As derived below, they depend only on the motions and positions of stars in the solar neighborhood. As of 1997, the most accurate values of these constants are = 14.82 ± 0.84 km s−1 kpc−1 and = -12.37 ± 0.64 km s−1 kpc−1. From the Oort constants, it is possible to determine the orbital properties
Astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation. It...

 of the Sun, such as the orbital velocity and period
Orbital period
The orbital period is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.There are several kinds of...

, and infer local properties of the Galactic disk, such as the mass density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

 and how the rotational velocity changes as a function of radius from the Galactic center.

Historical significance and background

By the 1920s, a large fraction of the astronomical community had recognized that some of the diffuse, cloud-like objects, or nebulae, seen in the night sky were collections of star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s located beyond our own, local collection of star clusters. These galaxies
Galaxy
A galaxy is a massive, gravitationally bound system that consists of stars and stellar remnants, an interstellar medium of gas and dust, and an important but poorly understood component tentatively dubbed dark matter. The word galaxy is derived from the Greek galaxias , literally "milky", a...

had diverse morphologies, ranging from ellipsoids to disks. The concentrated band of starlight that is the visible signature of the Milky Way was indicative of a disk structure for our galaxy; however, our location within our galaxy made structural determinations from observations difficult.

Classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

 predicted that a collection of stars could be supported against gravitational collapse by either random velocities
Velocity dispersion
In astronomy, the velocity dispersion σ, is the range of velocities about the mean velocity for a group of objects, such as a cluster of stars about a galaxy...

 of the stars or their rotation about its center of mass. For a disk-shaped collection, the support should be mainly rotational. Depending on the mass density, or distribution of the mass in the disk, the rotation velocity may be different at each radius from the center of the disk to the outer edge. A plot of these rotational velocities against the radii at which they are measured is called a rotation curve. For external disk galaxies, one can measure the rotation curve by observing the Doppler shifts of spectral features measured along different galactic radii, since one side of the galaxy will be moving towards our line of sight and one side away. However, our position in the Galactic midplane of the Milky Way, where dust in molecular clouds obscures
Extinction (astronomy)
Extinction is a term used in astronomy to describe the absorption and scattering of electromagnetic radiation by matter between an emitting astronomical object and the observer. Interstellar extinction—also called Galactic extinction, when it occurs in the Milky Way—was first...

 most optical light in many directions, made obtaining our own rotation curve technically difficult until the discovery of the 21 cm hydrogen line in the 1930s.

To confirm the rotation of our galaxy prior to this, in 1927 Jan Oort
Jan Oort
Jan Hendrik Oort was a Dutch astronomer. He was a pioneer in the field of radio astronomy. The Oort cloud of comets bears his name....

 derived a way to measure the Galactic rotation from just a small fraction of stars in the local neighborhood. As described below, the values he found for and proved not only that the Galaxy was rotating but also that it rotates differentially
Differential rotation
Differential rotation is seen when different parts of a rotating object move with different angular velocities at different latitudes and/or depths of the body and/or in time. This indicates that the object is not solid. In fluid objects, such as accretion disks, this leads to shearing...

, or as a fluid rather than a solid body.

Derivation

Consider a star in the midplane of the Galactic disk with Galactic longitude  at a distance from the Sun. Assume that both the Sun and the star have circular orbit
Circular orbit
A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis.Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions...

s around the center of the Galaxy at radii of and from the galactic center
Galactic Center
The Galactic Center is the rotational center of the Milky Way galaxy. It is located at a distance of 8.33±0.35 kpc from the Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius where the Milky Way appears brightest...

 and rotational velocities of and , respectively. The motion of the star along our line of sight, or radial velocity
Radial velocity
Radial velocity is the velocity of an object in the direction of the line of sight . In astronomy, radial velocity most commonly refers to the spectroscopic radial velocity...

, and motion of the star across the plane of the sky, or transverse velocity
Proper motion
The proper motion of a star is its angular change in position over time as seen from the center of mass of the solar system. It is measured in seconds of arc per year, arcsec/yr, where 3600 arcseconds equal one degree. This contrasts with radial velocity, which is the time rate of change in...

, as observed from the position of the Sun are then:


With the assumption of circular motion, the rotational velocity is related to the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

 by and we can substitute this into the velocity expressions:


From the geometry in Figure 1, one can see that the triangles formed between the galactic center, the Sun, and the star share a side or portions of sides, so the following relationships hold and substitutions can be made:

and with these we get

To put these expressions only in terms of the known quantities and , we take a Taylor expansion of about .


Additionally, we take advantage of the assumption that the stars used for this analysis are local, i.e. is small, and the distance d to the star is smaller than or , then
.

So:


Using the sine and cosine half angle formulae, these velocities may be rewritten as:


Writing the velocities in terms of our known quantities and two coefficients and yields:

where

At this stage, the observable velocities are related to these coefficients and the position of the star. It is now possible to relate these coefficients to the rotation properties of the galaxy. For a star in a circular orbit, we can express the derivative of the angular velocity with respect to radius in terms of the rotation velocity and radius and evaluate this at the location of the Sun:

so


is the Oort constant describing the shearing motion and is the Oort constant describing the rotation of the Galaxy. As described below, one can measure and from plotting these velocities, measured for many stars, against the galactic longitudes of these stars.

Measurements

As mentioned in an intermediate step in the derivation above:


Therefore, we can write the Oort constants and as:


Thus, the Oort constants can be expressed in terms of the radial and transverse velocities, distances, and galactic longitudes of objects in our Galaxy - all of which are, in principle, observable quantities.

However, there are a number of complications. The simple derivation above assumed that both the Sun and the object in question are traveling on circular orbits about the Galactic center. This is not true for the Sun (the Sun's velocity relative to the local standard of rest
Local standard of rest
In astronomy, the local standard of rest or LSR follows the mean motion of material in the Milky Way in the neighborhood of the Sun. The path of this material is not precisely circular...

 is approximately 13.4 km/s), and not necessarily true for other objects in the Milky Way either. The derivation also implicitly assumes that the gravitational potential of the Milky Way is axisymmetric and always directed towards the center. This ignores the effects of spiral arms
Spiral galaxy
A spiral galaxy is a certain kind of galaxy originally described by Edwin Hubble in his 1936 work The Realm of the Nebulae and, as such, forms part of the Hubble sequence. Spiral galaxies consist of a flat, rotating disk containing stars, gas and dust, and a central concentration of stars known as...

 and the Galaxy's bar
Barred spiral galaxy
A barred spiral galaxy is a spiral galaxy with a central bar-shaped structure composed of stars. Bars are found in approximately two-thirds of all spiral galaxies...

. Finally, both transverse velocity
Proper motion
The proper motion of a star is its angular change in position over time as seen from the center of mass of the solar system. It is measured in seconds of arc per year, arcsec/yr, where 3600 arcseconds equal one degree. This contrasts with radial velocity, which is the time rate of change in...

 and distance
Cosmic distance ladder
The cosmic distance ladder is the succession of methods by which astronomers determine the distances to celestial objects. A real direct distance measurement of an astronomical object is possible only for those objects that are "close enough" to Earth...

 are notoriously difficult to measure for objects which are not relatively nearby.

Since the non-circular component of the Sun's velocity is known, it can be subtracted out from our observations to compensate. We do not know, however, the non-circular components of the velocity of each individual star we observe, so they cannot be compensated for in this way. But, if we plot transverse velocity divided by distance against galactic longitude for a large sample of stars, we know from the equations above that they will follow a sine function. The non-circular velocities will introduce scatter around this line, but with a large enough sample the true function can be fit for and the values of the Oort constants measured, as shown in figure 2. is simply the amplitude of the sinusoid and is the vertical offset from zero. Measuring transverse velocities and distances accurately and without biases remains challenging, though, and sets of derived values for and frequently disagree.

Most methods of measuring and are fundamentally similar, following the above patterns. The major differences usually lie in what sorts of objects are used and details of how distance or proper motion are measured. Oort, in his original 1927 paper deriving the constants, obtained = 31.0 ± 3.7 km s−1 kpc−1. He did not explicitly obtain a value for , but from his conclusion that the Galaxy was nearly in Keplerian rotation (as in example 2 below), we can presume he would have gotten a value of around -10 km s−1 kpc−1. These differ significantly from modern values, which is indicative of the difficulty of measuring these constants. Measurements of and since that time have varied widely; in 1964 the IAU
International Astronomical Union
The International Astronomical Union IAU is a collection of professional astronomers, at the Ph.D. level and beyond, active in professional research and education in astronomy...

 adopted = 15 km s−1 kpc−1 and = -10 km s−1 kpc−1 as standard values. Although more recent measurements continue to vary, they tend to lie near these values.

The Hipparcos
Hipparcos
Hipparcos was a scientific mission of the European Space Agency , launched in 1989 and operated between 1989 and 1993. It was the first space experiment devoted to precision astrometry, the accurate measurement of the positions of celestial objects on the sky...

 satellite, launched in 1989, was the first space-based astrometric
Astrometry
Astrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...

 mission, and its precise measurements of parallax and proper motion have enabled much better measurements of the Oort constants. In 1997 Hipparcos data was used to derive the values = 14.82 ± 0.84 km s−1 kpc−1 and = -12.37 ± 0.64 km s−1 kpc−1; these measurements are probably among the most reliable available. The Gaia spacecraft, planned for launch in 2012, is an updated successor to Hipparcos; when it comes online the quality of available data should allow new levels of accuracy in measuring the Oort constants.

Meaning

The Oort constants can greatly enlighten one as to how the Galaxy rotates. As one can see and are both functions of the Sun's orbital velocity as well as the first derivative of the Sun's velocity. As a result, describes the shearing motion in the disk surrounding the Sun, while describes the angular momentum gradient in the solar neighborhood, also referred to as vorticity.

To illuminate this point, one can look at three examples that describe how stars and gas orbit within the Galaxy giving intuition as to the meaning of and . These three examples are solid body rotation, Keplerian rotation and constant rotation over different annuli. These three types of rotation are plotted as a function of radius (), and are shown in Figure 3 as the green, blue and red curves respectively. The grey curve is approximately the rotation curve of the Milky Way
Milky Way
The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky...

.

Solid body rotation

To begin, let one assume that the rotation of the Milky Way
Milky Way
The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky...

 can be described by solid body rotation, as shown by the green curve in Figure 3. Solid body rotation assumes that the entire system is moving as a rigid body with no differential rotation
Differential rotation
Differential rotation is seen when different parts of a rotating object move with different angular velocities at different latitudes and/or depths of the body and/or in time. This indicates that the object is not solid. In fluid objects, such as accretion disks, this leads to shearing...

. This results in a constant angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

, , which is independent of . Following this we can see that velocity scales linearly with , , thus
Using the two Oort constant identities, one then can determine what the and constants would be,


This demonstrates that in solid body rotation, there is no shear motion, i.e. , and the vorticity is just the angular rotation, . This is what one would expect because there is no difference in orbital velocity as radius increases, thus no stress between the annuli. Also, in solid body rotation, the only rotation is about the center, so it is reasonable that the resulting vorticity in the system is described by the only rotation in the system. One can actually measure and find that is non-zero ( km s−1 kpc−1.). Thus the galaxy does not rotate as a solid body in our local neighborhood, but may in the inner regions of the Galaxy.

Keplerian rotation

The second illuminating example is to assume that the orbits in the local neighborhood follow a Keplerian orbit, as shown by the blue line in Figure 3. The orbital motion in a Keplerian orbit is described by,
where is the Gravitational Constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

, and is the mass enclosed within radius . The derivative of the velocity with respect to the radius is,


The Oort constants can then be written as follows,


For values of Solar velocity, km/s, and radius to the Galactic center
Galactic Center
The Galactic Center is the rotational center of the Milky Way galaxy. It is located at a distance of 8.33±0.35 kpc from the Earth in the direction of the constellations Sagittarius, Ophiuchus, and Scorpius where the Milky Way appears brightest...

, kpc, the Oort's constants are km s−1 kpc−1, and km s−1 kpc−1. However, the observed values are km s−1 kpc−1 and km s−1 kpc−1. Thus, Keplerian rotation is not the best description the Milky Way
Milky Way
The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky...

 rotation. Furthermore, although this example does not describe the local rotation, it can be thought of as the limiting case that describes the minimum velocity an object can have in a stable orbit.

Flat rotation curve

The final example is to assume that the rotation curve of the Galaxy is flat, i.e. is constant and independent of radius, . The rotation velocity is in between that of a solid body and of Keplerian rotation, and is the red dottedline in Figure 3. With a constant velocity, it follows that the radial derivative of is 0,
and therefore the Oort constants are,


Using the local velocity and radius given in the last example, one finds km s−1 kpc−1 and km s−1 kpc−1. This is remarkably close to the actual measured Oort constants and tells us that the solar neighborhood is roughly rotating with the same linear velocity.

What one should take away from these three examples, is that with a remarkably simple model, the rotation of the Milky Way
Milky Way
The Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky...

 can be described by these two constants. The first two examples are used as constraints to the Galactic rotation, for they show the fastest and slowest the Galaxy can rotate at at a given radius. The flat rotation curve serves as an intermediate step between the two rotation curves, and in fact gives the most reasonable Oort constants as compared to current measurements.

Uses

One of the major uses of the Oort constants is to calibrate the galactic rotation curve. A relative curve can be derived from studying the motions of gas clouds in the Milky Way, but to calibrate the actual absolute speeds involved requires knowledge of V0. We know that:


Since R0 can be determined by other means (such as by carefully tracking the motions of stars near the Milky Way's central supermassive black hole), knowing and allows us to determine V0.

It can also be shown that the mass density can be given by:


So the Oort constants can tell us something about the mass density at a given radius in the disk. They are also useful to constrain mass distribution models for the Galaxy. As well, in the epicyclic approximation for nearly circular stellar orbits in a disk, the epicyclic frequency
Epicyclic frequency
In astrophysics, particularly the study of accretion disks, the epicyclic frequency is the frequency at which a radially displaced fluid parcel will oscillate. It can be referred to as a "Rayleigh discriminant"...

  is given by , where is the angular velocity
Angular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...

. Therefore, the Oort constants can tell us a great deal about motions in the galaxy.
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