understand the term luminosity as the total power of radiation emitted by a star

Published by Patrick Mutisya · 14 days ago

Standard Candles – Cambridge A‑Level Physics 9702

Standard Candles

Objective

Understand the term luminosity as the total power of radiation emitted by a star and how it is used in the standard‑candle method for measuring astronomical distances.

1. What is Luminosity?

The luminosity \$L\$ of a star is the total energy it radiates per unit time. It is an intrinsic property of the star, independent of the observer’s location.

In SI units, luminosity is measured in watts (W), where 1 W = 1 J s\(^{-1}\).

2. Relationship Between Luminosity, Flux and Distance

When radiation spreads uniformly in all directions, the energy passes through the surface of an expanding sphere. The flux \$F\$ (energy per unit area per unit time) measured by an observer at distance \$d\$ from the star is

\$F = \frac{L}{4\pi d^{2}}\$

Re‑arranging gives the distance in terms of measured flux and known luminosity:

\$d = \sqrt{\frac{L}{4\pi F}}\$

3. Standard Candles – The Concept

A standard candle is an astronomical object whose luminosity \$L\$ is known (or can be determined reliably). By measuring its apparent flux \$F\$, the distance \$d\$ can be calculated using the inverse‑square law above.

4. Common Standard Candles

  • Cepheid variable stars: Period–luminosity relation allows \$L\$ to be inferred from the pulsation period.

  • Type Ia supernovae: Peak luminosity is remarkably uniform, making them excellent for measuring distances to distant galaxies.

  • RR Lyrae stars: Less luminous than Cepheids but useful for distances within the Milky Way.

5. Example: Cepheid \cdot ariables

  1. Measure the pulsation period \$P\$ (in days).

  2. Use the period–luminosity relation \$ \log{10} L = a \log{10} P + b \$ (constants \$a\$, \$b\$ are calibrated from nearby Cepheids).

  3. Calculate \$L\$ and then the distance \$d\$ from the observed flux \$F\$.

6. Typical Luminosities of Standard Candles

ObjectTypical Luminosity \$L\$ (W)Absolute Magnitude \$M\$
Cepheid (P = 10 d)≈ \$2.0 \times 10^{31}\$≈ –5
RR Lyrae≈ \$5.0 \times 10^{30}\$≈ +0.5
Type Ia Supernova (peak)≈ \$1.0 \times 10^{36}\$≈ –19.3

7. Key Equations Summary

  • \$L\$ – luminosity (total power emitted) in watts.

  • \$F\$ – flux (apparent brightness) in W m\(^{-2}\).

  • \$d\$ – distance to the source in metres.

  • Inverse‑square law: \$F = \dfrac{L}{4\pi d^{2}}\$.

  • Distance from flux: \$d = \sqrt{\dfrac{L}{4\pi F}}\$.

  • Period–luminosity for Cepheids: \$\log{10} L = a \log{10} P + b\$.

8. Practical Considerations

When applying the standard‑candle method, remember:

  • Interstellar extinction can reduce the observed flux; corrections may be required.

  • Calibration of \$L\$ must be based on objects with independently known distances (e.g., parallax).

  • Systematic errors in the period–luminosity relation propagate into distance estimates.

9. Suggested Diagram

Suggested diagram: Geometry of a star emitting isotropically, showing a sphere of radius \$d\$, flux \$F\$ measured on the surface, and the relation \$F = L/(4\pi d^{2})\$.

10. Summary

Luminosity is the intrinsic power output of a star. By treating certain objects as standard candles—objects with known luminosity—we can convert a measured flux into a distance using the inverse‑square law. This technique underpins the cosmic distance ladder, allowing astronomers to map the scale of the universe from nearby Cepheids to distant Type Ia supernovae.