Standard candles
Section: Astronomy and Cosmology | Syllabus: Cambridge A Level Physics (9702)
Standard Candles
Luminosity
Luminosity (L) The total power of radiation emitted by a star, measured in watts (W).Luminosity is an intrinsic property of a star - it does not depend on how far away the observer is. The Sun has a luminosity of approximately L_ = 3.8 × 10^26 W.
Key PointA star may have high luminosity but appear dim because it is far away. Conversely, a nearby star with low luminosity may appear bright.
Radiant Flux Intensity
As light travels outward from a star, it spreads over an ever-increasing spherical surface. The radiant flux intensity (F) is the power received per unit area at a distance from the star.
F = (L/4π d^2)where F = radiant flux intensity (W m^-2), L = luminosity (W), d = distance from the star (m)
This is an inverse square law: if the distance doubles, the flux intensity decreases by a factor of four.
Figure: Inverse Square LawLight from a star spreads over a sphere of surface area 4π d^2. At distance d, the intensity is L/(4π d^2). At distance 2d, the area is 4 times larger, so intensity is 4 times smaller.
Standard Candles
Standard Candle An astronomical object of known luminosity that can be used to determine distances to galaxies.If we know the luminosity L of an object and can measure its radiant flux intensity F, we can calculate its distance:
Rearranging F = (L/4π d^2):d = √(L/4π F)
Cepheid Variable Stars
Cepheid variables are pulsating stars whose brightness varies regularly over time. The period of variation is directly related to their luminosity.
Period-Luminosity RelationshipDiscovered by Henrietta Swan Leavitt (c. 1908): Cepheid variables with longer periods have greater luminosities. By measuring the period, astronomers can determine the luminosity, and hence calculate the distance.
Figure: Cepheid Light CurveA graph of brightness vs time shows regular periodic variation. The period (typically 1-100 days) determines the star's luminosity.
Type Ia Supernovae
At very large distances, individual stars cannot be observed. Type Ia supernovae are used instead:
- Type Ia supernovae occur when a white dwarf star explodes
- All Type Ia supernovae reach approximately the same peak luminosity (~10^36 W)
- They are extremely bright and can be seen in distant galaxies
- Used to measure distances up to billions of light years
Worked Examples
Worked Example: Radiant Flux IntensityQuestion: The Sun has a luminosity of 3.8 × 10^26 W. Calculate the radiant flux intensity at a distance of 4 light years. (1 ly = 9.46 × 10^15 m)
SolutionDistance: d = 4 × 9.46 × 10^15 = 3.78 × 10^16 m
F = (L/4π d^2) = 3.8 × 10^264π × (3.78 × 10^16)^2
F = 3.8 × 10^261.80 × 10^34 = 2.1 × 10^-8 W m^-2
Worked Example: Distance Using Standard CandlesQuestion: A Type Ia supernova is observed with a peak radiant flux intensity of 9 × 10^-18 W m^-2. Assuming peak luminosity is 10^36 W, estimate the distance to the galaxy.
Solutiond = √(L/4π F) = √10^364π × 9 × 10^-18
d = √10^361.13 × 10^-16 = √8.85 × 10^51
d ≈ 9 × 10^25 m (about 10 billion light years)
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