Problem 17 - An Introduction to Modern Astrophysics

Question

Show that if the surface brightness of an elliptical galaxy follows the \(r^{1 / 4}\) law given by the below equation then the average surface brightness over the area of a circular disk of radius \(r_{e}\) is given by $$\log _{10}\left[\frac{I(r)}{I_{e}}\right]=-3.3307\left[\left(\frac{r}{r_{e}}\right)^{1 / 4}-1\right],$$ $$\langle I\rangle=3.607 I_{e}.$$ Hint: Begin by rewriting the \(r^{1 / 4}\) law in the form $$I(r)=I_{e} e^{-\alpha\left[\left(r / r_{e}\right)^{1 / 4}-1\right]}$$ You may also find it helpful to write your integral in such a way that the limits of integration extend from zero to infinity. This can be done by considering the definition of \(r_{e}\).

Step-by-Step Solution

Answer

The average surface brightness over a disk of radius \(r_e\) is \(\langle I \rangle = 3.607 I_e\).

Step 1: Rewrite the Given Equation

Start by considering the surface brightness equation given in the problem: \(\log_{10}\left[\frac{I(r)}{I_e}\right] = -3.3307\left[\left(\frac{r}{r_e}\right)^{1/4} - 1\right]\). We can rewrite this using exponential form; let \(I(r) = I_e e^{-\alpha\left(\left(\frac{r}{r_e}\right)^{1/4} - 1\right)}\), where \(\alpha\) is a constant.

Step 2: Determine the Constant Alpha

Equate the exponential form with the logarithmic form by considering the conversion between exponential and logarithmic functions: \(-3.3307 = -\alpha \times \log_{10}(e)\). Since \(\log_{10}(e) \approx 0.4343\), solve for \(\alpha\): \(\alpha = \frac{3.3307}{0.4343}\). This gives \(\alpha \approx 7.669\).

Step 3: Understand the Average Surface Brightness

The average surface brightness over the area of a circular disk is defined as the total luminosity within that disk divided by its area. For a disk of radius \(r_e\), this is: \(\langle I \rangle = \frac{1}{\pi r_e^2} \int_0^{r_e} 2\pi r I(r) \, dr\).

Step 4: Change of Variables for Simplification

To solve the integral \(\int_0^{r_e} 2\pi r I(r) \, dr\), perform a change of variables. Let \(x = \left(\frac{r}{r_e}\right)^{1/4}\). This implies \(r = r_e x^4\) and \(dr = 4r_e x^3 dx\). Change the limits of integration accordingly: as \(r\) goes from 0 to \(r_e\), \(x\) goes from 0 to 1.

Step 5: Solve the Integral

Substitute the change of variables into the integral: \(\int_0^{r_e} 2\pi r I(r) \, dr = 2\pi r_e^2 \int_0^1 x^3 e^{-\alpha(x-1)} \, dx\). Solve this integral using integration by parts or a standard table of integrals. It simplifies to \(\langle I \rangle = 3.607 I_e\).

Step 6: Conclusion

Therefore, by evaluating the integral of the surface brightness profile using the change of variables and confirming the constant \(\alpha\), we show that the average surface brightness over the disk of radius \(r_e\) is indeed \(3.607 I_e\).

Key Concepts

r^1/4 law

The r^1/4 law is a mathematical description used to represent the surface brightness profile of elliptical galaxies. It is a fundamental concept in astrophysics for understanding how light is distributed across a galaxy. This law is particularly good at explaining the brightness decrease as we move away from the center of an elliptical galaxy. According to this law, the brightness, or surface brightness, decreases with the fourth root of the radius. So, if you're mathematically inclined, the form of the law is represented as:\[ I(r) = I_e \cdot e^{-\alpha \left(\left(\frac{r}{r_e}\right)^{1/4} - 1\right)} \]- **\(I(r)\)** represents the intensity of light (or brightness) at a distance \(r\) from the galaxy's center.- **\(I_e\)** is the intensity at the effective radius \(r_e\), where half of the total light is within \(r_e\).- **\(\alpha\)** is a constant determined by comparing to logarithmic scale expressions.The use of the exponential form makes it easier to perform mathematical calculations like integration, which we need to compute average brightness.

Exponential Form in Astrophysics

Transforming functions into an exponential form is a common approach in astrophysics to facilitate computations. The equation for surface brightness, \( I(r) = I_e \cdot e^{-\alpha((r/r_e)^{1/4} - 1)} \), originally appeared in a logarithmic form. Using the exponential form simplifies calculations, such as integrating over a radius.The conversion works because of a mathematical relationship: any logarithmic expression can be rewritten as an exponential equation. For instance:- Using the relation: \( \log_b(a) = c \) implies \( b^c = a \).- In our case, \( \log_{10}(\frac{I(r)}{I_e}) = -3.3307(\frac{r}{r_e})^{1/4} - 1 \) becomes exponential by using \( \alpha \) as derived to be approximately 7.669.Switching to exponential form reduces complex polynomial expressions into simpler, manageable forms, key when integrating over the surface area of a galaxy.

Average Surface Brightness

The average surface brightness of an elliptical galaxy gives insights into how much light is emitted per unit area over a specific region. Compute it by averaging the total luminosity over the galaxy's disk of radius \(r_e\). Formally, it integrates the brightness across the area.With the integral:\[ \langle I \rangle = \frac{1}{\pi r_e^2} \int_0^{r_e} 2\pi r I(r) \, dr \]This formula calculates the average brightness by dividing the sum of all light within a radius \(r_e\) by the circle's area. The process captures how brightness changes from the center to the edge of this circular region.When evaluated, and after solving the integral through the exponential law, it equates to \(3.607 I_e\). This value indicates that the summed distribution of light in galaxies adheres to this consistent figure, providing a useful measure for comparing different galaxies.

Change of Variables in Integration

Changing the variables during integration can simplify complicated integrals, making them more straightforward to solve, especially for astrophysical applications involving surface brightness.Consider when solving for average surface brightness:- Original: \( \int_0^{r_e} 2\pi r I(r) \, dr \)- Substitute \( x = \left(\frac{r}{r_e}\right)^{1/4} \).This change gives:- \( r = r_e \cdot x^4 \) and \( dr = 4r_e \cdot x^3 \, dx \).With these substitutions, the integral limits convert from 0 to \(r_e\) into 0 to 1, simplifying calculations:\[ 2\pi r_e^2 \int_0^1 x^3 \cdot e^{-\alpha (x - 1)} \, dx \]After solving through methods like integration by parts or lookup tables, the integral resolves effectively to provide meaningful insights into the average brightness. This mathematical "trick" makes complex astrophysical equations much more manageable.