Astronomy 218I n t e r g a l a c t i c

D i s t a n c e s

Virial MassFor a system of masses interacting by Newton’s second law, the virial theorem tells us that twice the total kinetic energy plus the potential energy is conserved, 2K+U = 0.

The kinetic energy is simply K = ½M⟨𝑣2⟩. The potential energy requires knowledge of the radius of the system,

Empirically, for elliptical galaxies it has been found that this can be written in terms of the half mass radius, rh, the radius that contains half the galaxy’s observed mass.

Substituting these relations into the virial theorem and isolating the mass yields the virial mass,

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Weighing EllipticalsThe virial mass relation is quite similar to that derived for rotationally supported systems, but it faces several practical difficulties.

Since only one velocity component can be measured, we must assume isotropy ⟨𝑣2⟩ = 3 ⟨σ2⟩. Even for nearby galaxies, proper motions are < 10−4 arcsec yr−1.

We can directly measure only the half-light radius, and then in projection, so we must assume this equals rh.

For the dwarf spheroidal galaxy Leo I, d ≈ 250 kpc, the observed half-light radius rhʺ ≈ 240 arcsec ⇒ rh = 290 pc.

The corresponding σ = 8.8 km s−1 leads to M ≈ 4 × 107 M⊙. Dividing by L ≈ 5 × 106 L⊙ implies M/L ~ 8.

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These techniques for measuring mass can also be applied to finding black holes in the galaxy’s center, but separating the black hole’s mass from that of the surrounding stars requires looking very closely.

Comparing the black hole’s force to the typical bulge centripetal force reveals the radius of influence.

Interferometry with radio waves is a good way to peer deep into the obscured center of a galaxy.

Finding Black Holes

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Weighing Black HolesObservations like these have revealed that supermassive black holes lurk in the center of most large galaxies. The practical limit to these observations is the need to resolve the radius of influence. The mass of the central black hole is strongly correlated with the luminosity & mass of elliptical galaxies and of the bulge in spiral galaxies. These systems exhibit a strong correlation L ∝ σ4 and a stronger correlation Mbh ∝ σ4. The reason Mbh ~ 0.1 % MGalaxy/bulge is not understood.

SDSS J010013.02+280225.8

The Distance to AndromedaHubble’s observations of Cepheids in Andromeda did much more than settle the Shapley-Curtis debate.

His distance, d ~ 275 kpc, set the intergalactic distance scale and established spiral nebulae as the equivalent of the Milky Way.

Hubble’s confusion of metal-rich Cepheids in Andromeda for dimmer metal-poor W Virginis variables (Type II Cepheds) lead to an underestimate of the true distance to Andromeda of ~ 778 kpc.

Galactic Distance ScaleCepheids are the culmination of the galactic distance ladder. Since few are nearby, our knowledge of their luminosity builds on distance measuring techniques with less reach, starting with stellar parallax.

In Hubble’s day, (up to the Hipparcos mission in 1989) methods like moving cluster method and main sequence fitting were needed to extend our reach to include enough stars to calibrate spectroscopic parallax.

The reliance of the calibration of spectroscopic parallax on nearby main sequence stars excludes the brightest O & B stars, limiting its usefulness to ~10 kpc.

Standard CandleSpectroscopic parallax is an example of a “standard candle”, an object with a known luminosity. By measuring the flux, we know the distance.

or in magnitudes

d = 10 0.2(m-M)+1

In reality, none of the astronomical distance measures are truly standard. Their usefulness is determined by their “standardizability” (and their calibration).

For spectroscopic parallax, it is the detected spectral type what provides the standardization.

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Cepheid LimitFor Cepheids, the needed standardization is provided by the period-luminosity relationship, which was discovered by Henrietta Swan Leavitt in 1912 as a correlation between mV and period for Cepheids in the Small and Large Magellanic Clouds, before it was realized these were extra-galactic.

Detecting a Cepheid relies on resolving individual bright stars in a distant galaxy. In Edwin Hubble’s day, this limited their usefulness to ~ 1 Mpc. With modern ground-based telescopes, this limit is 4 Mpc.

One of the key projects driving the design of the Hubble Space telescope was the desire to extend the range over which Cepheid variables could be observed.

HST ultimately allowed Cepheids to be detected in galaxies up to 29 Mpc.

This greatly expanded the set of galaxies with well measured distances, providing an improved calibration of extragalactic distance measures.

Hubble Key Project

To go beyond the hundreds of galaxies within 30 Mpc, requires a distance measure brighter and more resolvable than the brightest Cepheids (L ~ 105 L☉).

The Luminosity Functions of Planetary Nebulae and Globular Clusters are among the methods used, but these offer only small additional range (< 50 Mpc).

Ultimately, only objects as bright as galaxies themselves can be seen at 1 Gpc, thus distance measures using entire galaxies are required.

Extragalactic Distances

Faber-Jackson RelationSandra Faber and Robert Jackson in 1976 noted the empirical relationship between V-band absolute magnitude and velocity dispersion in elliptical galaxies.

In terms of luminosity, the Faber-Jackson relation is

The Faber-Jackson can be derived from the Virial theorem and the assumption of a fixed, limiting surface brightness, B = L/4πR2

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In 1977, Brent Tully and Richard Fisher published a similar empirical relation for spiral galaxies, correlating the absolute B-band magnitude with the peak rotation speed of the galaxies disk.

For both Faber-Jackson and Tully-Fisher M ~ L ∝ 𝑣4.

For nearby galaxies, the kinematics can be spatially resolved, but in more distant galaxies, redshift and blueshifts become doppler broadening.

Tully-Fisher Relation

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The only other object that approaches the optical luminosity of a galaxy is a supernova. Core-collapse supernovae, which are classified observationally as Type II, Ib or Ic, vary a hundred-fold in peak brightness due to variations in progenitor mass and structure of the stellar envelope. In contrast, for thermonuclear supernova (Type Ia), which we believe result when a white dwarf exceeds the Chandrasekhar mass, all have similar brightness MV ~ MB ~ –19.3 ± 0.3.

Bright as a Galaxy

SN 1994D in NGC 4526

Standardized CandleThe Type Ia supernova’s brightness and nearly identical peak luminosity are nearly ideal for a standard candle. However, the factor of 2.5 in brightness between the brightest and dimmest is less than ideal. This can corrected to < 10% by noticing the correlation between peak brightness and decay time. The relative rarity of SN Ia, 1 per 100 years per galaxy, make it challenging to determine the distance to a specific galaxy, but > 100 are observed each year, with ~3 SN Ia per year within 25 Mpc.

In 1917, Vesto Slipher measured the line shifts of 25 nearby spiral nebulae (not yet recognized as galaxies).

The size of the redshifts was surprising; if due to relative motion, the implied velocities were as large as 1000 km s−1.

Slipher also found that 21/25 were redshifted, much more than the 1/2 one would expect from random motion.

The spiral nebulae seemed repelled by the Earth.

Galactic Redshift

Hubble’s lawIn 1929, Edwin Hubble combined redshifts with distances for a set of 25 galaxies (some more distant than Slipher).

The resulting plot revealed a linear correlation between redshift and distance.

With Cepheids limited to <1 Mpc, Hubble’s distances were based on using the brightest star in each galaxy as a standard candle.

However, these “stars” turned out to be compact HII regions in more distant galaxies, which are as much as 50× brighter.

500 km s−1 / M

pc

While modern data corrects the distance, the correlation remains. The correlation is quantified by the slope, called Hubble’s constant, H0.

𝑣r = cz = H0 d

with the HST Key project determining H0 = 70 ± 5 km s−1 Mpc-1 in 2001 from Cepheids in 18 galaxies.

With supernovae, Hubble’s law has been tested to 1 Gpc. For many years proceeding the HST Key project, a debate raged with firmly entrenched factions claiming H0 = 50 or 100, thus it was common to write H0 = 100h km s-1 Mpc-1.

Hubble’s Constant

H 0 = 72

± 7

km s-

1 Mpc

-1

Hubble’s law is the result of Universal Recession: all galaxies (except for a few nearby galaxies where local gravitational attraction is at play) are moving away from us, with the redshift telling us their distance.

The philosophical implications of the Hubble law are monumental, but we will hold those for the present.

We can also put Hubble’s law to practical use, as a way of measuring distance to the most distant objects.

While Hubble’s law breaks down locally, it actually works better the more distant the object; peculiar motions are overwhelmed by the recessional velocity at high redshift. In general astronomers quote the redshift of distant objects in place of their distance.

Universal Recession

By adding L ∝ 𝑣4 , supernovae and Hubble’s law to our galactic distance ladder, we can now measure distances to the edge of the Universe.

In the process of looking to these great distances, the finite speed of light causes us to look back in time, seeing the Universe as its younger self.

Cosmic Distance Ladder

Next TimeTurn in Homework #8

Loud Galaxies