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Oort constants

The Oort constants (discovered by Jan Oort) and are empirically derived parameters that characterize the local rotational properties of our galaxy, the Milky Way, in the following manner:

where and are the rotational velocity and distance to the Galactic Center, respectively, measured at the position of the Sun, and v and r are the velocities and distances at other positions in our part of the galaxy. As derived below, A and B depend only on the motions and positions of stars in the solar neighborhood. As of 2018, the most accurate values of these constants are = 15.3 ± 0.4 km s−1 kpc−1, = −11.9 ± 0.4 km s−1 kpc−1.[1] From the Oort constants, it is possible to determine the orbital properties of the Sun, such as the orbital velocity and period, and infer local properties of the Galactic disk, such as the mass density and how the rotational velocity changes as a function of radius from the Galactic Center.

Historical significance and background edit

 
Orbits of stars around the galactic disk with faster angular velocity closer to the Galaxy Center but colored as seen from the Sun and plotted against the galactic longitude together with the same graph using data from Gaia[2][3]

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 stars located beyond our own, local collection of star clusters. These galaxies 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 predicted that a collection of stars could be supported against gravitational collapse by either random velocities of the stars or their rotation about its center of mass.[4] 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 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 derived a way to measure the Galactic rotation from just a small fraction of stars in the local neighborhood.[5] As described below, the values he found for   and   proved not only that the Galaxy was rotating but also that it rotates differentially, or as a fluid rather than a solid body.

Derivation edit

 
Figure 1: Geometry of the Oort constants derivation, with a field star close to the Sun in the midplane of the Galaxy.

Consider a star in the midplane of the Galactic disk with Galactic longitude   at a distance   from the Sun. Assume that both the star and the Sun have circular orbits around the center of the Galaxy at radii of   and   from the Galactic Center and rotational velocities of   and  , respectively. The motion of the star along our line of sight, or radial velocity, and motion of the star across the plane of the sky, or transverse velocity, 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 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  , and we take:

 .[6]

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

 
Oort constants on a wall in Leiden
 

  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 edit

 
Figure 2: Measuring the Oort constants by fitting to large data sets. Note that this graph erroneously shows B as positive. A negative B value contributes a westerly component to the transverse velocities.

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 is approximately 13.4 km/s),[6] 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 and the Galaxy's bar. Finally, both transverse velocity and distance 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.[5] 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 adopted   = 15 km s−1 kpc−1 and   = −10 km s−1 kpc−1 as standard values.[7] Although more recent measurements continue to vary, they tend to lie near these values.[8][9][10]

The Hipparcos satellite, launched in 1989, was the first space-based astrometric mission, and its precise measurements of parallax and proper motion have enabled much better measurements of the Oort constants. In 1997 Hipparcos data were used to derive the values   = 14.82 ± 0.84 km s−1 kpc−1 and   = −12.37 ± 0.64 km s−1 kpc−1.[11] The Gaia spacecraft, launched in 2013, is an updated successor to Hipparcos; which allowed new and improved levels of accuracy in measuring four Oort constants   = 15.3 ± 0.4 km s−1 kpc−1,   = -11.9 ± 0.4 km s−1 kpc−1,   = −3.2 ± 0.4 km s−1 kpc−1[definition needed] and   = −3.3 ± 0.6 km s−1 kpc−1.[definition needed][1]

With the Gaia values, we find

 

This value of Ω corresponds to a period of 226 million years for the sun's present neighborhood to go around the Milky Way. However, the time it takes for the Sun to go around the Milky Way (a galactic year) may be longer because (in a simple model) it is circulating around a point further from the centre of the galaxy where Ω is smaller (see Sun#Orbit in Milky Way).

The values in km s−1 kpc−1 can be converted into milliarcseconds per year by dividing by 4.740. This gives the following values for the average proper motion of stars in our neighborhood at different galactic longitudes, after correction for the effect due to the Sun's velocity with respect to the local standard of rest:

Galactic longitude Constellation average proper motion mas/year approximate direction
Sagittarius B+A 0.7 north-east
45° Aquila B 2.5 south-west
90° Cygnus B−A 5.7 west
135° Cassiopeia B 2.5 west
180° Auriga B+A 0.7 south-east
225° Monoceros B 2.5 north-west
270° Vela B−A 5.7 west
315° Centaurus B 2.5 west

The motion of the sun towards the solar apex in Hercules adds a generally westward component to the observed proper motions of stars around Vela or Centaurus and a generally eastward component for stars around Cygnus or Cassiopeia. This effect falls off with distance, so the values in the table are more representative for stars that are further away. On the other hand, more distant stars or objects will not follow the table, which is for objects in our neighborhood. For example, Sagittarius A*, the radio source at the centre of the galaxy, will have a proper motion of approximately Ω or 5.7 mas/y southwestward (with a small adjustment due to the Sun's motion toward the solar apex) even though it is in Sagittarius. Note that these proper motions cannot be measured against "background stars" (because the background stars will have similar proper motions), but must be measured against more stationary references such as quasars.

Meaning edit

 
Figure 3: Diagram of the various rotation curves in a galaxy

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.

Solid body rotation edit

To begin, let one assume that the rotation of the Milky Way 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. This results in a constant angular velocity,  , 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.[11][7]). 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 edit

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, 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,   kpc,[6] 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.[11][7] Thus, Keplerian rotation is not the best description the Milky Way 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 edit

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 close to the actual measured Oort constants and tells us that the constant-speed model is the closest of these three to reality in the solar neighborhood. But in fact, as mentioned above,   is negative, meaning that at our distance, speed decreases with distance from the centre of the galaxy.

What one should take away from these three examples, is that with a remarkably simple model, the rotation of the Milky Way 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 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 edit

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.[6] 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),[12] knowing   and   allows us to determine V0.

It can also be shown that the mass density   can be given by:[6]

 

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.[6] As well, in the epicyclic approximation for nearly circular stellar orbits in a disk, the epicyclic frequency   is given by  , where   is the angular velocity.[13] Therefore, the Oort constants can tell us a great deal about motions in the galaxy.

See also edit

References edit

  1. ^ a b Bovy, J. (June 2017). "Galactic rotation in Gaia DR1". MNRAS. 468 (1): L63–L67. arXiv:1610.07610. Bibcode:2017MNRAS.468L..63B. doi:10.1093/mnrasl/slx027.
  2. ^ "Where do the stars go or come from? - Gaia - Cosmos". www.cosmos.esa.int. Retrieved 2022-06-18.
  3. ^ Gaia Collaboration; Drimmel, R.; Romero-Gomez, M.; Chemin, L.; Ramos, P.; Poggio, E.; Ripepi, V.; Andrae, R.; Blomme, R.; Cantat-Gaudin, T.; Castro-Ginard, A. (2022-06-14). "Gaia Data Release 3: Mapping the asymmetric disc of the Milky Way". arXiv:2206.06207 [astro-ph.GA].
  4. ^ pp. 312-321, §4.4, Galactic dynamics (2nd edition), James Binney, Scott Tremaine, Princeton University Press, 2008, ISBN 978-0-691-13027-9.
  5. ^ a b J. H. Oort (1927-04-14). "Observational evidence confirming Lindblad's hypothesis of a rotation of the galactic system". Bulletin of the Astronomical Institutes of the Netherlands. 3 (120): 275–282. Bibcode:1927BAN.....3..275O.
  6. ^ a b c d e f Binney, J.; Merrifield, M. (1998). Galactic Astronomy. Princeton: Princeton University Press. ISBN 978-0-691-02565-0. OCLC 39108765.
  7. ^ a b c Kerr, F. J; Lynden-Bell, D. (15 August 1986). "Review of Galactic Constants". Monthly Notices of the Royal Astronomical Society. 221 (4): 1023–1038. Bibcode:1986MNRAS.221.1023K. doi:10.1093/mnras/221.4.1023.
  8. ^ Branham, Richard (September 2010). "Kinematics and velocity ellipsoid of the F giants". Monthly Notices of the Royal Astronomical Society. 409 (3): 1269–1280. Bibcode:2010MNRAS.409.1269B. doi:10.1111/j.1365-2966.2010.17389.x.
  9. ^ Olling, Rob; Dehnen, Walter (10 December 2003). "The Oort constants measured from proper motions". The Astrophysical Journal. 599 (1): 275–296. arXiv:astro-ph/0301486. Bibcode:2003ApJ...599..275O. doi:10.1086/379278. S2CID 29988865.
  10. ^ Bobylev, Vadim; Bajkova, Anisa (November 2010). "Galactic parameters from masers with trigonometric parallaxes". Monthly Notices of the Royal Astronomical Society. 408 (3): 1788–1795. arXiv:1006.5152. Bibcode:2010MNRAS.408.1788B. doi:10.1111/j.1365-2966.2010.17244.x. S2CID 118533079.
  11. ^ a b c Feast, M.; Whitelock, P. (November 1997). "Galactic Kinematics of Cepheids from HIPPARCOS Proper Motions". MNRAS. 291 (4): 683–693. arXiv:astro-ph/9706293. Bibcode:1997MNRAS.291..683F. doi:10.1093/mnras/291.4.683.
  12. ^ Eisenhauer, F.; et al. (November 2003). "A Geometric Determination of the Distance to the Galactic Center". The Astrophysical Journal. 597 (2): 121–124. arXiv:astro-ph/0306220. Bibcode:2003ApJ...597L.121E. doi:10.1086/380188. S2CID 16425333.
  13. ^ Sparke, L; Gallagher, J (2007). Galaxies in the Universe. Cambridge University Press. ISBN 978-0-521-67186-6.

External links edit

  •   Media related to Oort constants at Wikimedia Commons

oort, constants, discovered, oort, displaystyle, displaystyle, empirically, derived, parameters, that, characterize, local, rotational, properties, galaxy, milky, following, manner, v0r0, dvdr, v0r0, dvdr, displaystyle, begin, aligned, frac, left, frac, frac, . The Oort constants discovered by Jan Oort A displaystyle A and B displaystyle B are empirically derived parameters that characterize the local rotational properties of our galaxy the Milky Way in the following manner A 12 V0R0 dvdr R0 B 12 V0R0 dvdr R0 displaystyle begin aligned amp A frac 1 2 left frac V 0 R 0 frac dv dr Bigg vert R 0 right amp B frac 1 2 left frac V 0 R 0 frac dv dr Bigg vert R 0 right end aligned where V0 displaystyle V 0 and R0 displaystyle R 0 are the rotational velocity and distance to the Galactic Center respectively measured at the position of the Sun and v and r are the velocities and distances at other positions in our part of the galaxy As derived below A and B depend only on the motions and positions of stars in the solar neighborhood As of 2018 the most accurate values of these constants are A displaystyle A 15 3 0 4 km s 1 kpc 1 B displaystyle B 11 9 0 4 km s 1 kpc 1 1 From the Oort constants it is possible to determine the orbital properties of the Sun such as the orbital velocity and period and infer local properties of the Galactic disk such as the mass density and how the rotational velocity changes as a function of radius from the Galactic Center Contents 1 Historical significance and background 2 Derivation 3 Measurements 4 Meaning 4 1 Solid body rotation 4 2 Keplerian rotation 4 3 Flat rotation curve 5 Uses 6 See also 7 References 8 External linksHistorical significance and background edit nbsp Orbits of stars around the galactic disk with faster angular velocity closer to the Galaxy Center but colored as seen from the Sun and plotted against the galactic longitude together with the same graph using data from Gaia 2 3 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 stars located beyond our own local collection of star clusters These galaxies 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 predicted that a collection of stars could be supported against gravitational collapse by either random velocities of the stars or their rotation about its center of mass 4 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 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 derived a way to measure the Galactic rotation from just a small fraction of stars in the local neighborhood 5 As described below the values he found for A displaystyle A nbsp and B displaystyle B nbsp proved not only that the Galaxy was rotating but also that it rotates differentially or as a fluid rather than a solid body Derivation edit nbsp Figure 1 Geometry of the Oort constants derivation with a field star close to the Sun in the midplane of the Galaxy Consider a star in the midplane of the Galactic disk with Galactic longitude l displaystyle l nbsp at a distance d displaystyle d nbsp from the Sun Assume that both the star and the Sun have circular orbits around the center of the Galaxy at radii of R displaystyle R nbsp and R0 displaystyle R 0 nbsp from the Galactic Center and rotational velocities of V displaystyle V nbsp and V0 displaystyle V 0 nbsp respectively The motion of the star along our line of sight or radial velocity and motion of the star across the plane of the sky or transverse velocity as observed from the position of the Sun are then Vobs r Vstar r Vsun r Vcos a V0sin l Vobs t Vstar t Vsun t Vsin a V0cos l displaystyle begin aligned amp V text obs r V text star r V text sun r V cos left alpha right V 0 sin left l right amp V text obs t V text star t V text sun t V sin left alpha right V 0 cos left l right end aligned nbsp With the assumption of circular motion the rotational velocity is related to the angular velocity by v Wr displaystyle v Omega r nbsp and we can substitute this into the velocity expressions Vobs r WRcos a W0R0sin l Vobs t WRsin a W0R0cos l displaystyle begin aligned amp V text obs r Omega R cos left alpha right Omega 0 R 0 sin left l right amp V text obs t Omega R sin left alpha right Omega 0 R 0 cos left l right end aligned nbsp 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 Rcos a R0sin l Rsin a R0cos l d displaystyle begin aligned amp R cos left alpha right R 0 sin left l right amp R sin left alpha right R 0 cos left l right d end aligned nbsp dd and with these we get Vobs r W W0 R0sin l Vobs t W W0 R0cos l Wd displaystyle begin aligned amp V text obs r left Omega Omega 0 right R 0 sin left l right amp V text obs t left Omega Omega 0 right R 0 cos left l right Omega d end aligned nbsp To put these expressions only in terms of the known quantities l displaystyle l nbsp and d displaystyle d nbsp we take a Taylor expansion of W W0 displaystyle Omega Omega 0 nbsp about R0 displaystyle R 0 nbsp W W0 R R0 dWdr R0 displaystyle left Omega Omega 0 right left R R 0 right frac d Omega dr Bigg vert R 0 nbsp dd Additionally we take advantage of the assumption that the stars used for this analysis are local i e R R0 displaystyle R R 0 nbsp is small and the distance d to the star is smaller than R displaystyle R nbsp or R0 displaystyle R 0 nbsp and we take R R0 d cos l displaystyle R R 0 d cdot cos left l right nbsp 6 dd So Vobs r R0dWdr R0d cos l sin l Vobs t R0dWdr R0d cos2 l Wd displaystyle begin aligned amp V text obs r R 0 frac d Omega dr Bigg vert R 0 d cdot cos left l right sin left l right amp V text obs t R 0 frac d Omega dr Bigg vert R 0 d cdot cos 2 left l right Omega d end aligned nbsp Using the sine and cosine half angle formulae these velocities may be rewritten as Vobs r R0dWdr R0dsin 2l 2Vobs t R0dWdr R0d cos 2l 1 2 Wd R0dWdr R0dcos 2l 2 12R0dWdr R0 W d displaystyle begin aligned amp V text obs r R 0 frac d Omega dr Bigg vert R 0 d frac sin left 2l right 2 amp V text obs t R 0 frac d Omega dr Bigg vert R 0 d frac left cos left 2l right 1 right 2 Omega d R 0 frac d Omega dr Bigg vert R 0 d frac cos left 2l right 2 left frac 1 2 R 0 frac d Omega dr Bigg vert R 0 Omega right d end aligned nbsp Writing the velocities in terms of our known quantities and two coefficients A displaystyle A nbsp and B displaystyle B nbsp yields Vobs r Adsin 2l Vobs t Adcos 2l Bd displaystyle begin aligned amp V text obs r Ad sin left 2l right amp V text obs t Ad cos left 2l right Bd end aligned nbsp where A 12R0dWdr R0B 12R0dWdr R0 W displaystyle begin aligned amp A frac 1 2 R 0 frac d Omega dr Bigg vert R 0 amp B frac 1 2 R 0 frac d Omega dr Bigg vert R 0 Omega end aligned nbsp 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 W vrdWdr R0 d v r dr R0 V0R02 1R0dvdr R0 displaystyle begin aligned amp Omega frac v r amp frac d Omega dr Bigg vert R 0 frac d v r dr Bigg vert R 0 frac V 0 R 0 2 frac 1 R 0 frac dv dr Bigg vert R 0 end aligned nbsp dd so nbsp Oort constants on a wall in LeidenA 12 V0R0 dvdr R0 B 12 V0R0 dvdr R0 displaystyle begin aligned amp A frac 1 2 left frac V 0 R 0 frac dv dr Bigg vert R 0 right amp B frac 1 2 left frac V 0 R 0 frac dv dr Bigg vert R 0 right end aligned nbsp A displaystyle A nbsp is the Oort constant describing the shearing motion and B displaystyle B nbsp is the Oort constant describing the rotation of the Galaxy As described below one can measure A displaystyle A nbsp and B displaystyle B nbsp from plotting these velocities measured for many stars against the galactic longitudes of these stars Measurements edit nbsp Figure 2 Measuring the Oort constants by fitting to large data sets Note that this graph erroneously shows B as positive A negative B value contributes a westerly component to the transverse velocities As mentioned in an intermediate step in the derivation above Vobs r Adsin 2l Vobs t Adcos 2l Bd displaystyle begin aligned amp V text obs r A d sin left 2l right amp V text obs t A d cos left 2l right B d end aligned nbsp Therefore we can write the Oort constants A displaystyle A nbsp and B displaystyle B nbsp as A Vobs rdsin 2l B Vobs td Acos 2l displaystyle begin aligned amp A frac V text obs r d sin left 2l right amp B frac V text obs t d A cos left 2l right end aligned nbsp 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 is approximately 13 4 km s 6 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 and the Galaxy s bar Finally both transverse velocity and distance 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 A displaystyle A nbsp is simply the amplitude of the sinusoid and B displaystyle B nbsp is the vertical offset from zero Measuring transverse velocities and distances accurately and without biases remains challenging though and sets of derived values for A displaystyle A nbsp and B displaystyle B nbsp frequently disagree Most methods of measuring A displaystyle A nbsp and B displaystyle B nbsp 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 A displaystyle A nbsp 31 0 3 7 km s 1 kpc 1 He did not explicitly obtain a value for B displaystyle B nbsp 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 5 These differ significantly from modern values which is indicative of the difficulty of measuring these constants Measurements of A displaystyle A nbsp and B displaystyle B nbsp since that time have varied widely in 1964 the IAU adopted A displaystyle A nbsp 15 km s 1 kpc 1 and B displaystyle B nbsp 10 km s 1 kpc 1 as standard values 7 Although more recent measurements continue to vary they tend to lie near these values 8 9 10 The Hipparcos satellite launched in 1989 was the first space based astrometric mission and its precise measurements of parallax and proper motion have enabled much better measurements of the Oort constants In 1997 Hipparcos data were used to derive the values A displaystyle A nbsp 14 82 0 84 km s 1 kpc 1 and B displaystyle B nbsp 12 37 0 64 km s 1 kpc 1 11 The Gaia spacecraft launched in 2013 is an updated successor to Hipparcos which allowed new and improved levels of accuracy in measuring four Oort constants A displaystyle A nbsp 15 3 0 4 km s 1 kpc 1 B displaystyle B nbsp 11 9 0 4 km s 1 kpc 1 C displaystyle C nbsp 3 2 0 4 km s 1 kpc 1 definition needed and K displaystyle K nbsp 3 3 0 6 km s 1 kpc 1 definition needed 1 With the Gaia values we find dvdr R0 A B 3 4 km s kpcV0R0 W A B 27 2 km s kpc displaystyle begin aligned amp frac dv dr Bigg vert R 0 A B 3 4 text km s kpc amp frac V 0 R 0 Omega A B 27 2 text km s kpc end aligned nbsp This value of W corresponds to a period of 226 million years for the sun s present neighborhood to go around the Milky Way However the time it takes for the Sun to go around the Milky Way a galactic year may be longer because in a simple model it is circulating around a point further from the centre of the galaxy where W is smaller see Sun Orbit in Milky Way The values in km s 1 kpc 1 can be converted into milliarcseconds per year by dividing by 4 740 This gives the following values for the average proper motion of stars in our neighborhood at different galactic longitudes after correction for the effect due to the Sun s velocity with respect to the local standard of rest Galactic longitude Constellation average proper motion mas year approximate direction0 Sagittarius B A 0 7 north east45 Aquila B 2 5 south west90 Cygnus B A 5 7 west135 Cassiopeia B 2 5 west180 Auriga B A 0 7 south east225 Monoceros B 2 5 north west270 Vela B A 5 7 west315 Centaurus B 2 5 westThe motion of the sun towards the solar apex in Hercules adds a generally westward component to the observed proper motions of stars around Vela or Centaurus and a generally eastward component for stars around Cygnus or Cassiopeia This effect falls off with distance so the values in the table are more representative for stars that are further away On the other hand more distant stars or objects will not follow the table which is for objects in our neighborhood For example Sagittarius A the radio source at the centre of the galaxy will have a proper motion of approximately W or 5 7 mas y southwestward with a small adjustment due to the Sun s motion toward the solar apex even though it is in Sagittarius Note that these proper motions cannot be measured against background stars because the background stars will have similar proper motions but must be measured against more stationary references such as quasars Meaning edit nbsp Figure 3 Diagram of the various rotation curves in a galaxyThe Oort constants can greatly enlighten one as to how the Galaxy rotates As one can see A displaystyle A nbsp and B displaystyle B nbsp are both functions of the Sun s orbital velocity as well as the first derivative of the Sun s velocity As a result A displaystyle A nbsp describes the shearing motion in the disk surrounding the Sun while B displaystyle B nbsp 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 A displaystyle A nbsp and B displaystyle B nbsp 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 R displaystyle R nbsp 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 Solid body rotation edit To begin let one assume that the rotation of the Milky Way 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 This results in a constant angular velocity W displaystyle Omega nbsp which is independent of R displaystyle R nbsp Following this we can see that velocity scales linearly with R displaystyle R nbsp v r displaystyle v propto r nbsp thus dvdr vr W displaystyle begin aligned amp frac dv dr frac v r Omega end aligned nbsp Using the two Oort constant identities one then can determine what the A displaystyle A nbsp and B displaystyle B nbsp constants would be A 12 W0R0R0 W R0 0B 12 W0R0R0 W R0 W0 displaystyle begin aligned amp A frac 1 2 left frac Omega 0 R 0 R 0 Omega Bigg vert R 0 right 0 amp B frac 1 2 left frac Omega 0 R 0 R 0 Omega Bigg vert R 0 right Omega 0 end aligned nbsp This demonstrates that in solid body rotation there is no shear motion i e A 0 displaystyle A 0 nbsp and the vorticity is just the angular rotation B W displaystyle B Omega nbsp 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 A 14 displaystyle A 14 nbsp km s 1 kpc 1 11 7 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 edit 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 v GMr displaystyle v sqrt frac GM r nbsp where G displaystyle G nbsp is the gravitational constant and M displaystyle M nbsp is the mass enclosed within radius r displaystyle r nbsp The derivative of the velocity with respect to the radius is dvdr 12GMR3 12vr displaystyle frac dv dr frac 1 2 sqrt frac GM R 3 frac 1 2 frac v r nbsp The Oort constants can then be written as follows A 12 V0R0 v2r R0 3V04R0B 12 V0R0 v2r R0 1V04R0 displaystyle begin aligned amp A frac 1 2 left frac V 0 R 0 frac v 2r Bigg vert R 0 right frac 3V 0 4R 0 amp B frac 1 2 left frac V 0 R 0 frac v 2r Bigg vert R 0 right frac 1V 0 4R 0 end aligned nbsp For values of Solar velocity V0 218 displaystyle V 0 218 nbsp km s and radius to the Galactic Center R0 8 displaystyle R 0 8 nbsp kpc 6 the Oort s constants are A 20 displaystyle A 20 nbsp km s 1 kpc 1 and B 7 displaystyle B 7 nbsp km s 1 kpc 1 However the observed values are A 14 displaystyle A 14 nbsp km s 1 kpc 1 and B 12 displaystyle B 12 nbsp km s 1 kpc 1 11 7 Thus Keplerian rotation is not the best description the Milky Way 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 edit The final example is to assume that the rotation curve of the Galaxy is flat i e v displaystyle v nbsp is constant and independent of radius r displaystyle r nbsp 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 v displaystyle v nbsp is 0 dvdr 0 displaystyle frac dv dr 0 nbsp and therefore the Oort constants are A 12 V0R0 0 R0 12 V0R0 B 12 V0R0 0 R0 12 V0R0 displaystyle begin aligned amp A frac 1 2 left frac V 0 R 0 0 Bigg vert R 0 right frac 1 2 left frac V 0 R 0 right amp B frac 1 2 left frac V 0 R 0 0 Bigg vert R 0 right frac 1 2 left frac V 0 R 0 right end aligned nbsp Using the local velocity and radius given in the last example one finds A 13 6 displaystyle A 13 6 nbsp km s 1 kpc 1 and B 13 6 displaystyle B 13 6 nbsp km s 1 kpc 1 This is close to the actual measured Oort constants and tells us that the constant speed model is the closest of these three to reality in the solar neighborhood But in fact as mentioned above A B displaystyle A B nbsp is negative meaning that at our distance speed decreases with distance from the centre of the galaxy What one should take away from these three examples is that with a remarkably simple model the rotation of the Milky Way 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 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 editOne 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 6 We know that V0 R0 A B displaystyle V 0 R 0 A B nbsp 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 12 knowing A displaystyle A nbsp and B displaystyle B nbsp allows us to determine V0 It can also be shown that the mass density rR displaystyle rho R nbsp can be given by 6 rR B2 A22pG displaystyle rho R frac B 2 A 2 2 pi G nbsp 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 6 As well in the epicyclic approximation for nearly circular stellar orbits in a disk the epicyclic frequency k displaystyle kappa nbsp is given by k2 4BW displaystyle kappa 2 4B Omega nbsp where W displaystyle Omega nbsp is the angular velocity 13 Therefore the Oort constants can tell us a great deal about motions in the galaxy See also editDifferential rotation Milky Way Galaxy rotation curve VorticityReferences edit a b Bovy J June 2017 Galactic rotation in Gaia DR1 MNRAS 468 1 L63 L67 arXiv 1610 07610 Bibcode 2017MNRAS 468L 63B doi 10 1093 mnrasl slx027 Where do the stars go or come from Gaia Cosmos www cosmos esa int Retrieved 2022 06 18 Gaia Collaboration Drimmel R Romero Gomez M Chemin L Ramos P Poggio E Ripepi V Andrae R Blomme R Cantat Gaudin T Castro Ginard A 2022 06 14 Gaia Data Release 3 Mapping the asymmetric disc of the Milky Way arXiv 2206 06207 astro ph GA pp 312 321 4 4 Galactic dynamics 2nd edition James Binney Scott Tremaine Princeton University Press 2008 ISBN 978 0 691 13027 9 a b J H Oort 1927 04 14 Observational evidence confirming Lindblad s hypothesis of a rotation of the galactic system Bulletin of the Astronomical Institutes of the Netherlands 3 120 275 282 Bibcode 1927BAN 3 275O a b c d e f Binney J Merrifield M 1998 Galactic Astronomy Princeton Princeton University Press ISBN 978 0 691 02565 0 OCLC 39108765 a b c Kerr F J Lynden Bell D 15 August 1986 Review of Galactic Constants Monthly Notices of the Royal Astronomical Society 221 4 1023 1038 Bibcode 1986MNRAS 221 1023K doi 10 1093 mnras 221 4 1023 Branham Richard September 2010 Kinematics and velocity ellipsoid of the F giants Monthly Notices of the Royal Astronomical Society 409 3 1269 1280 Bibcode 2010MNRAS 409 1269B doi 10 1111 j 1365 2966 2010 17389 x Olling Rob Dehnen Walter 10 December 2003 The Oort constants measured from proper motions The Astrophysical Journal 599 1 275 296 arXiv astro ph 0301486 Bibcode 2003ApJ 599 275O doi 10 1086 379278 S2CID 29988865 Bobylev Vadim Bajkova Anisa November 2010 Galactic parameters from masers with trigonometric parallaxes Monthly Notices of the Royal Astronomical Society 408 3 1788 1795 arXiv 1006 5152 Bibcode 2010MNRAS 408 1788B doi 10 1111 j 1365 2966 2010 17244 x S2CID 118533079 a b c Feast M Whitelock P November 1997 Galactic Kinematics of Cepheids from HIPPARCOS Proper Motions MNRAS 291 4 683 693 arXiv astro ph 9706293 Bibcode 1997MNRAS 291 683F doi 10 1093 mnras 291 4 683 Eisenhauer F et al November 2003 A Geometric Determination of the Distance to the Galactic Center The Astrophysical Journal 597 2 121 124 arXiv astro ph 0306220 Bibcode 2003ApJ 597L 121E doi 10 1086 380188 S2CID 16425333 Sparke L Gallagher J 2007 Galaxies in the Universe Cambridge University Press ISBN 978 0 521 67186 6 External links edit nbsp Media related to Oort constants at Wikimedia Commons Retrieved from https en wikipedia org w index php title Oort constants amp oldid 1146743109, wikipedia, wiki, book, books, library,

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