Astronomy 218O u r P l a c e i n t h e

M i l k y W a y

Orbiting SunThe Sun’s proper motion relative to the Galactic center, μ” = 5.8 milliarcseconds yr−1 = 2.8 × 10−8 rad yr−1 and its distance from the Galactic center, R0 = 8 kpc, indicate an orbital velocity of 𝑣0 = 225 kpc Gyr−1 = 220 km s−1.

The Sun’s orbital period around the Galactic Center is then

= 0.22 Gyr

Maintaining this orbit requires that the mass inside of the Sun’s orbit must be

= 9.3 × 1010 M☉

This 93 billion M☉ for the inner galaxy compares to 23 billion L☉, indicating that the galaxy’s M/L > M☉/L☉.

1 > 31౮1 > 61/4 LQD336 LQD (ZSѝ2 ғ ౬Ӥ31

ౌӤ , ౌ౦ > ౚ4 3 > )2/76 · 21:"6*4

)3/3 · 219ZS*3 ౌӤ

Measurements of the position and motion of gas clouds throughout the Milky Way reveals that the solar environment includes several spiral arms.

Scutum-Centaurus Arm Norma Arm Sagittarius Arm Central Bar Orion Spur Perseus Arm Outer Arm

Spiral Arms

Galactic CoordinatesThe Galactic plane defines a galactic coordinate system, in the same way the Equator defines the terrestrial latitude and longitude.

The line from the Sun to Sagittarius A* defines the zero of galactic longitude, ℓ, which increases counter-clockwise viewed from the north.

The galactic latitude, b, is measured from the plane of the galaxy to the object using the Sun as vertex.

One interpretation of the spiral arms would be the winding up of an initial linear pattern.

However, the differential rotation of the galaxy would have wound up the spiral pattern so tightly that it would effectively disappear, contrary to observations.

Thus the spiral arms are not tied to disk material.

Differential Rotation

Density WavesInstead spiral arms are density waves, similar to the over density of cars in a traffic jam. The jam persists even though individual cars move steadily through it.

Such density waves persist long after the event that triggered them is forgotten.

In the case of galaxies, the waves are regions of locally higher density, which accelerate stars into them and decelerate them as they leave, perpetuating the over density.

Not just the density of stars becomes enhanced as they move through the spiral arms, but clouds of gas and dust are actually compressed.

Their increased density triggers star formation.

This affects the propagation of spiral arms as the formation of stars leads to supernovae and stellar winds which further compress the gas and dust, leading to waves of ongoing star formation.

Spiral Star Formation

It is this ongoing star formation which makes the spiral arms so visible. Newly formed massive stars, and their attendant HII regions and supernova remnants, line the spiral arms, shining bright and dying before leaving the arm.

In contrast, older stars move through the arms and have a smoother angular distribution.

As one moves to progressively longer wavelengths, the spiral arms become less noticeable. The actual over-density is small, only a few %.

Spiral Lights

Orion SpurThe Orion Spur is also referred to as the Orion-Cygnus Arm or the Orion Arm.

The Sun sits near the center of the Local Bubble, close to the inner rim of the arm, Facing in the leading direction points in the direction of Deneb and the constellation Cygnus. Facing in the trailing direction points toward Betelgeuse and the constellation Orion.

We can measure the relative motions of nearby stars in two steps.

The space velocity 𝑣 can be decomposed into 2 components, relative to the line of sight.

The radial component, 𝑣r, along the line of sight, can be measured by doppler shift.

The tangential (or transverse) component, 𝑣t, across the line of sight, can be measured by proper motion, μ.

To achieve the desired accuracy, these measurements must be corrected for the Earth’s orbital velocity (𝑣orb ≈ 30 km s−1) and the Earth’s rotational velocity (𝑣rot ≈ 0.5 km s−1), as well as any binary motions of the distant star.

Local Stellar Motions

𝑣𝑣r

𝑣t

dμ

Examination of the radial velocities of the 40 stars closest to the Sun, d < 5 pc, results in this figure.

The number of redshifted stars approximately equals the number of blueshifted stars.

Kapteyn’s star, at a distance of 3.9 pc, is a noticeable outlier, with a velocity of 250 km s−1 away from the Sun.

Ignoring Kapteyn’s star, the root-mean-square velocity dispersion is ~ 35 km s−1 in radial velocity.

Local Radial Velocity

6.3. STELLAR MOTIONS 137

Figure 6.11: Components of a star’s velocity relative to the Sun.

itself. This can be done by averaging the radial velocity of the star over anentire orbital period.

The radial velocity of forty stellar systems within 5 parsecs of the Sun isplotted in Figure 6.12. Notice a few interesting results:

1 2 3 4 5!100

0

100

200

d [pc]v r [

km/s

]

Figure 6.12: Radial velocity of stars within 5 pc of the Sun.

• The radial velocities show approximately equal numbers of blueshifts(vr < 0) and redshifts (vr > 0).

• There is one notable outlier on the plot: Kapteyn’s star, alias CD-45!1841, alias Gliese 191, which is moving away from the Sun with

Proper MotionMeasuring the motion of a star across the line of sight is much more time-consuming than measuring the radial velocity via the Doppler Shift. Time must pass for the star to move perceptively on the sky.

In the small angle limit, the proper motion μ = 𝑣t/d. If μ is measured in rad yr−1 and d in parsecs,

μ is more naturally measured in arcseconds yr−1, with 1 radian = 206,265 arcseconds. Comparison of 𝑣t to 𝑣r favors km s−1 with 1 pc yr−1 = 976,582 km s−1. Simplifying

౮౬QD ZSѝ2 > ਁ

QD ਂ ਁ SBE ZSѝ2 ਂ

ਁ ౮౬LN Tѝ2 ਂ > 5/85 ਁ

BSDTFD ZSѝ2 ਂ ਁ QD ਂ > 5/85 ਁ ͖͖ ਂ

Examination of the tangential velocities of the 40 stars closest to the Sun results in this figure.

The red line marks μ = 5 arcsec yr−1.

Again Kapteyn’s star shows it’s uniqueness with a large ʋt producing a large proper motion, μ = 8.7 arcsec yr−1

Barnard’s star, despite its smaller tangential velocity, has the highest proper motion, μ = 10.4 arcsec yr−1.

61 Cygni, μ = 5.2 arcsec yr−1, has the highest proper motion of any star visible to the naked eye.

Local Tangential Velocity

140 CHAPTER 6. OUR GALAXY

For Barnard’s Star, as an example,

vt = 4.74!

10.358

0.547

"

km s!1 = 89.8 km s!1 . (6.24)

The tangential velocity of stellar systems within 5 parsecs of the Sun is shownin Figure 6.14. The high proper motion of Barnard’s Star is due both to the

1 2 3 4 50

50

100

150

d [pc]v t [k

m/s

]

Figure 6.14: Tangential velocity of stars within 5 pc of the Sun. Stars abovethe dotted line have µ"" > 5 arcsec yr!1.

fact that it is very close to us and to the fact that its tangential velocity ishigher than the average for nearby stars.

If the average tangential velocity vt doesn’t vary with distance from theSun, then on average, nearby stars will have a higher proper motion thanmore distant stars. One way to search for nearby stars, if you don’t have thetime or patience to measure parallaxes for every star in the sky, is to startby looking at stars with high proper motion. The history books state that inthe year 1838, Friedrich Wilhelm Bessel measured the parallax of 61 Cygni.What they sometimes don’t tell you is why Bessel chose that star to observe:it is, after all, a humble fifth magnitude star. Bessel, in fact, chose 61 Cygnibecause it has the highest proper motion of any star visible to the naked eye,µ"" = 5.2 arcsec yr!1.12 Of course, any star, no matter its distance, will have

12The unusual nature of Kapteyn’s Star was first recognized in the year 1897, when thisotherwise boring M dwarf was found to have a proper motion of nearly 9 arcseconds peryear. At the time, this was the largest proper motion known, defeated only when E. E.Barnard measured the proper motion of Barnard’s Star in 1916.

Combining the radial and tangential velocities for the 40 closest stars reveals that 39/40 belong to a group with a mean relative velocity 𝑣 ~ 50 km s−1.

These velocities will rearrange their proximity to the Sun on a timescale of t ~ 5 pc / 50 pc Myr−1 ~ 0.1 Myr. For example, Gliese 710, currently 19 pc away, will pass within ⅓ pc in 1.4 Myr, brightening from mv = 9.7 to 0.9.

Kapteyn’s star, with twice the velocity of any other nearby star, is in a class by itself, the nearest halo star.

Local Space Velocity

6.4. LOCAL STANDARD OF REST 141

zero proper motion if it’s moving straight toward us or straight away from us.An example of a nearby star with small proper motion is Gliese 710, whichhas vr = !13.9 km s!1 and µ"" = 0.014 arcsec yr!1. At the moment, Gliese710 is over 19 parsecs away from us, and has mV = 9.7. In 1.4 Myr, however,is will be only 0.34 parsecs away (about a fourth the present distance toProxima Centauri) and have mV = 0.9.

Putting it all together, the total speed relative to the Sun,

v = (v2r + v2

t )1/2 , (6.25)

is called the “space motion” or “space velocity”. A plot of the space motionfor nearby stars (Figure 6.15) reveals that Kapteyn’s Star again stands out

1 2 3 4 50

100

200

300

d [pc]v

[km

/s]

Figure 6.15: Space velocity of stars within 5 pc of the Sun. The outlier atd " 3.9 pc, v " 300 km s!1 is Kapteyn’s star.

like the proverbial sore thumb. Its space motion of v " 300 km s!1 is twicethat of Barnard’s Star, the next speediest star in the solar neighborhood. Theaverage space motion of stars within 5 parsecs of the Sun is v # 50 km s!1 #50 pc Myr!1. This indicates that the list of stars within 5 parsecs of us willbe thoroughly revised on timescales t # 5 pc/50 pc Myr!1 # 0.1 Myr.

6.4 The Local Standard of Rest

If the disk of our galaxy were perfectly orderly, with all the stars on exactlycircular orbits in the same plane, it would be simple to compute the expected

Local Standard of RestThe aspherical and time varying distribution of mass within the Galaxy prevents stars from moving in closed elliptical orbits. Instead, stars follow complex rosette patterns.

These patterns make actual stellar orbits a poor basis for a coordinate system.

We create an idealized reference frame, the Local Standard of Rest, which moves in a circular orbit at ʋ0 = 220 km s−1 from the Sun’s location at R0 = 8 kpc.

Coordinates relative to the LSR are (R,θ,z), with the Sun at (R0, 0, 0 kpc). Velocities are (Π,Θ,Z), with 𝑣LSR = (0, 𝑣0, 0).

Sun’s Peculiar MotionThe LSR is moving steadily in the direction of Cygnus.

But, the LSR is not the orbit of the Sun. Using a statistical analysis of the radial velocities of neighboring stars, we can determine the Sun’s motion. Relative to the LSR, the Sun has a peculiar velocity (𝑢⊙, 𝑣⊙, 𝑤⊙)

= (Π−Π⊙,Θ−Θ⊙,Z−Z⊙) = (−10.4, 14.8, 7.3) km s−1

Thus the Sun is moving inward (toward the Galactic Center) at 10 km s−1, forward (faster than the LSR in the azimuthal direction) at 15 km s−1 and northward (toward the north Galactic pole) at 7 km s−1.

This places the Sun’s apex or direction of motion toward the constellation Hercules and its antapex, opposite direction from the apex, toward the constellation Columba.

Combining tangent point method observations toward the Galactic center with stellar observations in the outer Galaxy reveals the rotation curve of the Galaxy.

The expectation was that beyond a radius that contained the mass of the galaxy, the velocity should diminish with radius following Kepler’s law.

Instead mass continues to increase with radius, well into the region where the light is exponentially declining. Something dim but massive lurks in the Galaxy.

Galactic Rotation Curve

M ∝ R

M(2R0) ~ 3 M(R0)

The visible structure of the Galactic Disk extends ~15 kpc from the center.

Beyond this radius, atomic gas (molecular gas is found for R≤10 kpc) and halo stars map the “missing” mass, whose density is ∝ R−2.

But the mass is present, it is the light that is missing.

What could the “dark matter” be? Many suggestions have been raised; red dwarves, white dwarves, brown dwarves, stellar-mass black holes, elementary particles, ….

Dark Matter

8 kpc

15 kpc

One possibility is that dim red dwarfs are even more numerous that suspected. Many searches, most recently by Hubble, have looked for such red dwarfs, but turned up too few to account for dark matter.

Our best accounting of the full stellar population yields M/L ~ 10 while the rotation curve indicates M/L ~ 50.

Not Red Dwarfs

Our study of stars suggests several other candidates for dim but massive objects, black holes, neutron stars, white dwarves and brown dwarves.

This very dimness makes them difficult to observe directly. However, gravitational lensing can allow us to directly observe their mass by bending more light into the line of sight.

Observations have found numerous MACHOS (Massive Compact Halo Objects), mostly low-mass white dwarves, but only ~20% of the missing mass.

Not Machos

Not NeutrinosThe universe is in fact flooded with neutrinos, relics of the Big Bang and the lives and deaths of stars.

Neutrinos interact extremely rarely, and we’ve learned from neutrino oscillations that they have mass, both promising characteristics for the dark matter.

Neutrino oscillation measurements reveal mass differences Δm122 = 0.00008 eV2 and Δm232 = 0.003 eV2, but not the masses. A number of other experiments place limits on the mass of the neutrino < 1 eV. At this mass, neutrinos are not the dark matter.

Particle physics predicts a myriad of similar, as yet undetected, Weakly Interacting, Massive Particles(WIMPs) with masses > 100 mp ~ 100 GeV that could be the dark matter.

Next TimeRotation Curves and Missing Mass