4. Kinematics of the Milky Way4.1. Differential Rotation and Oort’s ConstantsThe flatness of the MW is suggestive a general rotation about an axis normal to the GP. Observations of both stars and IS gas have shown it to be differential the angular velocity ω of rotation depends on the distance from the GC (unlike solid body rotation): near the Sun the rotational velocity decreases with radius – while it increases towards the GC!

• In 1924, Oort discovered the galactic halo, a group of stars orbiting the Milky Way but outside the main disk.

• In 1927, he calculated that the center of the Milky Way was 5,900 parsecs (19,200 light years) from the Earth in the direction of the constellation Sagittarius.

• In 1932, by measuring the motions of stars in the Milky Way he was the first to find evidence for dark matter, when he found the mass of the galactic plane must be more than the mass of the material that can be seen.

• He showed that the Milky Way had a mass 100 billion times that of the Sun.

• In 1950, he suggested that comets came from a common region of the Solar System (now called the Oort cloud).

• He found that the light from the Crab Nebula was polarized, and produced by synchrotron emission.

Jan Oort, 1900- 1992

• After the Second World War he began work in the new field of radio astronomy, using an old radar antenna from the Germans.

• In the 1950s, he raised funds for a new radio telescope in Dwingeloo

• Thanks to his efforts the Westerbork Synthesis Radio Telescope

was built in 1970

Studied in Groningen with Kapteyn

1927, Jan Oort derived a series of relations that have become the framework in the description of the differential rotation curve

Note!! Major assumption - all orbits in disk are circular around GC

- the Sun (at O) and a star (at S) are orbiting around GC (at C)- the star S is at a distance d from Sun and lies in the GP at a

galactic longitude ℓ wrt the GC with a galactocentric distance R

- the Sun is at R0 and has velocity V0

- and β are auxiliary angles

The relative velocity of the star S wrt the sun is the difference between the projections of the circular velocities along the LOS:

(see Fig. a and c, prev. page)

With being the angle between the LOS to the star and its velocity in space.

Geometry finds the angle from S to C and O to be +90°. The sine theorem then tells us:

The angular velocities of the star and the Sun are ω = V/R and ω0 = V0/R0 respectively. Entering this and the relation of cos

V0

The tangential component of the relative velocity can be obtained in a similar way:

Problems: - d … the distance to a star (or gas cloud) is difficult to measure, except very nearby (trig. parallaxes) or Cepheids

- Extinction at visual wavelengthsBecause of the distance limitation at optical wavelengths, Oort derived a set of approximate

equations for VR and VT that are valid in region near the SunDespite this limitation, these formulae provide a surprising number of information about large-

scale structure of the Galaxy

Approximation (Taylor expansion): for d << R0 the difference in angular velocities will be small (and β will be small), and the velocity gradient ω(R) around ω0(R0) will be small, the radial velocity of a star can be related to the velocity of the Sun with a 1st order Taylor expansion (which is an approximation for the value of a function in the vicinity of the point, based on the derivatives at that point):

Eq. (1)

Eq. (2)

Thus, to the first order, the difference between ω and ω0 is:

And the approximate value of ω is:

Now, the derivative of (dω/dR) is given by:

Substituting the approximation for ω -ω0 (eq. 3) with the derivative (eq. 4) in equation (1) and (2) for VR and VT respectively, and at the same time making use of the identity ω=V/R gives us the following approximations (to 1st order) after some rearrangements:

Eq. (3)

Eq. (4)

First order Taylor expansion:

From the figure on p.2 it is clear that (and the small angle approximation cosβ≈1 with β in radian)

So the radial velocity can be rewritten as

With the trigonometric relation sinℓ cosℓ = 1/2sin2ℓ

A is the first Oort constant:

A is an angular velocity that is a measure of shear in a galaxy. Shear tells how an extended object is stretched out as it experiences differential rotation of the disk.

So we have a simple relation of VR = A d sin (2ℓ) which is good locally out to a few kpc.

It is a useful expression: the distance to an object is a function of local motions (incorporated in A) and the galactic longitude, which is observed. No other assumptions need to be made (except circular motions). The expressions shows (proves – see page 2) that we have a double sine wave relation between velocity and longitude for local objects.

In an analogous way Oort defined a second constant: now called the Oort constant B going back to the tangential velocity as given at the bottom of p.5, and using the same

trigonometric approximation:

The definition of the second Oort constant in the following way:

Reduces the expressions of the approximate radial and tangential velocities in the neighbourhood of the Sun to the two very simple equations of:

Interpretation of the functions and values of the constants

Double sine wave as a function of ℓ because directions to ℓ =0°, ℓ=180° are perpendicular to circular

motions and hence show no radial velocity component

Objects at ℓ=90° and ℓ=270° are essentially in the same orbit, thus have same VR and we see no radial velocity component in that LOS

Not that the quadrants change from being systematic blueshifted to red to blue to red again for I, II, III and IV

Except at small angles below 90° above 270°

Independently, the proper motions of stars also form a double sine wave; it has its maximum at 0° and 180° and minimum (maximum negative velocity) at ℓ=90° and ℓ=270°

But the overall sine wave has the offset from zero which is exactly the Oort constant B:

Recall:

With μ = VT/d we get:

So the same as VR, but offset by 90° in ℓ and B in mean velocity

Two VR curves for stars at distances of d = 1kpc and 2 kpc respectivelyIf the distance to the object is known, the amplitude of

the curve is given by the Oort constant A

VR = A d sin (2ℓ)

As mentioned on the previous sheet:

(but )

The mean offset of the sinewave from zero is given by the Oort constant B

1927, on the basis of this kind of analysis, Oort established the differential rotation of the MW.

Based on an extensive data set the IAU has recommended the following values for the Oort’s constants:

A = 15 km s-1 kpc-1; B = -10 km s-1 kpc-1 ; IAU

Based on the Hipparcos mission, the most recent best values are:

A = 14.8 ± 0.8 km s-1 kpc-1: B = -12.4 ± 0.6 km s-1 kpc-1 ; Hipparcos

Some further interesting relations between the Oort constant:

and the derivations of the local parameters of Galactic rotation

the difference between Oort’s constants gives the angular velocity at the location of the Sun

Adding the constants as given on sheet 6 gives the local derivative of the circular velocity:

We can also relate the velocity dispersions in the R and θ directions to Oort’s constants :

And determine the Sun’s orbital velocity around the galaxy, Prev, from Oort’s constants:

= -0.24 kpc km-1 s = 7.4 x 1015 s = 2.4 x 108 yr Considering the age of the Sun at ~5 x 109 yr, this implies that the Sun has has made

about 20 revolutions around the Galactic Centre. At the end of the previous revolution, the Carboniferous era had ended on earth and the

first mammals would soon appear

To determine the large-scale velocity structure of the Galactic disk, we have to return to the general expressions of VR and VT (can not use Taylor expansion). For this the 21cm observations of the neutral hydrogen have been an essential tool:

gas with the maximum observable radial velocity will lie at the point Pk (point T in the figure on p.2), where the LOS is tangent to a circle

At Pk the distance to the GC will be minimal

At Pk the relative radial velocity will be maximum

For the regarded LOS with longitude ℓ; they are:

And the distance from the Sun to the star/point Pk will be:

When d increases further, the radial velocity will decrease monotonically

If we restrict our observations again to the solar neighbourhood, for values close to but less than 90°, or close to but greater than 270°, we have d << R, R ~ R0 and we can use the Taylor expansion:

Important for det. of rot. curve

Typical intensity distribution of HI along a LOS for a given value of ℓ: the width of the profile is due to the Doppler shifts and the peaks due to HI clouds that have higher column densities

↑ Various line profiles at 21 cm in the Galactic Plane at various galactic longitudes (labelled on the left of each profile).

Note: The methods breaks down for |ℓ| < 20° because of non-circular motions, probably due to the Gal. Bar.

Composite result; very important for spiral structure determination

But disentanglement of velocity profiles not straightforward; apart from the tangent point (4 in fig. above) where R = R0sinℓ we do not know at what distance the density peaks lie (HI clouds/spiral arms)

Requires knowledge of rotation curve, i.e. V=V(R) = ω(R)

4.2. The Rotation Curve of the Milky Way – and Evidence of Dark MatterSystematic observations of the gas for different Galactic longitude will allow the derivation of the rotation curves within the solar circle for 0° < ℓ < 90 ° and 270 ° < ℓ < 360 ° :from the determination of the maximum velocity at the

tangent point C, we immediately get the distance to the GC (Rmin) and therewith a function

V(R) for R < R0

Unfortunately this technique does not work for 90° < ℓ < 270° as there is no unique orbit for which a maximum or characteristic velocity can be observed (e.g we really have no idea where the emission peak A lies in space if we do not know the relation between V and R)

In reality, the method breaks down close to 90° and 270° because the method becomes very insensitive to changes in velocity (i.e. the radial component hardly changes)

Nor, as mentioned on previous slide, within ℓ < 20°

To measure V(R) for R > R0, we must rely on other objects in the GP, such as Cepheids. Problem: they become increasingly sparser (and more dust absorbed towards GC)

The data point suggest that the rotation curve does

not decrease for R > R0

As expected from Newtonian mechanics if most of the mass were interior to the solar circle:

Keplerian motion V ~ 1/R²A significant amount of mass

mass exist for R > R0

Surprise - as most of the luminous matter resides within R0

Similar evidence of flat rotation curves (beyond optical light radius) has been found in other galaxies as early on as in the late 70-’s by e.g. V. Rubin, not from HI but from H observations for a galaxy at v = 4800 km/s using slit spectography

Since the 80’s the systematic mapping of HI rotation curves became a business all by itself (later also H)

All spiral galaxies show flat (or rising) rotation curves way beyond the optical disk, and most remain like that as far out as measurements can be obtained.

Since Galactic rotation depends on the distribution of mass, a great deal can be learned about the matter in galaxies by studying these curves.

In the innermost region, out to a few kpc, we do have rigid-body rotation,

the rigid body rotation near the GC implies that the mass must be nearly spherically distributed and the density nearly constant. Recall centripetal force = grav. force:

V prop. to R, i.e. the angular velocity ω is const., and V increases linearly with R, thus Rω = V.

But beyond a few kpc’s rotation curves remain flat.

Flat rotation curves suggest that the bulk of the mass in the outer portion of the Galaxy is spherically distributed with a density law proportional to R-2

So we observe V(R) = V ≈ const:

Differentiating with respect to radius gives:

And assuming equation of mass conservation in a spherical symmetric system from stellar theory:

Recall that the R-2 density dependence is very different as determined in our star counts where we had a description of the luminous stellar halo that varied as R-3.5 does much more rapid than indicated by flat rotation curve.most of the form of the matter is non-luminous!! The dark matter is noticeable only through

its gravitational effect on luminous matter, nearby satellites and grav. lensing!!

Modified DM density law:Forces the density profile to approach a const. value (rather than diverge) at the center

Note: for R >> a, the R-2 dependence is obtained whereas for R << a, ρ ~ const.

This model is also used for other galaxies with individual values for a and ρ0

NFW DM halo profile:

(As on sheet 50, chap.3)

Proposed by Navarro, Frenk & White (1996) who found from numerical simulations of the so-called CDM (Cold Dark Matter) cosmology model, the formation of a wide range of sizes and mass scales of DM halos from dwarf galaxies to rich clusters of galaxies. All resulting mass profiles (including DM) could be described by the above ‘universal’ profile for appropriate values of ρ0 and a.

The NFW profile is very similar to the previous value, i.e. it behaves like 1/R² over much of the halo, but is shallower in the centre (~1/R) and steeper in the outer parts (1/R³)

Even with an NFW profile the overall mass keeps increasing with increasing R, i.e. there is no border to the DM halo and the mass increases infinitely. That is not realistic. Other galaxies exist in our Universe, but their halos may be overlapping, certainly in dense regions we may have separate luminous objects but merging DM halos.

A Component Model of the Milky Way (and other spiral galaxies)

- Take the mass distribution from known components (thin disk with stars, gas, and dust), Gal. Bulge plus Gal. halo form stars (denoted spheroid)

- calculate (inversely) the expected rotation curves V(R) from the known matter distribution M(<R) = V(R)²R / G V(R) ~ M-1/2

- Sum those up and determine the unaccounted part of the velocity curve

Leads to the matter distribution of the dark halo

Rotation Curves and Dark Matter distribution in other galaxies:

Bosma 1981, AJ 86, 1825

Recall: radial force equation: V²(R)/R = GM(<R) / R²→ M(<R) = V(R)²R / G

Increase of M as a function of R

Instead of Keplerian decrease V(R) prop. R-½ at R a few times hR

Taking the same rot. curves by Bosma thecumulative mass distribution as a function of R keeps growing

→ requires a with R increasing amount of Dark Matter

5. The Galactic Centre (CO 22.4 or 24.4)Observations of GC remain difficult because of about 30 mag’s of visual extinction.

With the pec. velocity of the Sun northwards of the GP of (wsun =7.2 km/s), the Sun will reach an appreciable height of 85 pc above the GP in about 15 million years – will be mostly above the obscuring material of the MW .

Data based on wavelengths longer than 1μm (IR and radio) or X-ray and gamma.

5.1. The mass distribution near the Galactic CentreVery useful: K-band observations (sensitive to Pop I K and M giants)Star density increases as R-1.8 to a galactocentric radius of ~ 1pc (in agreement with

expectation from solid body rotation - as observed in that region) because of high star density, collisions are frequent (once every ~106 yrs) results in isothermal velocity distribution (stars behave as particles in an isothermal gas) velocity distribution approx. Maxwellian

The observed density distribution and velocity distribution are consistent with behaviour expected of “isothermal stellar gas”

But for galactocentric distances below about 2pc the velocities increase substantially either an incredible increase in star density (more like a density increase as R-2.7)

Existence of a great deal of mass in small volume:

Schoedel & Genzel et al. 2002 have measured various orbits of stars close to GC See orbit of star S2 (and movie MW_BH.mpg)

Has orbital period of P = 15.2 yr, e = 0.87,

Perigalacticon distance of 120 AU (17 ℓ-hrs) a few times Pluto’s orbit

From Kepler’s 3rd law the mass can be calculated (homework)

M = 3.7 ± 0.2 106Msun

Within 0.6mpc (1mpc = 1.2ℓd)

This enormous mass density excludes the possibility of the matter being a ball of heavy degenerate fermions (“neutrino ball”) – or a cluster of dark astrophysical object (neutron stars as the life time of such a cluster would be too short < 105yr). These would all require orbits with larger radii than observed for S2.

supermassive black holeMore recent data; 6 further determinations of orbits of stars circling around the SMBH, and observations radio and mm as well as variability of Xray and NIR (flares below 60 min durations) limits the size of the mass overdensity to 0.024mpc (5AU) of only 60 RS

Recall definition of Schwarzschild radius: R SgrA* = 0.08 AU = 16 Rsun

1992 - 2007

1995 - 2010

Schoedel & Genzel et al. 2002 have measured various orbits of stars close to GC See orbit of star S2 (and movie MW_BH.mpg)

Has orbital period of P = 15.2 yr, e = 0.87,

Perigalacticon distance of 120 AU (17 ℓ-hrs) a few times Pluto’s orbit

From Kepler’s 3rd law the mass can be calculated (homework)

M = 3.7 ± 0.2 106Msun

Within 0.6mpc (1mpc = 1.2ℓd)

This enormous mass density excludes the possibility of the matter being a ball of heavy degenerate fermions (“neutrino ball”) – or a cluster of dark astrophysical object (neutron stars as the life time of such a cluster would be too short < 105yr). These would all require orbits with larger radii than observed for S2.

supermassive black holeMore recent data; 6 further determinations of orbits of stars circling around the SMBH, and observations radio and mm as well as variability of Xray and NIR (flares below 60 min durations) limits the size of the mass overdensity to 0.024mpc (5AU) of only 60 RS

Recall definition of Schwarzschild radius: R SgrA* = 0.08 AU = 16 Rsun

Schoedel et al. 2003 ApJ 596, 2005 (RvA&A 18)

Unlikely to be - a dense massive

star cluster- a cluster of BH

of 3Msun each

Mass distribution as derived from observations of orbits

Dark cluster that could marginally fit the orbit delineated by S2 – but has a life time of 105yrs only

Pretty strong and tight evidence compared to existence of SMBH at centers of other galaxies:

NOTE: Best observational evidence in other galaxies for M32

- central mass of 3 106Msun - HST: confined within ~0.3pc- Density of 10-12 kg m-3 (best terr. vacua)

18 orders of magnitude smaller than mean density inside RS would require

IRS 16 a extremely luminous object at the heart of the GC?Distinct luminosity peak within ~01.pc: IRS 16 (infrared source) lunar occultation resolved it into 15 individual sources: most likely extremely hot stars of extreme luminosities ~106Lsun (O and B???) and lots of dust (UV fluxes are absorbed and re-radiated in IR) Wolf Rayet stars?

Interlude: Wolf Rayet stars1867; discovered the French astronomers Charles Wolf and Georges Rayet using visual spectrometery at Paris Observatory

WR stars are very hot (25’000 - 50’000K) evolved, massive stars (> 20 Msun), with strong stellar winds: speeds up to 2000 km/s and huge mass loss: 10-5Msun/yr (compared to 10-14 of our Sun)

Wolf-Rayet stars are a normal stage in the evolution of massive stars. Due to their strong emission lines they are relatively easy to identify in nearby galaxies. They look like pinwheel nebulae.

But they are very rare:About 150 Wolf-Rayets are known in MW, ~100 are known in the LMC, only 12 in SMC

Artistic impression of a WR

Excerpts from Press release:

- The study captures the Quintuplet stars just before disintegrating in supernovae explosions.

"Only a few pinwheels are known in the galaxy," . We've found five all next to each other in the same cluster. No one has seen anything like this before.“

- the swirling dust in pinwheel stars is key to the presence of the most evolved massive stars and points to the presence of pairs of stars.

"The only way that pinwheels can form is if they have two stars, swirling around each other.

For instance, HST found huge young star clusters (10 more massive than normal ones, close to GC ~ at about 30pc from center)

Science, August 18 2006HST image of the Quintuplet cluster. Each of the five bright red stars is now thought to be a pinwheel nebula.

Wolf-Rayet star may develop into a "collapsar" stage in its death throes - when the core of the star collapses to form a black hole, sucking in the surrounding material thought to be the precursor of a long gamma-ray burst.

The Radio Source in Sagittarius1930, first radio emission discovered from Gal. Centre by K. Jansky

- Nuclear disk of neutral gas in a region of a few 100pc to 1kpc (slightly tilted wrt GP)- Galactic lobes of ionized gas (observed in 20cm radio emission produced by synchrotron

radiation (b)) with dimensions of 10 x 200pc, mostly perpendicular to the GP (ejected material?) with some filaments heavily curved

- the radiation is linearly polarized magnetic field of 10-6T – 10-8 T

- the inner 8pc contains the radio source SgrA; with VLA observations angular resolutions of 0.4 milli-arcsec can be achieved ~ 2AU

- The largest feature of the complex is a circumnuclear molecular ring, doughnut-shaped with an inner radius of 2pc and outer radio of 8pc slightly inclined to GP (~20°)

The molecular circumnuclear ring

Observed at various wavelengths associated with several atoms and molecules (H I, H2, C II, O I, OH, CO, HCN, and CS)

- doughnut-shaped structure with inner radius 2pc to out radius 8pc and thickness increasing from 0.5pc 2pc

- rotating velocity at about 110 km/s

- estimated mass within 2-5kpc of M ~ 1 – 3 x 104 Msun (from collisionless excitation of molecules and intensity of the emission

This ring has very different characteristics compared to other dust clouds.

- T ~ 300K 400K from 4.5 pc to inner edge at ~ 2pc

- molecular hydrogen density: 1.5 – 3 x 1010m-3

Factor of 20-30 hotter, and 30-100 denser (Recall: normal GMC: T ~ 15K and NH2 ~ 108m-3)

Very sharp edge of inner ring: particle density about 10-100 times lower (such a discontinuity tends to disappear on short time scales – hence not stable must have occurred < 105yrs ago)

The gas in this cavity is largely ionized (unlike neutral ring) energy required to clear out such a cavity is of the order of a typical SN explosion (1044J)

The ring is clumpy (again not a long lived situation) and has clumps with 2000K – consistent with heating by shocks

Other features in the SgrA complex:SgrA East: non-thermal source with shell-like appearance accepted interpretation: SNR that happened between 100-5000 yrs ago

SgrA West: unusual (ultracompact) HII region (thermal) that looks like a mini spiral (see also 3 sheets ago)

SgrA*: unresolved radio point source (see VLA 6cm image below), near SgrA West

The whole Sgr A complex lies close to the luminosity peak of the cluster IRS16

SgrA* lies only 1” from IRS16 center

SgrA* is presumed to be the actual center (SMBH) of Milky Way (see next sheet

The western arc: ionized edge of the inner edge of the circumnebular ring, rotating at 100km/s

The other components appear ionized filaments of matter rotating and falling towards its center

The ionized gas within the cavity vel to 700km/s at 0.1pc - WR winds??? Pinwheel nebulae

Most recent results (18 Aug. 2006 Science )

An X-ray Source in Sgr A:To within the error bars the radio Sgr A West ring (including SgrA*) coincides with small continuous X-ray source

Characteristic T ~ 108K and high luminosities 1028 – 2 x 1031 W for soft (2-6keV), resp. hard (10keV – 10MeV) XraysHighly variable on short time-scales diameter of object(s) is less than 0.1pc

The SMBH in SgrA*Indication of SMBH:

- a central mass density of M = 3.7 ± 0.2 106Msun R S = 0.08 AU = 16 Rsun (below resolution limit with VLA interferometry : ~2AU)- variability of Xray and NIR (flares below 60 min durations) limits the size of the mass

overdensity to 0.024mpc (5AU) of only 60 RS SMBH only likely explanations

- Producing the IR radation of the ring and maintaining the degree of ionization in cavity requires LUV ~ 107Lsun (TBB ~ 35’000K); (the gas absorbs some of the UV radiation, ionizing it unusual HII region SgrA West)

- Question: can a SMBH produce the observed IR luminosity:- From particle number density and velocity structure in hot cavity

accretion rate on Gal.Centre of- the accreted matter will release grav. energy:

Determine (Newtonian) energy release from particle of mass M spiraling in through an accretion disk from initial radius ri to final radius of rf.

According to Virial Theorem: radiated energy should be ½ change in potential energy

Assuming ri>>rf and rf = Rs

We get – interestingly enough – a result that is both independent of the mass and radius of the black hole:

Thus – the observed accretion rate is more than sufficient to explain the produced luminosity seen in SgrA and SgrA*.

Other evidence in support of SMBH being at center of Galaxy:

If SMBH is center of MW grav. well it be at rest wrt Galaxy center, i.e. motionless:

- Proper motion of SgrA* wrt quasars only reflects peculiar motion of Sun

- It does not move wrt other objects in its neighbourhood (must be much more massive than the surrounding objects)

High energy emission lines from near Galactic Center:

Gamma-ray → a significant high energy emission line was detected near GC at 511 keV:

electron – positron collision will cause mutual annihilation production of 2 photons with energy 511 keV (the rest mass energy of an electron or positron). This has been observed with quite high flux!

Since it is believed that BH can produce positrons, this was regarded as evidence of a BH in SgrA* earlier on (in the early 90’s)

But the observed flux at L511 ~1031 W ~ 5 x 104 Lsun requires a smaller BH (at most a few 100 solar masses): accretion disk temperatures increase with decreasing radius and only very hot accretion disks can produce observed photon rate

New GRANAT observations with higher resolution imaging capability found another X-ray source some 45’ (~300pc) from SgrA* that was the producer if this 511keV emission a 1-stellar mass BH candidate.

Pretty strong and tight evidence compared to existence of SMBH at centers of other galaxies:

NOTE: Best observational evidence in other galaxies for M32

- central mass of 3 106Msun

- HST: confined within ~0.3pc

- Density of 10-12 kg m-3 (best terr. vacua)

18 orders of magnitude smaller than mean density inside RS would require

IRS 16 a extremely luminous object at the heart of the GC?Distinct luminosity peak within ~01.pc: IRS 16 (infrared source) lunar occultation resolved it into 15 individual sources: most likely extremely hot stars of extreme luminosities ~106Lsun (O and B???) and lots of dust (UV fluxes are absorbed and re-radiated in IR) Wolf Rayet stars?

Artistic impression of a WRWR are massive thus short-lived (< 10 million yrs); recently SF??? But gas and dust density is very low at GC, no other indication for SF- -> Are the luminous objects in IRS 16 WR stars?

- Some high velocity gas (~700km/s) is seen around IRS 16 (ejected from WR?), but could be infalling gas as velocity does not match expectation values for acceleration by WR wind

Whatever IRS 16 is, it does not have sufficient mass to account for the rise in orbital velocities close to the GC

No evidence of the mass overdensity in the form of luminous matter