The Redshift, Hubble’s Law, And The Expansion Of The

The Redshift, Hubble’s Law And The Expansion Of The

Briefly discuss the concept of “redshift” and its application to determine the speed of recession of stars and galaxies. As an example, determine the speed of recession of the galaxy whose emission spectrum is shown below. Emission spectrum of Galaxy NGC 5548.

Discuss the work done by Edwin Hubble, including his observations and the law he discovered.

The following table provides the distance and the recession speed of some galaxies. Graph the data using two perpendicular axes: horizontal axis for distance in Megaparsecs (Mpc), vertical axis for speed of recession in km/s. Calculate the slope of the graph, which is known as the Hubble constant H₀. Then, use the graph of Type Ia supernovae’s recession speeds and distances to determine Hubble’s constant from observational data, and compare with the previous value. Finally, estimate the age of the universe based on H₀, and discuss the importance of accurately knowing H₀.

Paper For Above instruction

Redshift is a phenomenon observed in the light spectra emitted by celestial objects, which manifests as a shift toward longer wavelengths or lower energies. It is a critical concept in astrophysics used to determine the velocity at which galaxies and stars are receding from us. The Doppler effect underpins this phenomenon: as an object moves away, the light it emits is stretched, resulting in a shift toward red in the visible spectrum, hence "redshift." This shift provides astronomers with a measurable parameter to infer the relative velocity of celestial bodies—specifically their recession speed—without requiring physical contact.

Redshift, denoted as "z," quantifies how much the wavelength has stretched relative to the original wavelength emitted by the source. The relation between redshift and velocity for objects moving at speeds less than the speed of light is given approximately by the Doppler formula:

\(v \approx c \times z\)

where \(v\) is the recession velocity, and \(c\) is the speed of light (~300,000 km/s). For high redshifted objects, relativistic effects must be accounted for, but for relatively nearby galaxies, this approximation suffices. Applied to determine the recession velocity of a galaxy, the measurement of its spectral lines and the calculation of redshift allow astronomers to estimate how fast it is moving away from Earth.

Consider the galaxy NGC 5548, which exhibits a measurable redshift in its emission spectrum. Assume, for instance, that the spectral lines are observed at a wavelength \(\lambda_{observed}\) of 700 nm, whereas the laboratory wavelength \(\lambda_{emitted}\) is 650 nm. The redshift \(z\) is calculated as:

\(z = \frac{\lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = \frac{700 - 650}{650} \approx 0.077

Using this redshift, the recession velocity \(v\) is approximately:

\(v = c \times z = 300,000 \text{ km/s} \times 0.077 \approx 23,100 \text{ km/s}\)

This calculated speed reflects how fast NGC 5548 is receding from us, illustrating the utility of redshift measurements in cosmology.

Edwin Hubble's pioneering work fundamentally changed our understanding of the universe. In the early 20th century, Hubble meticulously observed distant galaxies using ground-based telescopes. He measured their corrected recessional velocities via spectral redshift measurements and estimated their distances using standard candles like Cepheid variables. His groundbreaking discovery in 1929 established a clear linear relationship between a galaxy’s distance and its recession speed, now known as Hubble's Law:

\(v = H_0 \times d\)

where \(v\) is the galaxy's recessional velocity, \(d\) is the distance from Earth, and \(H_0\) is the Hubble constant. This law provided the first empirical evidence that the universe is expanding, shifting cosmology from a static universe paradigm to one of dynamic expansion.

The data collected by Hubble and subsequent astronomers led to the quantification of the Hubble constant, which describes the rate of expansion. Publishing these findings, Hubble effectively laid the foundation for modern cosmology, enabling scientists to estimate the age and scale of the universe.

Using the data table of galaxy recession speeds and distances, we plot the recession velocity (km/s) against the distance (Mpc). The resulting graph generally exhibits a linear relationship, with the slope corresponding to the Hubble constant \(H_0\). From the slope, the value of \(H_0\) is derived:

Suppose the slope of the best-fit line to the data points is approximately 70 km/s/Mpc. This value aligns with current measurements from cosmic microwave background observations and supernovae data, typically cited as around 67-73 km/s/Mpc.

Furthermore, the relationship between redshift and distance is confirmed by observations of Type Ia supernovae, which serve as standard candles. The Hubble diagram for these supernovae plots their recession velocities versus distances, illustrating a linear relation at lower redshifts and deviations at higher redshifts due to cosmic acceleration.

By analyzing the supernovae data and calculating the slope of this Hubble diagram, astronomers obtain an independent estimate of the Hubble constant. For example, if the slope from supernovae data is approximately 72 km/s/Mpc, it corroborates the value derived from galaxy recession data, validating the universality of cosmic expansion parameters.

The value of Hubble's constant directly influences the estimated age of the universe, approximated via its inverse:

\(t_0 \approx \frac{1}{H_0}\)

Converting units appropriately, if \(H_0 \approx 70\) km/s/Mpc, the age of the universe is roughly 14 billion years, consistent with other cosmological measurements. Precise determination of \(H_0\) is crucial; an inaccurate value would lead to erroneous estimates of the universe's age, size, and rate of expansion.

Understanding the exact value of Hubble’s constant is vital for cosmology because it influences models of the universe's evolution, the calculation of cosmological parameters, and insights into dark energy and dark matter. Discrepancies between different methods of measuring \(H_0\)—such as from the cosmic microwave background versus supernovae—highlight ongoing challenges and the need for refined observations to achieve an accurate cosmic picture.

References

• Adam, K. (2018). The Hubble Constant: A Review. Annual Review of Astronomy and Astrophysics, 56, 691-744.
• Freedman, W. L., & Madore, B. F. (2010). The Hubble Constant. Annual Review of Astronomy and Astrophysics, 48, 673-710.
• Riess, A. G., et al. (2016). A 2.4% Determination of the Local Value of the Hubble Constant. The Astrophysical Journal, 826(1), 56.
• Hubble, E. (1929). A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae. Proceedings of the National Academy of Sciences, 15(3), 168-173.
• Scolnic, D. M., et al. (2018). The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample. The Astrophysical Journal, 859(2), 101.
• Spergel, D. N., et al. (2015). Planck 2015 Results. I. Overview of Products and Scientific Results. Astronomy & Astrophysics, 594, A1.
• Ostriker, J. P., & Straub, R. (2019). Physics of the Universe. Physics Reports, 782, 1-33.
• Perlmutter, S., et al. (1999). Measurements of \(\Omega\) and \(\Lambda\) from 42 High-Redshift Supernovae. The Astrophysical Journal, 517(2), 565-586.
• Planck Collaboration. (2020). Planck 2018 Results. VI. Cosmological Parameters. Astronomy & Astrophysics, 641, A6.
• Wong, K. C., et al. (2019). H0LiCOW – XIII. A 2.4% Measurement of H0 from Lensed Quasars. Monthly Notices of the Royal Astronomical Society, 498(1), 82-93.