Problem 2 - An Introduction to Modern Astrophysics

Question

(a) The absolute magnitude of \(\mathrm{M} 101,\) an \(\mathrm{Sc}\) galaxy, is -21.51 in the \(B\) band. Using Eq. (11) estimate its isophotal radius \(\left(R_{25}\right)\) at 25 B-mag arcsec \(^{-2}\). $$\log _{10} R_{25}=-0.249 M_{B}-4.00.$$ (b) Use the Tully-Fisher relation (Eq. 7 ) to estimate the rotational velocity of M101. $$M_{B}=-11.0 \log _{10} V_{\max }+3.31$$ (c) Estimate the angular rotation speed of a star at \(R_{25},\) in units of arcsec \(\mathrm{yr}^{-1}\) (d) Could van Maanen have detected the rate of rotation of M101? How long would it take for the galaxy to rotate through \(1 " ?\)

Step-by-Step Solution

Answer

\(R_{25} \approx 22.7\,\text{kpc}\), \(V_{\max} \approx 180\,\text{km/s}\); Very slow rotation, \(\sim 25,974\,\text{yrs/arcsec}\).

Step 1: Calculate the isophotal radius using Eq. (11)

To find the isophotal radius \( R_{25} \) at 25 B-mag arcsec\(^{-2}\), use the formula provided: \[ \log_{10} R_{25} = -0.249 M_{B} - 4.00 \]Substitute the given absolute magnitude \( M_{B} = -21.51 \).\[ \log_{10} R_{25} = -0.249(-21.51) - 4.00 \]Calculate the value inside the parenthesis:\[ \log_{10} R_{25} = 5.356 - 4.00 \]\[ \log_{10} R_{25} = 1.356 \]Finally, solve for \( R_{25} \) by exponentiating both sides:\[ R_{25} = 10^{1.356} \approx 22.7 \text{ kpc} \]

Step 2: Estimate rotational velocity using the Tully-Fisher relation

We apply the Tully-Fisher relation to estimate the rotational velocity \( V_{\max} \) for galaxy M101:\[ M_{B} = -11.0 \log_{10} V_{\max} + 3.31 \]Substitute \( M_{B} = -21.51 \) into the equation:\[ -21.51 = -11.0 \log_{10} V_{\max} + 3.31 \]Re-arranging gives:\[ -24.82 = -11.0 \log_{10} V_{\max} \]Divide by \(-11.0\):\[ \log_{10} V_{\max} = 2.256 \]Exponentiate to find \( V_{\max} \):\[ V_{\max} = 10^{2.256} \approx 180 \text{ km/s} \]

Step 3: Estimate angular rotation speed at \( R_{25} \)

The angular rotation speed, \( \omega \), of a star at \( R_{25} \), can be approximated by the formula:\[ \omega = \frac{V_{\max}}{R_{25}} \]Convert \( R_{25} \) from kpc to arcsec, assuming 1 kpc = 206265 arcsec (based on distance relationship in astronomical units):\[ R_{25} = 22.7 \times 206265 \approx 4.68 \times 10^{6} \text{ arcsec} \]Substituting the values:\[ \omega \approx \frac{180}{4.68 \times 10^{6}} \approx 3.85 \times 10^{-5} \text{ arcsec/yr} \]

Step 4: Determine detectability by van Maanen and time for 1 arcsec rotation

To see if van Maanen could detect the rotation of M101, note his ability to detect minute motion. Van Maanen could detect changes as small as about 0.01 arcsec/yr, though this varies.Given the angular speed \( \omega = 3.85 \times 10^{-5} \text{ arcsec/yr} \), detecting this slow rate is improbable with early 20th-century technology.Calculate the time for \(1\) arcsec of rotation:\[ t = \frac{1\,\text{arcsec}}{3.85 \times 10^{-5}\,\text{arcsec/yr}} \approx 25,974 \text{ yrs} \]

Key Concepts

Isophotal Radius

The isophotal radius, commonly denoted as \( R_{25} \), is an important metric in the study of galaxies. It defines the boundary where the brightness of a galaxy falls to a specified level, often 25 B-mag arcsec\(^{-2}\). This boundary acts as a useful scale for comparing galaxies—it allows astronomers to say, "This is how far the light of this galaxy extends with a certain brightness."
The concept behind the isophotal radius is rooted in photometry, the science of measuring light. Photometric measurements help astronomers understand a galaxy's size and spatial structure. This parameter is often calculated using observations and specific formulas. In our case, the formula involves the absolute magnitude \( M_B \) of a galaxy.

• Absolute magnitude \( M_B \) measures the intrinsic brightness of a galaxy in the B-band.
• Isophotal radius helps inform about the scale and extent of the galaxy's light.

Calculating \( R_{25} \) involves taking the logarithm and exponentiating it back. For M101, substituting the given values gives us an isophotal radius of approximately 22.7 kpc. These calculations not only help in measuring the physical size of a galaxy but also provide insights into its luminosity profile.

Tully-Fisher Relation

The Tully-Fisher Relation is a fascinating relationship in astrophysics that connects the rotational velocity to the luminosity of a spiral galaxy. Named after astronomers R. Brent Tully and J. Richard Fisher, this empirical relation signifies that the brighter a galaxy, the faster it spins.
This relationship is expressed mathematically and often refined for various specific analyses:

• The rotational velocity, \( V_{\max} \), is a critical factor that determines the dynamics of galaxies.
• Absolute magnitude, \( M_B \), serves as a representation of total brightness.

This relation allows astronomers to estimate a galaxy's distance, just by measuring its rotational velocity. It's especially useful for galaxies that are too far to rely on parallax or direct measurement methods.
In the exercise above, for galaxy M101, solving the equation using given data provides a maximum rotational velocity \( V_{\max} \) of approximately 180 km/s. This velocity estimation using the Tully-Fisher Relation gives us crucial insights into the mass distribution within a galaxy and its dynamism as it spirals in space.

Rotational Velocity

Rotational velocity is the speed at which an object, such as a star within a galaxy, moves around the galaxy's center. In the context of a galaxy, it's a vital component in understanding the dynamics and distribution of mass in galaxies.
Rotational velocity can offer insights into:

• The mass of a galaxy, based on how stars and other matter are moving.
• The gravitational forces acting within the galaxy.

When considering M101, its rotational velocity was determined to be 180 km/s using the Tully-Fisher Relation. This velocity tells us about the strong gravitational attraction exerted by the mass contained within this vast structure.
Observations of rotational velocity assist astronomers in mapping the velocity curve of galaxies. By doing so, it is possible to infer the presence of unseen mass, like dark matter, which might not emit light but still has gravitational effects on visible matter.

Angular Rotation Speed

Angular rotation speed, often denoted as \( \omega \), is crucial in understanding how fast a galaxy or component thereof spins. This speed is essential for anyone studying the kinetics of celestial bodies like stars or even entire galaxies.
Angular rotation speed is often given in units such as arcsec/yr, indicating how much the object rotates over time. For M101, this was calculated in the original problem. Here's how it's significant:

• It helps gauge the time it would take for the galaxy to complete one entire rotation or to move through one arcsecond of the sky.
• Variations in angular rotation speed across different parts of a galaxy can indicate differing internal forces or historical cosmic events impacting the galaxy's evolution.

The calculated angular speed for M101 suggests it moves through a very small angle annually. With a value of \(3.85 \times 10^{-5}\) arcsec/yr, this slow speed means that it would take almost 25,974 years for the galaxy to rotate through just one arcsecond. Even though this rate is imperceptible to the human eye on short timescales, it is crucial for long-term studies of galactic motion and provides the historic perspective on understanding celestial movements over millennia.