# 3.4: The Cosmic Distance Ladder

- Page ID
- 5655

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## From Parallaxes to Hubble's Law

How do we know distances to objects in space? No method for determining distances works at all scales. The analogy with a ladder is that we have different steps allow us to bridge one method for measuring distances to the next one. First we find distances to the closest objects. Then we use information about those nearby objects to infer distances to more distant objects.

- For the closest objects - the moon or Venus, we can use radar ranging. This allowed scientists to figure out the distance to the Sun. Once we know the distance to the Sun, we can figure out how much energy is being emitted each second (i.e., the "luminosity") in the following way: the Sun emits equal amounts of energy in all directions. We can use a photon counter to determine how much light is being emitted at various wavelengths. Each photon has an energy: \(E=hν\) and luminosity, \(L\), is the total energy per wavelength per second. Next, we have to determine what fraction of energy is intercepted by the photon counter - this will scale as the area of a spherical shell centered on the Sun with a radius equal to the Earth-Sun distance:
- \(L_{\odot}=L_{collector}\cdot\frac{4\pi\,R^2}{A_{collector}}\)
- where \(L_{\odot}\) is the luminosity of the Sun
- \(L_{collector}\) is the energy per second intercepted by the photon collector
- \(R\) is the distance from the Sun to the collector (on Earth)

- For stars in our part of the Milky Way galaxy, we can use trigonometric parallax measurements to determine distances. Once we know the distance, we can measure the energy per second with a photon counting device and use this to calculate the luminosity of the stars. This helped us to understand that stars with different types of spectra have different luminosities. As soon as that correlation was established we could turn this around to take the next step on the distance ladder.
- For stars that are farther away in our galaxy, we can obtain spectra to derive spectral types and luminosity classes. Once we have the spectral type, we also know the true luminosity of the star, \(L_{\ast}\), from our work on the previous rung of the distance ladder. Now, we measure the apparent brightness with our photon-counter and flip the equation above and use \(L_{\ast}\) to calculate R (distance). The apparent brightness will be fainter when the object is farther away. This is an example of a
**standard candle**- we know the intrinsic brightness of an object (a star in this case) and use the difference between the intrinsic brightness and the apparent brightness to estimate the distance. - Cepheid variables: Henrietta Leavitt discovered a relationship between periodicity in the brightness variations of Cepheid stars and their true luminosity (described in Chapter 1.1). Cepheids are another example of a
**standard candle**that allowed us to derive distances to nearby galaxies. - The Tully-Fisher relation found that the faster a galaxy is spinning, the brighter it is. Like the Faber-Jackson relation for estimating distances to galaxies, the technique allows astronomers to relate an observation (here, the spin rate of a galaxy) that is reasonably easy to make to the distance to that object.
- A Type Ia supernovae is a carefully regulated process - mass from an evolving binary companion is funneled onto a white dwarf. When the white dwarf accretes enough mass to hits the Chandrasekhar limit of about 1.4 M
_{sun}, a Type 1a supernova occurs and the white dwarf evolves into a neutron star. A Type Ia supernova is always the same physical process, so it is always the same brightness. Type 1a supernovae are lucky events that provide very bright standard candles. - Redshift - Hubble found a relation between the redshift of a galaxy and its distance. In this case, a spectrum is obtained and the measured shift of spectral lines is the redshift that reveals the velocity of a galaxy. If spectral lines are shifted toward redder wavelengths, the galaxy is moving away from us; if spectral lines are shifted toward bluer wavelengths, the galaxy has a component of radial velocity that is coming toward us. The precision of redshift measurements has been improved (e.g., by Freedman et al.) and shown to be a reliable distance indicator at the largest scales in our universe.