Astronomy 3500Astronomy 3500Galaxies and CosmologyGalaxies and Cosmology

Examine the nature and morphology of galaxies of stars and their distribution in space, and to use that knowledge to understand the nature of the universe as a whole. Topics include the Milky Way Galaxy, the various morphological

types of galaxies and their spatial distribution, exotic objects including active galaxies, and

what the characteristics of galaxies and clusters of galaxies tell us about the nature of

the universe as a whole.

Emphasis is placed on the development of critical judgment to separate observational information from proposed physical models.

24. The Milky Way Galaxy24. The Milky Way GalaxyGoalsGoals:1. Summarize the basic observational

characteristics of the Galaxy and how they were established, distinguishing observational fact from physical models.

2. Introduce the dynamical equations for Galactic rotation and what they tell us about the nature of the Milky Way.

3. Characterize basic features of the Galaxy, the Galactic centre, the central bulge, the disk with its spiral features, and the halo.

Picturing the Galaxy:The proper study of the Milky Way Galaxy probably begins in 1610, when Galileo first discovered that the Milky Way consists of “innumerable” faint stars. In 1718 Halley discovered the proper motions of Arcturus, Sirius, and Aldebaran, and by 1760 Mayer had published proper motions for some 80 stars based upon comparisons of their recorded positions. His results established that the Sun and stars are not at rest relative to one another in the Galaxy.

The obvious problem with trying to map our Galaxy from within is that the Sun is but one of many billions of stars that populate it, and our vantage point in the disk 8-9 kpc from the Galactic centre makes it difficult to detect objects in regions obscured by interstellar dust. But attempts have been made frequently.

In 1785 William Herschel derived the first schematic picture of the Galaxy from optical “star gauging” in 700 separate regions of the sky. He did it by making star counts to the visual limit of his 20 foot (72-inch diameter) telescope. He assumed that r ~ N1/3 (i.e. N ~ r3), and obtained relative thicknesses for the Galactic disk in the various directions sampled. No absolute dimensions were established. By 1817 Hershel had adopted a new picture of the Galaxy as a flattened disk of nearly infinite extension (similar to the modern picture).

In 1837 Argelander, of the Bonn Observatory and orginator of the BD catalogue, was able to derive an apex for the solar motion from studying stellar proper motions. His result is very similar to that recognized today. Also in 1837, Frederick Struve found evidence for interstellar extinction in star count data, which was considered necessary at that time to resolve Herschel’s “infinite universe” with Olber’s paradox (which had been published in 1823).By the turn of the century many astronomers felt that a concerted, detailed effort should be made to establish reliable dimensions for the Milky Way. The task was initiated by Kapteyn in 1905 with his plan to study in a systematic fashion 206 special areas, each 1° square, covering most of the sky — the well known “Selected Areas” for Galactic research. By then, separately-pursued research programs into the nature of the Milky Way system often produced distinctly different results.

In 1918, for example, Shapley noted the asymmetric location of the centre of the globular cluster system with respect to the Sun, and suggested that it coincided with the centre of the Galaxy. But the distance to the Galactic centre found in such fashion was initially overly large because of distance scale problems.

Longitude distributionof globular clusters.

Kapteyn and van Rhijn published initial results from star counts in 1920, namely a Galaxy model with a radius of ~4.5 kpc along its major plane and a radius of ~0.8 kpc at the poles. Kapteyn published an alternate model in 1922 in with the Sun displaced from the centre, yet by less than the distance of ~15 kpc to the centre of the globular cluster system established by Shapley.

The issue reached a turning point in 1920 with the well known Shapley-Curtis debate on the extent of the Galactic system. The merits of the arguments presented on both sides of this debate have been the subject of considerable study over the years, but it was years later before the true extragalactic nature of the spiral nebulae was recognized. Although Shapley was considered the “winner” of the debate, it was Curtis who argued the correct points. A big step was Hubble’s 1924 derivation of the distance to the Andromeda Nebula using Cepheid variables. Somewhat less well-known is Lindblad’s 1926 development of a mathematical model for Galactic rotation. Lindblad’s model was developed further in 1927-28 by Oort, who demonstrated its applicability to the radial velocity data for stars. Finally, in 1930 Trumpler provided solid evidence for the existence of interstellar extinction from an extremely detailed study of the distances and diameters of open star clusters.

Perhaps the best “picture” of the Galaxy is that sketched by Sergei Gaposhkin from Australia, as published in Vistas in Astronomy, 3, 289, 1957. The lower view is Sergei’s attempt to step outwards by 1 kpc from the Sun.

Sergei Gaposhkin’s drawing is crucial for the insights it provides into the size and nature of the Galactic bulge, that spheroidal (or bar-shaped?) distribution of stars surrounding the Galactic centre. Keep in mind that all such attempts rely heavily upon the ability of the human eye (and brain) to distinguish a “grand design” from the confusing picture posed by the interaction of dark dust clouds, bright gaseous nebulae, and rich star fields along the length of the Milky Way (see below).

The present picture of the Galaxy has the Sun lying ~20 pc above the centreline of a flattened disk, ~8.5±0.5 kpc from the Galactic centre. The spheroidal halo is well established, but the existence of a sizable central bar and the nature of the spiral arms are more controversial.

Another schematic representing the present view of the Galaxy.

An outdated picture of the Galaxy by the instructor prior to 2010 had the Sun lying ~20 pc above the centreline of a flattened disk, ~9±1 kpc from the Galactic centre. The spheroidal halo is well established, and there is an obvious warping of the Galactic disk in the direction of the Magellanic Clouds that is best seen in the fourth Galactic quadrant.

Star Count Analysis:Define, for a particular area of sky:N(m) = total number of stars brighter than magnitude m per square degree of sky, andA(m) = the total number of stars of apparent magnitude m ±½ in the same area (usually steps of 1 mag are used).N(m) increases by the amount A(m)m for each increase m in magnitude m.

dN(m) = A(m) dm,or A(m) = dN(m)/dm.

Star counts in restricted magnitude intervals are usually made over a restricted area of sky subtending a solid angle = . The entire sky consists of 4 steradians = 4 (radian)2 = 4 (57.2957795)2 square degrees = 41,252.96 square degrees ≈ 41,253 square degrees. Thus, 1 steradian = 41,253/4 square degrees = 3283 square degrees.

In order to consider the density of stars per unit distance interval of space in the same direction, it is necessary to consider the star counts as functions of distance, i.e. N(r), A(r). If the space density distribution is D(r) = number of stars per cubic parsec at the distance r in the line of sight, then: . If D(r) = constant = D, then:

Cumulative star counts in a particular area of sky should therefore increase as r3 for the case of a uniform density of stars as a function of distance. For no absorption:m – M = 5 log r – 5, 0.2(m – M) + 1 = log r, or r = 10[0.2(m – M) + 1] .Thus,

,if M and D are constant.

r

drrDrrN0

2 )()( 3

0 312

0

2)( DrdrrDDdrrrNrr

CmmM

MmMm

D

DDmN

6.06.06.0

31000

6.03

1000312.031

101010

1010)(

i.e. log N(m) = 0.6m + C.A(m) = dN(m)/dm = d/dm [10C 100.6m] = (0.6)(10C)

(loge10)100.6m = C'100.6m.Denote lo = the light received from a star with m = 0. l(m) = lo10–0.4m [m1–m2 = –2.5 log b1/b2].or –0.4 m = log l(m)/lo .The total light received from stars of magnitude m is therefore given by:L(m) = l(m) A(m) (per unit interval of sky)

= loC'10–0.4m + 0.6m = loC'100.2m .The total light received by all stars brighter than magnitude m is given by:Ltot(m) = ∫ L(m')dm' = loC' ∫100.2m'dm' = K 100.2m, where K is a constant.

Thus, Ltot(m) diverges exponentially as m increases (Olber’s Paradox).

The results from actual star counts in various Galactic fields are:i. Bright stars are nearly uniformlydistributed between the pole andthe plane of the Galaxy, but faintstars are clearly concentratedtowards the Galactic plane.ii. Most of the light from the regionof the Galactic poles comes fromstars brighter than m ≈ 10, whilemost of the light from the Galacticplane comes from fainter stars(maximum at m ≈ 13).iii. Increments in log A(m) are lessthan the value predicted for auniform star density, no interstellarextinction, and all stars of the sameintrinsic brightness.

It implies that D(r) could decreasewith increasing distance (a featureof the local star cloud that couldvery well be true according to thework of Bok and Herbst), orinterstellar extinction could bepresent (or both!). The existenceof a local star density maximumis also confirmed by the star densityanalysis of McCuskey (right).

Actual star counts were done in the past using (m, log ) tables (magnitude, parallax), which were simple to use with experience. For each value of m, the entries reach a maximum at some value of log k. The summation of the entries for each column gives the values for the predicted counts. The values can be compared with actual star counts in a particular area, which are usually much smaller. They must be reduced by the values for the apparent density function (k) for each shell. It is therefore necessary to reconstruct the (m, log ) table including an estimated (k) function. A solution for the observed counts generally requires a number of iterations with a variable (k) function until a best match is obtained. Experience is particularly helpful. Once a solution for (k) is obtained, one still needs to know a(r) to obtain D(r) from the results. Such a(r) estimates can come from various sources, e.g. Neckel & Klare (A&AS, 42, 251, 1980).

An Example of a m-log Table.As tied to Van Rhijn’s luminosity function.

The use of star counts inside and around the Veil Nebula in Cygnus (part of the Cygnus Loop) to determine the distance to the dust cloud and the amount of extinction it produces at photographic (blue) wavelengths.

Well-recognized characteristics of the Galaxy:1. Gould’s Belt, consisting of nearby young stars (spectral types O and B) defining a plane that is inclined to the Galactic plane by 15 to 20. Its origin is uncertain. The implication is that the local disk is bent or warped relative to the overall plane of the Galaxy. This is not to be confused with the warping of the outer edges of the Galaxy.

2. An abundance gradient exists in the Galactic disk and halo, consistent with the most active pollution by heavy elements occurring in the densest regions of these parts of the Galaxy. See results below from Andrievsky et al. A&A, 413, 159, 2004 obtained from stellar atmosphere analyses of Cepheid variables.

The abundance gradient is also seen in the halo according to the distribution of globular clusters of different metallicity relative to the Galactic centre (below).

3. The orbital speed of the Sun about the Galactic centre is about 251±9 km s–1, as determined from the measured velocities of local group galaxies, as well as from a gap in the local velocity distribution of stars corresponding to “plunging disk” stars (Turner 2014, CJP, 92, 959). This fact is actually NOT “well recognized” by most astronomers.

4. The Galactic bulge is spheroidal, although some researchers believe it displays a boxy structure at infrared wavelengths suggestive of a central bar viewed nearly edge on. A mapping(right) of Milky Wayplanetary nebulae in Galactic co-ordinates(Majaess et al. MNRAS,398, 263, 2009) suggests amore spheroidal structuretypical of galaxies like NGC 4565 (top). Thenature of the Galacticbulge is still unclear. Thesurface brightness followsa de Vaucouleurs law.

13307.3

)(log

41

10ee r

r

I

rI

5. The Galaxy is a spiral galaxy. But does it have 2 arms or 4, and can it be matched by a logarithmic spiral? A “grand design” spiral pattern is not obvious in the plot of the projected distribution of Cepheids (points) and young open clusters (circled points) below (Majaess et al. 2009).

A schematic representationof what are considered tobe major spiral features.

How would you connectthe points?

Most recent studies considerthe Cygnus feature to be aspur or minor arm, and thePerseus feature is consideredto be a major arm!

There is an “Outer PerseusArm” in many deep surveys.It lies >4 kpc from the SunIn the direction of theGalactic anticentre.

6. The Galactic disk is warped, presumably from a gravitational interaction with the Magellanic Clouds. The warp is evident in 21cm maps of neutral hydrogen restricted (by radial velocity) to lie at large distances from the Galactic centre (below).

7. The Galaxy has a magnetic field that appears to be coincident with its spiral arms (or features), with the likely geometry of the magnetic field lines running along the arms. Weak fields of ~tens of Gauss are typically measured. The evidence for the presence of a magnetic field comes from the detection of interstellar polarization in the direction of distant stars (see below).

8. Note features in the textbook that are NOT included in the list:

spiral structure

the Milky Way’s central bar

3-kpc expanding arm

dark matter halo

evidence of dark matter

Can you understand why?

Kinematics of the Milky Way:The Galactic co-ordinate system is defined such that the Galactic midplane is defined by main plane of 21cm emission. The zero-point is defined by the direction towards the Galactic centre (GC), which is assumed to be coincident with Sagittarius A*.

The Galaxy’s rotation is observed to be clockwise as viewed from the direction of the north Galactic pole (NGP). Galactic co-ordinates are Galactic longitude, l, measured in the direction of increasing right ascension from the direction of the GC, and Galactic latitude, b, measured northward (positive) or southward (negative) from the Galactic plane.

The velocity system for objects in the Galaxy is defined by:Θ = Rdθ/dt, the velocity in the direction of Galactic rotationΠ = dR/dt, the velocity towards the Galactic anticentreZ = dz/dt, the velocity out of the Galactic plane.

The equations of motion are derived relative to the Local Standard of Rest (LSR), a fictitious object centred on the Sun and orbiting the Galaxy at the local circular velocity; the Sun orbits at a faster rate. The radial velocity of an object in the Galactic plane is given by:vR = Θ cos α – Θ0 cos (90°–l)

= Θ cos α – Θ0 sin l .where Θ is the circular velocity at distance R from the Galactic centre and Θ0 is the circular velocity at R0, the Sun’s distance from the Galactic centre.

By the Sine Law:

So .

Therefore,

since

Outside the Galactic plane theradial velocity becomes:

.

00

cos90sinsin

RRR

l

lR

Rsincos 0

lR

lRR

R

llR

RvR

sin

sin

sinsin

00

0

00

00

RRand0

0

0

blRvR cossin00

The observed tangential velocity of the object relative to the LSR is given by:vT = Θ sin α – Θ0 cos l (where vT is positive in the direction of Galactic rotation).But R sin α = R0 cos l – d, where d is the distance to the object.So and

These are the general equationsof Galactic rotation.

R

dl

R

R cossin 0

dlR

dR

lRR

R

lR

RdlR

RvT

cos

cos

coscos

00

0

00

0

000

If Ω decreases with increasing distance from the Galactic centre, then for any given value of l in the 1st (0° < l < 90°) and 4th (270° < l < 360°) quadrants, the maximum value of Ω occurs at the tangent point along the line-of-sight, i.e. at Rmin = R0 sin l. In that case, d = R0 cos l, so:

Rmin = R0 cos (90° – l) = R0 sin l .vR(max) = Θ(Rmin) – Θ0 sin l .

Approximations to the general formulae can be made for relatively nearby objects, where d << R0, in which case:

and .

But

And, for d << R0, R0 R ≈ d cos l .

So, for nearby objects in the Galactic plane vR becomes:

000

0

22

021

00RRR dR

dRR

dR

dRR

dR

dRR

ddR

dRRdd

R000

0

20

0

0

2

00

1

so,1

RdR

d

RdR

d

RdR

d

RRdR

d

dR

d

RR

or vR = Ad sin 2l = Ad sin 2l cos2 b, outside the plane, where:

is Oort’s constant A.

lddR

d

R

llddR

d

R

lldRdR

d

RR

lRdR

d

RRRRv

R

R

R

RR

2sin

cossin

cossin1

sin1

0

0

0

0

0

021

0

0

20

0

00

20

0

000

00

021

RdR

d

RA

For the tangential velocity:

or vT = Ad cos 2l + Bd, where:

is Oort’s constant B.

ddR

d

RlAd

dR

lddR

d

R

dR

lddR

d

R

dlRdR

d

RRRRv

R

R

R

RT

0

0

0

0

0

021

0

0

0

021

0

02

0

0

020

0

000

2cos

2cos1

cos

cos1

00

021

RdR

d

RB

Expectations from theequations of motion arethat radial velocities(solid line) and propermotions(dashed line)for nearby stars shoulda double sine wave variation with Galacticlongitude. They do.

The proper motionrelationship is amodified version ofthe vT relation:

74.4

2cos

so2cos

BlA

BlAdv

l

T

In the 1st Galactic quadrant (0° < l < 90°) stars are receding from the Sun.In the 2nd Galactic quadrant (90° < l < 180°) stars are approaching from the Sun.In the 3rd Galactic quadrant (180° < l < 270°) stars are receding from the Sun.In the 4th Galacticquadrant (270° < l < 360°)stars are approachingthe Sun.

Note that:

Also:

So evaluation of Oort’s constants permits one to specify the velocity gradient and local vorticity of local Galactic rotation. It can also provide a solution for R0 if the local rotational velocity can be found.

Can that be done?

.andso

and

0021

0

021

0

021

0

00

ABdR

dRA

dR

d

RB

dR

d

RA

R

RR

BAdR

dBA

R R

0

and0

00

Use of the equations for Galactic rotation is predicated upon the establishment of an accurate value for the Sun’s motion relative to the LSR. That is not an easy chore because of the nature of stellar orbits in the Galaxy, which are neither circular nor elliptical, but more like a roseate pattern.

The general direction of theSun’s motion relative tonearby stars is readily detectedin stellar proper motions, andlies roughly towardsRA = 18h and Dec = +30°,i.e. towards the constellationof Hercules.

A typical orbit for a star in the Galaxycan be pictured as epicyclic motion offrequency κ superposed on circularmotion of frequency Ω. When κ = 2Ωthe orbit is an ellipse.Since κ(R) ≠ 2Ω(R) in most cases,the orbits are roseate, somethinglike what is produced by a spirograph.Cyclical motion perpendicular tothe Galactic plane also occurs.

The random motion of nearbystars relative to each other produces the observed velocitydispersions for various stellargroups.

Stars in the Galactic bulge appear to exhibit no preferreddirection or orbital inclination,so define a spheroidal distribution.

The Local Standard of Rest: In the gravitational field of the Galaxy, stars near the Sun orbit the Galactic centre with velocities that are close to the local circular velocity Θc. A star at the Sun’s location that describes roughly a circular orbit about the Galactic centre has velocity components:

(Π, Θ, Z) = (0, Θc, 0) km/s.A velocity system centred on such a fictitious object is used to define the LSR. That is, the LSR is defined by an axial system aligned with the Π, Θ, and Z axes and with an origin describing a circular orbit about the Galactic centre with a velocity Θc. Nearby stars have peculiar velocities relative to the LSR described by:

u = Π – ΠLSR = Π,v = Θ – ΘLSR = Θ – Θc ,w = Z – ZLSR = Z .

The peculiar velocity of the Sun is therefore given by: (u, v, w) = (Π, Θ–Θc, Z) . The velocity of any star with respect to the Sun has three components:i. a peculiar velocity relative to the star’s LSR,ii. the peculiar velocity of the Sun with respect to the Sun’s LSR, andiii. the differential velocity of the LSR at the star with respect to the solar LSR resulting from differential Galactic rotation (usually negligible). Since (iii) is indeed negligible for d ≤ 100 pc, the observed velocity of a star relative to the Sun is given by the velocity vector (U*, V*, W*), where:

U* = u* – u = Π* – Π , V* = v* – v = Θ* – Θ , W* = w* – w = Z* – Z .

For any particular group of stars belonging to the disk and having nearly identical kinematic properties, one can define a kinematic centroid of their velocities by:

For disk stars not drifting either perpendicular to the Galactic plane or in the direction of the Galactic centre, it is reasonable to expect that:

N

ii

N

ii

N

ii

wN

w

vN

v

uN

u

1**

1**

1**

1

1

1

0and0 ** wu

However, <v*> ≠ 0, since a typical group of stars lags behind the solar LSR. Stars chosen spectroscopically include objects of various origins, unless the group is so young that the stars have not had time to travel far from their places of formation. The increasing density gradient in the disk of the Galaxy towards the Galactic centre implies that the majority of stars in the solar neighbourhood originated at points lying on average closer to the Galactic centre, i.e. most are currently near apogalacticon. Since the apogalacticon velocity of stars in elliptical orbits is less than the local circular velocity c, any mix of elliptical orbits for nearby stars implies the majority travel at less than c in the direction of the Sun’s orbital motion. It follows that locally-defined kinematic groups of stars should, in general, tend to lag behind the LSR motion. Asymmetric drift is well-observed in such groups, and must be taken into account in any determination of the Sun’s LSR velocity.

From both qualitative and mathematical arguments the asymmetric drift for any group of stars must depend directly upon the nature of the orbits for the stars in the group. For stars in nearly circular orbits no asymmetric drift is expected, while for stars having a mix of very eccentric orbits the asymmetric drift should be fairly large. A group of stars having the latter properties should also exhibit a fairly large dispersion in the component of their orbital motions directed along the line-of-sight to the Galactic centre, the velocities, whereas stars in strictly circular orbits have no such component of their orbital motion. The correlation of asymmetric drift with the dispersion in velocities, 2, proves to be a valuable tool for determining the exact parameters for the Sun’s LSR velocity.

See the solution on the next slide, from work by the instructor (Turner 2014, Can.J.Phys., 92, 959.)

u = +11.1 ±0.5 km/sv = +4.4 ±0.6 km/sw = +7.3 ±0.2 km/s

corresponding toS = 14.0 km/s towardslLSR = 21°.5, bLSR = +31°.6.

Once the solar LSR velocity is established, it is possible to establish likely distances to Galactic objects from their radial velocities, i.e. using the equations of Galactic equations of motion.

But all such efforts have used an incorrect solar LSR velocity.

The rotation curve of the Galaxy is observed to be flat like those of other disk galaxies, although perhaps not as irregular as the solution obtained by Clemens. Note the rigid body rotation near the Galactic centre.

Beginning in the 1970s a model for the propagation of spiral features was proposed using a standing logarithmic spiral density wave (“density wave model”), but it rarely gives good agreement with what is observed for our Galaxy’s spiral characteristics, except for somefeatures.

The spiral characteristicsof galaxies like M51 canoften be linked to gravitational interactionwith a close companioncombined withdifferential rotation inthe galactic plane. Couldthat also be the case forthe Milky Way?

Methods of Establishing Distances:

i. Stellar parallaxes (trigonometric parallaxes, statistical parallaxes, and secular parallaxes are used).

ii. Cluster parallaxes (see lab manual for 1st year).

iii. Spectroscopic parallaxes (calibrated from i and ii).

iv. Radial velocities and proper motions (approximate only). Used for moving clusters.

v. Eclipsing binary light and radial velocity solutions.

The Galactic Centre:All the evidence indicates the existence of a supermassive object at the centre of the Galaxy, denoted as Sgr A*. It is the source of X-rays and a cluster of interesting stars orbiting it at high velocities. The large mass (~4106 M) and compact nature of the object implies it is a black hole, and appears to be typical of what is found at the centres of many nearby massive galaxies.

Orbits of GC starsaccording to Ghez et al.(2004).

Example Problem. Find the mass of the Galaxy given the local circular velocity of 251 km/s at the Sun’s location roughly 8.5 kpc from the Galactic centre.Solution.Use Kepler’s 3rd Law in Newtonian fashion,i.e. (MG + M) = a3/P2, for a in A.U. and P in years.For an orbital speed of 251 km/s and orbital radius of 8.5 kpc the orbital period is:

The semi-major axis is:

So the mass of the Galaxy is:

yr100805.2

s/yr103.1558km/s251

km/AU10496.1AU/pc206265pc85002

km/s251

2

8

7

8

R

P

AU107533.1AU/pc206265pc8500 9a

Sun

11Sun28

39

G 102451.1100805.2

107533.1MMM

So ~1011 M is derived for the mass of the Galaxy internal to the Sun. If the orbital velocity curve is flat to ~16 kpc from the Galactic centre, then one can redo the calculations to find that ~21011 M is derived for the mass of the Galaxy internal to ~16 kpc from the centre. Where did the extra ~1011 M come from, or is it proper to apply Kepler’s 3rd Law in situations like this? Recall that it applies to the case of a two-body situation only, not to a multi-body situation.Mass/Light Ratios, M/L:The mass-luminosity relation varies roughly as L ~ M4 (M3 for cool stars), so the mass-to-light ratio should vary as M/L ~ 1/M3 or 1/M2. The typical star near the Sun is a cool M-dwarf with a mass of only 0.25 M or less, implying a typical mass-to-light ratio for our Galaxy of ~16. Since most stars are probably less massive than that, the actual mass-to-light ratio for the Galaxy could be in excess of ~25 or so.

i. Stellar Parallaxesi. Stellar ParallaxesStellar parallax is the displacement in a star’s position in the sky with respect to the stellar background arising from the orbital motion of the Earth about the Sun. Denoted by the angle π, it is defined to be the angle subtended by 1 A.U. (the semi-major axis of the Earth’s orbit) at the distance of the star. In practice one can observe the annual displacement of a star resulting from Earth’s orbit about the Sun as 2π. In the skinny triangle approximation, 1 A.U. is the chord length subtended at the star by the angle π, measured in radians. In this case, the chord length ≈ the arc length subtended at the star = dπ, where d is the distance to the star. Since π < 1" for all stars, the equation can be written as an equality, i.e. 1 A.U. = dπ, or:

In order to take advantage of trigonometric stellar parallaxes measured in arcseconds, it is useful to define a unit of distance corresponding to that angle. Thus, the parsec is the distance to an object when 1 astronomical unit (A.U.) subtends an angle of 1 arcsecond:

Since all stars should exhibit parallax, measured values (trigonometric parallaxes) are of two types:πrel = relative parallax, is the annual displacement of a star measured relative to its nearby companionsπabs = absolute parallax, is the true parallax of a star, or what is measured for itIn the past, all parallaxes were relative parallaxes, and were adjusted to absolute via:πabs = πrel + correction

The definition of parallax and parsec = distance at which one Astronomical Unit (A.U.) subtends an angle of 1 arcsecond.

Note, by definition:

1 pc = 206265 A.U.

How parallax is measured.

The complication of the parallactic ellipse. In practice all parallaxes are measured using only points near maximum parallax displacement.

The concept of relative parallax is also illustrated.

The estimated frequency of an average 11th magnitude star (upper) and an average 16th magnitude star (lower) as a function of distance (visual magnitudes dashed line, photographic magnitudes solid line). 11th magnitude stars peak for a distance of ~250 pc, corresponding to a correction factor of 0".004. 16th magnitude stars peak for a distance of ~800 pc, corresponding to a correction factor of 0".00125.

The distance to any star or object with a measured absolute parallax is given by:

The relative uncertainty in distance is given by:

Typical corrections to absolute were +0".003 to +0".005 for old refractor parallaxes, but are roughly +0".001 for more recent reflector parallaxes from the U.S. Naval Observatory. Space-based parallaxes from the Hipparcos mission are all absolute parallaxes; they were measured relative to all other stars observed by the satellite. Their uncertainties are less than 1 mas (milliarcsecond), i.e. <0".001, although systematic errors of order 0".001 or more are suspected in many cases. The Gaia mission will measure parallaxes in similar fashion.

Example: What is the distance to the star Spica (α Virginis), which has a measured parallax according to Hipparcos of πabs = 12.44 ±0.86 mas?

Solution.

The distance to Spica is given by the parallax equation, i.e.

The uncertainty is:

The distance to Spica is 80.39 ±5.56 parsecs.

ii. Cluster Parallaxesii. Cluster ParallaxesThe primary purpose of photometry is to obtain information equivalent to spectroscopic data in a smaller amount of observing time. It is also a highly efficient method of studying variable stars. Stellar continua vary with spectral type, and different photometric systems are designed to sample selected portions of such continua either for the equivalent of spectral classification or for estimating third dimensions for stars, e.g. metallicity, etc. Basically any photometric system must be capable of determining a star’s spectral class, corrected for interstellar extinction, without serious problems from luminosity differences, or population effects. Systems are classified on the basis of the widths of the wavelength bands used to define them. Broad band systems use passbands from 300 to 1000Å wide (e.g. the UBV system), intermediate band systems 100 to 300Å wide (e.g. the Strömgren uvby system), and narrow band systems less than 100Å wide (e.g. Hβ photometry).

ii. Cluster Parallaxesii. Cluster ParallaxesTypically such parallaxes (or distances) are derived from photometry of stars in a cluster. Many photometric systems exist, but a knowledge of the Johnson UBV system helps to understand how other systems function.For early-type stars, the Balmer discontinuity (λ3647) and Paschen jump (λ8206) result from the opacity of H, and are modified by electron scattering in hot O-type stars and H– (negative H ion) opacity in cool GK stars. The Balmer discontinuity is very sensitive to both Teff and log g, so many systems use filter sets to isolate stellar continua on either side of it.

Johnson’s UBV System.The UBV system is designed to give magnitudes that are similar to those on the old International photographic and photovisual system, with a magnitude added on the short wavelength side of the Balmer discontinuity in order to give luminosity discrimination.

Filter λeff(nm) Δλ(nm)U 365 68B 440 98V 550 89

U−B is sensitive to gravity and reddening.

B−V is sensitive to temperature andreddening.

The diagram plotted here is the standard two-colour (or colour-colour) diagram for the UBV system. Plotted are the observed intrinsic relation for main-sequence stars of the indicated spectral types, the intrinsic locus for black bodies radiating at temperatures of 4000K, 5000K, 6000K, 8000K, etc., and the general effect on star colours arising from the effects of interstellar reddening.

Intrinsic colours vary according to both the temperature and luminosity of stars, roughly as indicated here.

Line blanketing affects the observed colours of late-type stars, low Z stars exhibiting an ultraviolet excess relative to high Z stars. The relationship is calibrated relative to the Hyades relation at BV = 0.60.

Blanketing lines vary in slope with spectral type, Blanketing lines vary in slope with spectral type, which is why ultraviolet excesses are normalized which is why ultraviolet excesses are normalized to to BV = 0.60.

Examples of star clusters with uniformly reddened stars.

Examples of star clusters exhibiting small (left) and large (right) amounts of differential reddening.

The reddening law can be described by:

where X is the slope and Y the curvature. Observations indicate that Y = 0.02 ±0.01 while X varies with R locally. See Turner 2014, Can.J.Phys., 92, 1696. In the Galactic plane X can vary from 0.62 to 0.83, outside the plane perhaps only X = 0.83 is applicable.

UBV intrinsic colours are presently tied to models for non-rotating stars (left), but differential reddening still dominates the observed colours (right).

The zero-age main sequence, as constructed from overlapping the main sequences for various open clusters, all tied to Hyades stars using the moving cluster method.

The present-day zero-age main sequence (ZAMS) for solar metallicity stars.

Why it is important to correct for differential reddening in open clusters: removal of random scatter from the colour-magnitude diagram.

Differences between the core and halo regions of open clusters. Note also the existence of main-sequence gaps.

Typical open cluster colour-magnitude diagrams corrected for extinction. Note the pre-main sequence stars in NGC 2264 (right) and the main-sequence gaps in NGC 1647 (left).

Many young clusters are also associated with very beautiful H II regions.

ZAMS fitting can be done by matching a template ZAMS (right) to the unreddened observations for a cluster (left). The precision is typically no worse than ±0.05 (~2.5%).

Main-sequence fitting can be good to a precision of ±0.1 magnitude (±5%) in V0MV (or better) after dereddening.

An example of the importance of ZAMS fitting for open clusters. The luminous and peculiar B2 Oe star P Cygni belongs to an anonymous open cluster, so its reddening and luminosity can be found using cluster stars.

The reddening (left) and ZAMS fit (right) for the P Cygni cluster, an example of the usefulness of cluster studies.

Kinematic Method of Calibration ― Moving Clusters

Open clusters are ideally suited to the calibration of stellar luminosities since they contain such a wide variety of stars of different spectral types and luminosity classes. The general method of using a calibrated zero-age main-sequence (ZAMS) to derive cluster distances is outlined by Blaauw in Basic Astronomical Data. However, the necessary zero-point calibration involves the independent determination of the distance to a nearby cluster whose unevolved main-sequence stars serve to establish the MV versus (B–V)0 (or spectral type) relation over a limited portion of the ZAMS. The distance to such a zero-point cluster can be derived using the moving cluster method, or one of its many variants.

The requirements for use of the moving cluster method are a sizable motion of the cluster both across the line-of-sight and in the line-of-sight. The technique is most frequently discussed for the case of the Hyades star cluster, which is the standard cluster used for the construction of the empirical stellar ZAMS. The constellations Ursa Major and Scorpius also contain moving clusters, and the Pleiades have attracted considerable attention as a moving cluster. In general, all stars in a moving cluster move together through space with essentially identical space velocities (their peculiar velocities are invariably much smaller by comparison). Once the direction to the cluster convergent point (divergent points are equivalent!) is established on the celestial sphere, the geometry of the situation is established. In particular, the angle between the star’s space velocity and radial velocity is fixed.

Motion of Hyades members.

The geometryof proper motion.

Moving cluster geometry.

The Hyades.

A star’s space velocity is given by: v2 = vT2 + vR

2, and vT = 4.74 μd. But, vR = v cos θ, and vT = v sin θ. Thus, the distance to a star, d*, is given by:

Once the radial velocity, vR (in km/s), of a moving cluster star, its proper motion μ(in "/yr), and angular distance from the cluster convergent point, θ, are known, its distance (in pc) can be obtained from the above equation. The dispersion in radial velocity for stars in an open cluster is typically quite small, no larger than ±1 to ±2 km/s. Therefore, for a cluster of stars of common distance, the ratio (tan θ)/μ must remain constant across the face of the cluster. It means that those stars lying closest to the cluster convergent point have the smallest proper motions, while those lying furthest from the convergent point have the largest proper motions.

For nearby clusters that feature allows one to determine the relative distances to stars in the cluster by comparing the individual stellar proper motions, μ*, with the mean cluster proper motion, μC, at that value of θ, i.e. using:

The technique has been used extensively for stars in the Hyades cluster, which has a line-of-sight distance spread on the order of 10% or more of its mean distance.

The moving cluster method can also be pictured in a more general manner, as noted by Upton (AJ, 75, 1097, 1970). In this case, the motion of a cluster like the Hyades away from the Sun results in an apparent decrease in the cluster’s angular dimensions, even though its actual dimensions are unchanged. By geometry and the assumption that the cluster’s actual diameter, D, is relatively constant, we have D ≈ rθ, where θ is the cluster’s angular diameter. The cluster’s distance is denoted r to avoid confusion with the derivative notation.

In practical terms:

The terms evaluated:

Since proper motion gradients are easier to measure accurately than the location of the convergent point, the method is somewhat superior to the convergent point method. The cluster convergent point is not lost by the method, since it is located by the points where the proper motion gradients become zero. Upton noted, however, that it was necessary to transform the α, δ motions of cluster stars on the celestial sphere into their Cartesian (flat surface) equivalents in order to obtain meaningful proper motion gradients. For the Hyades cluster, he also found it necessary to account for line-of-sight distance spread.

A further modification to the general method can be made using the original equations given previously, namely:

i.e. given the mean proper motion of a moving cluster, one can find its distance from the gradient in radial velocities across the face of the cluster.

The last technique is very difficult to apply, since it requires extremely accurate radial velocities for cluster members that are not biased by the systematic effects of binary companions. The method has been applied in a very sophisticated fashion to the Hyades cluster by Gunn et al. (AJ, 96, 198, 1988), with fairly good results. 1990 estimates for the distance modulus of the Hyades cluster by all of the various methods used (including trigonometric parallaxes) lie in the range 3.15 to 3.40, or 42.7 pc to 47.9 pc.

iii. Spectroscopic Parallaxesiii. Spectroscopic ParallaxesThese are normally done using tables of absolute magnitude as a function of spectral type, e.g. Turner ZAMS file.

Example: How distant is Spica (α Virginis), a B1 III-IV star with apparent visual magnitude V = 0.91, given that B1 III-IV stars typically have an absolute magnitude of MV = –4.1 ±0.3 (Turner ZAMS)?

Solution.The distance modulus for Spica is given by:

Thus:

The uncertainty is:

The spectroscopic parallax distance to Spica is 100.5 ±13.9 parsecs. Note the small disagreement with the star’s Hipparcos parallax distance of 80.39 ±5.56 parsecs.

iii. Statistical Parallaxesiii. Statistical ParallaxesTables of absolute magnitude for stars as a function of spectral type and luminosity class are constructed in a variety of ways. It is not unusual for several different techniques to be used in the compilation of one table, including trigonometric parallax, cluster parallax, and, finally, statistical parallax.

Recall the relation for tangential velocity,

for the proper motion μ in "/yr and the parallax π in arcseconds. It can be rewritten as:

There is a statistical method of establishing the mean parallax for a group of stars having common properties by making use of the data for their positions in the sky, their proper motions, and their radial velocities. The resulting statistical parallax for such a group is:

where the angled brackets denote mean values.

In general, a randomly distributed group of stars should have <vT> ≈ <vR>, i.e. one component of space velocity should be similar to another. Thus,

That is a simplification of the true method. In practice, the true situation is complicated by the Sun’s peculiar motion relative to the group. Denote (upsilon) = the -component in the direction of the solar antapex (i.e. the -component resulting from the Sun’s motion), (tau) = the -component perpendicular to the direction of the solar antapex (i.e. the -component resulting from the star’s peculiar velocity), and (lamda) = the angular separation of the star from the solar apex. Then:

is the secular parallax, the parallax inferred from the Sun’s space motion, where vSun is the solar motion relative to the local standard of rest (LSR), while:

is the statistical parallax, the parallax inferred from the statistical properties of stellar space velocities.Upsilon components apply when the Sun’s motion dominates group random velocities; tau components apply when group motions dominate. Both techniques have been applied to B stars and RR Lyrae variables, which are too distant for direct measurement of distance by standard techniques. Both classes of object are also relatively uncommon in terms of local space density, yet luminous enough to be seen to large distances. Because of general perturbations from smooth Galactic orbits predicted in density-wave models of spiral structure, the assumptions used for statistical and secular parallaxes may not be strictly satisfied for many statistical samples of stars.

An example for the B3 V stars, as published by Turner 2012, Odessa Astron. Publ., 25, 29 ― was a SMU lab.

Sample QuestionsSample Questions1. The star Delta Tauri is a member of the Hyades moving cluster. It has a proper motion of 0".115/year, a radial velocity of 38.6 km/s, and lies 29º.1 from the cluster convergent point.a. What is the star’s parallax?b. What is the star’s distance in parsecs?c. Another Hyades cluster member lies only 20º.0 from the convergent point. What are its proper motion and radial velocity?

Answer: The relationship applying to all stars in a moving cluster is:

With the values given previously, the distance is found from:

The star’s parallax is:

And its space velocity is:

The other star must share the same space velocity as Delta Tauri, so its proper motion and radial velocity are:

2. The open cluster Bica 6, at l = 167º, has a radial velocity of 57 km/s, or 48 km/s relative to the LSR. Its distance from main-sequence fitting is 1.6 kpc. Is the cluster’s motion in the Galaxy consistent with Galactic rotation?

Answer. See predictions at right.

The cluster shouldhave a LSR velocityof −9 km/s if itcoincided withGalactic rotation.The cluster’s actual LSR velocity of 48 km/s is therefore completely inconsistent with Galactic rotation. Can you think of a reason why?

3. Interstellar neutral hydrogen gas at l = 45º has a radial velocity of +30 km/s relative to the LSR. What are its distance from Earth and the Galactic centre if R0 = 8.5 kpc and θ0 = 251 km/s?

Answer. Recall:

For a flat rotation curve, θ = θ0 = 251 km/s, so:

lRR

RlRvR sinsin0

0000

kpc2709884.70.1375328

kpc1

kpc1375328.0km/s/kpc99.45.8

251

km/s251

11

km/s/kpc99.445sin5.8

30

sin

1

00

0

R

R

lR

v

RRR

For plane triangles the cosine law is:

Here:

Angle A = 45ºSide b = 8.5 kpc = R0

Side a = 7.27 kpc = RSide c = distance d to cloud

So:

Yielding:

Which is a quadratic equation with solution:

Abccba cos2222

45cos5.825.827.7 222 dd

03971.19020815.122 dd

kpc10.10andkpc1.92

givingkpc,1799507.80104075.62

3971.194020815.12020815.12

21

2

d

d