26.4 The Extragalactic Distance Scale

Astronomers use clever tricks to measure vast cosmic distances. Cepheid variables and Type Ia supernovae act as cosmic yardsticks, letting us gauge distances to faraway galaxies. These methods form the backbone of our cosmic distance ladder.

The ###Tully-Fisher_relation_0### adds another tool to the mix, linking a galaxy's spin to its brightness. While less precise, it helps fill gaps in our cosmic map. Together, these techniques unveil the universe's vast scale and expansion.

Measuring Distances to Galaxies

Cepheid variables for galactic distances

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• Variable stars exhibit periodic changes in brightness over time
  • Cepheid variables pulsate radially causing their luminosity to vary (Delta Cephei)
  • The period of pulsation directly relates to the star's intrinsic luminosity
• The period-luminosity relationship of Cepheids allows astronomers to determine their absolute magnitude
  • Longer periods indicate higher luminosities while shorter periods indicate lower luminosities (1-100 days)
• Comparing the absolute magnitude to the apparent magnitude, the distance to the Cepheid can be calculated using the distance modulus formula: $m - M = 5 \log_{10}(d) - 5$
  • $m$ = apparent magnitude, $M$ = absolute magnitude, $d$ = distance in parsecs
• Cepheids measure distances up to ~100 million light-years encompassing nearby galaxies (Andromeda galaxy, Large Magellanic Cloud)
  • The presence of Cepheids in a galaxy allows astronomers to determine its distance
• Cepheids are an important step in the cosmic distance ladder, bridging nearby and distant measurements

Type Ia supernovae as standard candles

• Type Ia supernovae occur when a white dwarf in a binary system accretes matter from its companion star
  • The white dwarf reaches the Chandrasekhar limit of ~1.4 solar masses and undergoes a thermonuclear explosion
• Type Ia supernovae have a consistent peak luminosity making them standard candles
  • The peak absolute magnitude of type Ia supernovae is approximately -19.3 (5 billion times brighter than the Sun)
• Measuring the apparent magnitude of a type Ia supernova at its peak, astronomers can calculate its distance using the distance modulus formula
• Type Ia supernovae are much brighter than Cepheids allowing them to measure distances to galaxies billions of light-years away (SN 1997ff, 11 billion light-years)
  • This method has been crucial in determining the expansion rate of the universe and providing evidence for dark energy
• The redshift of distant supernovae is used to determine their velocity and distance, contributing to the Hubble-Lemaître law

Tully-Fisher relation vs other methods

• The Tully-Fisher relation is an empirical relationship between a spiral galaxy's rotational velocity and its intrinsic luminosity
  • Faster rotating galaxies tend to be more luminous while slower rotating galaxies are less luminous
• Measuring a spiral galaxy's rotational velocity, astronomers can estimate its absolute magnitude using the Tully-Fisher relation
  • The distance to the galaxy can then be calculated using the distance modulus formula
• The Tully-Fisher relation measures distances to spiral galaxies that are too far for Cepheids to be resolved
  • It can be used for galaxies up to ~100 million light-years away (NGC 4258, 23.5 million light-years)
• Compared to Cepheids and type Ia supernovae, the Tully-Fisher relation is less precise
  • The scatter in the Tully-Fisher relation is larger than the period-luminosity relationship of Cepheids or the standard candle nature of type Ia supernovae
• The Tully-Fisher relation is still a valuable tool for estimating distances to spiral galaxies when other methods are not applicable
  • It provides an independent check on distances determined by other techniques (Cepheids, Type Ia supernovae)

Additional Distance Measurement Techniques

• Parallax: Used for measuring distances to nearby stars by observing their apparent shift against background stars
• Main sequence fitting: Compares the main sequence of a star cluster to a standard main sequence to determine the cluster's distance
• These methods, along with Cepheids, supernovae, and the Tully-Fisher relation, form the cosmic distance ladder, allowing astronomers to measure distances across the universe