Stellar radii

1. Overview

The physical size of a star (its radius) cannot be measured directly using standard imaging because stars are so distant they appear as unresolved point sources, even through the world’s most powerful telescopes. To overcome this, physicists apply the principles of black-body radiation and thermodynamics. By analyzing the light from a star, we can determine its surface temperature (using Wien’s Displacement Law) and its total power output or luminosity (using the Inverse Square Law of radiation). By combining these two properties within the Stefan–Boltzmann Law, the radius of the star can be calculated with high precision. This method allows us to distinguish between different classes of stars, such as tiny white dwarfs and gargantuan red supergiants.

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Key Definitions

• Black Body: An idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. When in thermal equilibrium, it emits a characteristic distribution of radiation (a black-body spectrum) that depends only on its absolute temperature.
• Luminosity ($L$): The total power of electromagnetic radiation emitted by a star. It is the total energy emitted per unit time across all wavelengths. (Unit: Watts, $\text{W}$ or $\text{J s}^{-1}$)
• Radiant Flux Intensity ($F$): The radiant power passing normally through a surface per unit area. In astronomy, this is the power received from a star per unit area on Earth. (Unit: $\text{W m}^{-2}$)
• Wien’s Displacement Law: A law stating that the peak wavelength ($\lambda_{\text{max}}$) of the spectrum of radiation emitted by a black body is inversely proportional to its thermodynamic temperature ($T$).
• Stefan–Boltzmann Law: The total energy radiated per unit surface area of a black body per unit time (the intensity) is directly proportional to the fourth power of its thermodynamic temperature ($T^4$).
• Thermodynamic Temperature ($T$): The absolute temperature measured in Kelvin ($\text{K}$), where the zero point is absolute zero.

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Content

3.1 The Nature of Black-Body Radiation

Stars are composed of hot, dense plasma. Although they are not "perfect" black bodies (due to absorption lines in their atmospheres), they are excellent approximations. A black-body spectrum is continuous, meaning it contains all wavelengths, but the energy is not distributed evenly.

The Black-Body Curve (Intensity-Wavelength Graph): If you plot the intensity of radiation against wavelength for a star, the resulting curve has a distinct shape:

1. The Peak: There is a specific wavelength ($\lambda_{\text{max}}$) where the intensity is at its maximum.
2. Temperature Dependence: As the temperature of a star increases:
   • The total intensity (the area under the curve) increases significantly.
   • The peak wavelength ($\lambda_{\text{max}}$) shifts toward shorter wavelengths (the ultraviolet/blue end of the spectrum).
   • The curve becomes "sharper" and taller.

[DIAGRAM DESCRIPTION: A graph with Intensity on the y-axis and Wavelength ($\lambda$) on the x-axis. Three curves are shown:

• Curve A (Hot star, 12,000 K): Very high peak, $\lambda_{\text{max}}$ is in the UV/Blue region.
• Curve B (Medium star, 6,000 K): Moderate peak, $\lambda_{\text{max}}$ is in the Visible (yellow) region.
• Curve C (Cool star, 3,000 K): Low, broad peak, $\lambda_{\text{max}}$ is in the Infrared/Red region. A dashed line (Wien's Law) connects the peaks, showing the $1/T$ shift.]

3.2 Wien’s Displacement Law

Wien’s Law allows us to determine the surface temperature of a star simply by measuring its color (specifically, its peak emission wavelength).

The Equation: $$\lambda_{\text{max}} \propto \frac{1}{T}$$ $$\lambda_{\text{max}} T = b$$

Where:

• $\lambda_{\text{max}}$ = Peak wavelength ($\text{m}$)
• $T$ = Thermodynamic temperature ($\text{K}$)
• $b$ = Wien’s displacement constant $\approx 2.90 \times 10^{-3} \text{ m K}$

Important Note: $\lambda_{\text{max}}$ does not mean the "maximum wavelength" emitted; it refers to the wavelength of maximum intensity.

3.3 The Stefan–Boltzmann Law

While Wien’s Law gives us the temperature, the Stefan–Boltzmann Law relates that temperature to the star's total power output (Luminosity) and its physical size (Surface Area).

For a black body, the power emitted per unit surface area is $\sigma T^4$. Since stars are spherical, their surface area is $A = 4\pi r^2$.

The Equation: $$L = 4\pi r^2 \sigma T^4$$

Where:

• $L$ = Luminosity ($\text{W}$)
• $r$ = Radius of the star ($\text{m}$)
• $\sigma$ = Stefan–Boltzmann constant ($5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$)
• $T$ = Thermodynamic temperature ($\text{K}$)

Key Proportionalities:

• $L \propto r^2$ (A star with twice the radius has four times the luminosity, if $T$ is constant).
• $L \propto T^4$ (A star with twice the temperature has sixteen times the luminosity, if $r$ is constant).

3.4 The Step-by-Step Process to Estimate Stellar Radius

In a typical exam scenario, you are required to synthesize several concepts to find the radius $r$. The logical flow is:

1. Determine Luminosity ($L$):
   • Measure the Radiant Flux Intensity ($F$) at Earth.
   • Determine the Distance ($d$) to the star (e.g., via stellar parallax).
   • Use the Inverse Square Law: $L = 4\pi d^2 F$.
2. Determine Temperature ($T$):
   • Observe the star's spectrum and identify the peak wavelength ($\lambda_{\text{max}}$).
   • Use Wien’s Law: $T = \frac{2.90 \times 10^{-3}}{\lambda_{\text{max}}}$.
3. Calculate Radius ($r$):
   • Rearrange the Stefan–Boltzmann Law: $r = \sqrt{\frac{L}{4\pi \sigma T^4}}$.

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Worked Example 1 — Temperature from Peak Wavelength

An astronomer observes a distant star and determines that its emission spectrum peaks at a wavelength of $420 \text{ nm}$. Calculate the surface temperature of the star.

1. Identify the equation: $$\lambda_{\text{max}} T = 2.90 \times 10^{-3} \text{ m K}$$

2. Convert units: $\lambda_{\text{max}} = 420 \text{ nm} = 420 \times 10^{-9} \text{ m}$

3. Substitute and solve: $$(420 \times 10^{-9}) \times T = 2.90 \times 10^{-3}$$ $$T = \frac{2.90 \times 10^{-3}}{420 \times 10^{-9}}$$ $$T = 6904.76... \text{ K}$$

4. Final Answer: $$T \approx 6900 \text{ K (2 s.f.)}$$

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Worked Example 2 — Comparing Two Stars (Ratio Method)

Star X has a surface temperature $T$ and radius $r$. Star Y has a surface temperature $2T$ and a radius $0.5r$. Determine the ratio of the luminosity of Star Y to the luminosity of Star X ($\frac{L_Y}{L_X}$).

1. Set up the luminosity equation for both: $L_X = 4\pi r^2 \sigma T^4$ $L_Y = 4\pi (0.5r)^2 \sigma (2T)^4$

2. Simplify the expression for $L_Y$: $L_Y = 4\pi (0.25r^2) \sigma (16T^4)$ $L_Y = (0.25 \times 16) \times (4\pi r^2 \sigma T^4)$ $L_Y = 4 \times L_X$

3. Final Answer: The ratio $\frac{L_Y}{L_X} = 4$. (Star Y is four times as luminous as Star X).

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Worked Example 3 — Full Radius Calculation

The star Rigel has a luminosity of $4.62 \times 10^{31} \text{ W}$. Its surface temperature is measured to be $12,100 \text{ K}$. Calculate the radius of Rigel and express it as a multiple of the Sun's radius ($R_{\odot} = 6.96 \times 10^8 \text{ m}$).

1. State the equation and rearrange for $r$: $$L = 4\pi r^2 \sigma T^4 \Rightarrow r = \sqrt{\frac{L}{4\pi \sigma T^4}}$$

2. Substitute the known values: $L = 4.62 \times 10^{31} \text{ W}$ $\sigma = 5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$ $T = 12,100 \text{ K}$

$$r = \sqrt{\frac{4.62 \times 10^{31}}{4 \times \pi \times (5.67 \times 10^{-8}) \times (12100)^4}}$$

3. Intermediate steps (to avoid calculator error): $T^4 = (12100)^4 \approx 2.1436 \times 10^{16}$ Denominator $= 4 \times \pi \times (5.67 \times 10^{-8}) \times (2.1436 \times 10^{16}) \approx 1.527 \times 10^{10}$ $r = \sqrt{\frac{4.62 \times 10^{31}}{1.527 \times 10^{10}}} = \sqrt{3.025 \times 10^{21}}$ $r = 5.50 \times 10^{10} \text{ m}$

4. Compare to Solar Radius: Ratio $= \frac{5.50 \times 10^{10}}{6.96 \times 10^8} \approx 79.0$

5. Final Answer: $$r = 5.50 \times 10^{10} \text{ m (or } 79 R_{\odot})$$

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Key Equations

Equation	Symbols	Status
$\lambda_{\text{max}} \propto \frac{1}{T}$	$\lambda_{\text{max}}$: Peak wavelength; $T$: Thermodynamic temp	Memorise
$\lambda_{\text{max}} T \approx 2.90 \times 10^{-3} \text{ m K}$	$T$: Thermodynamic temperature	Constant on sheet
$L = 4\pi \sigma r^2 T^4$	$L$: Luminosity; $r$: Radius; $\sigma$: Stefan-B const	On Data Sheet
$F = \frac{L}{4\pi d^2}$	$F$: Radiant flux; $d$: Distance to star	On Data Sheet
$T(\text{K}) = \theta(^\circ\text{C}) + 273.15$	Conversion from Celsius to Kelvin	Memorise

Constants from Data Sheet:

• Stefan-Boltzmann constant ($\sigma$): $5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$
• Wien's constant ($b$): $2.90 \times 10^{-3} \text{ m K}$

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Common Mistakes to Avoid

• ❌ Wrong: Using temperature in Celsius ($^\circ\text{C}$). ✅ Right: Always convert to Kelvin ($\text{K}$). A star at $5000^\circ\text{C}$ is $5273 \text{ K}$. This is critical because the $T^4$ term amplifies any error.
• ❌ Wrong: Forgetting to square the radius ($r^2$) or the distance ($d^2$) in the flux equation. ✅ Right: Double-check that you have applied the Inverse Square Law correctly.
• ❌ Wrong: Forgetting the power of 4 in the Stefan–Boltzmann Law. ✅ Right: $L$ is proportional to $T^4$. If the temperature doubles, the luminosity increases by a factor of $2^4 = 16$.
• ❌ Wrong: Confusing $\lambda_{\text{max}}$ with the "edge" of the spectrum. ✅ Right: $\lambda_{\text{max}}$ is the wavelength of the peak intensity. The star still emits light at many other wavelengths.
• ❌ Wrong: Using nanometers ($\text{nm}$) directly in Wien's Law without converting to meters. ✅ Right: $1 \text{ nm} = 10^{-9} \text{ m}$. Always use SI units ($\text{m}$) to match the units of Wien's constant ($\text{m K}$).

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Exam Tips

1. The "Show That" Questions: Exams often ask you to "show that the radius is approximately $X$". Always show the substitution of values into the rearranged formula $r = \sqrt{L / (4\pi \sigma T^4)}$ before giving the final answer.
2. Logarithmic Scales: Be careful if the black-body curve is presented on a log-log graph. The peak wavelength $\lambda_{\text{max}}$ is still the highest point, but the shape of the curve may look different.
3. Proportionality Reasoning: If a question asks how $L$ changes when both $r$ and $T$ change, use the ratio method (as shown in Worked Example 2). It is much faster and less prone to calculation error than calculating new absolute values.
4. Sanity Check:
   • Temperature: Most stars have surface temperatures between $2,000 \text{ K}$ and $40,000 \text{ K}$.
   • Radius: Stellar radii are usually between $10^7 \text{ m}$ (white dwarfs) and $10^{12} \text{ m}$ (supergiants). If your answer is $150 \text{ m}$, you likely forgot the $10^{-8}$ in the Stefan-Boltzmann constant or the $10^{-9}$ in the wavelength.
5. Defining Luminosity: If asked to define luminosity, do not just say "brightness." You must use the phrase "total power" or "total energy emitted per unit time" to earn the mark.
6. The "Color" of Stars:
   • Blue stars are hot (short $\lambda_{\text{max}}$, high energy photons).
   • Red stars are cool (long $\lambda_{\text{max}}$, lower energy photons).
   • Always link this back to Wien's Law: $\lambda_{\text{max}} \propto 1/T$.