CIE AS/A Level Physics 25.1 Standard candles Study Notes- 2025-2027 Syllabus
CIE AS/A Level Physics 25.1 Standard candles Study Notes – New Syllabus
CIE AS/A Level Physics 25.1 Standard candles Study Notes at IITian Academy focus on specific topic and type of questions asked in actual exam. Study Notes focus on AS/A Level Physics latest syllabus with Candidates should be able to:
- understand the term luminosity as the total power of radiation emitted by a star
- recall and use the inverse square law for radiant flux intensity ( F ) in terms of the luminosity ( L ) of the source \( F = L/(4 \pi d^2) \)
- understand that an object of known luminosity is called a standard candle
- understand the use of standard candles to determine distances to galaxies
Luminosity of a Star
Luminosity is a fundamental property used in astrophysics to describe how much energy a star emits.
Definition of Luminosity
Luminosity is the total power of electromagnetic radiation emitted by a star in all directions per second.
- It is measured in watts (W).
- It describes the true brightness of a star, independent of distance.
Even if two stars appear equally bright, the one farther away may actually have a much higher luminosity.
What Determines a Star’s Luminosity?
- Its surface temperature (hotter → more luminous)
- Its radius (larger → more luminous)
- Its nuclear fusion rate in the core
Related Formula (Stefan–Boltzmann Law) (Not required here but useful context)
\( \mathrm{L = 4\pi R^2 \sigma T^4} \)
Shows that luminosity increases strongly with temperature.
Example
What does luminosity tell you about a star?
▶️ Answer / Explanation
It tells you how much total power the star emits as radiation each second, independent of how far away it is.
Example
Two stars appear equally bright from Earth. Star A is twice as far away as Star B. Which star has the greater luminosity?
▶️ Answer / Explanation
If Star A is farther away but appears just as bright, it must emit much more power than Star B.
Therefore, Star A has greater luminosity.
Example
A star has a luminosity of \( \mathrm{3.8\times10^{26}\ W} \). Explain what this value means physically.
▶️ Answer / Explanation
This means the star emits \( \mathrm{3.8\times10^{26}} \) joules of energy every second in the form of electromagnetic radiation.
This is the star’s true power output, regardless of the observer’s distance.
Inverse Square Law for Radiant Flux Intensity
The radiant flux intensity (also called apparent brightness) of a star decreases with distance according to the inverse square law.
Formula
\( \mathrm{F = \dfrac{L}{4\pi d^2}} \)
- \( \mathrm{F} \) = radiant flux intensity (W m\(^{-2}\)) — power received per unit area
- \( \mathrm{L} \) = luminosity of the star (W)
- \( \mathrm{d} \) = distance from the observer to the star (m)
This equation shows that brightness decreases with the square of the distance.
If distance doubles → intensity becomes one quarter.
Meaning of the Inverse Square Law
- Light spreads out uniformly over a sphere of radius \( \mathrm{d} \).
- The surface area of a sphere is \( \mathrm{4\pi d^2} \).
- As distance increases, the same energy is spread over a larger area → intensity decreases.
Example
A star has luminosity \( \mathrm{L = 1.0\times10^{26}\ W} \). Find the radiant flux intensity at a distance of \( \mathrm{2.0\times10^{11}\ m} \).
▶️ Answer / Explanation
Use \( \mathrm{F = \dfrac{L}{4\pi d^2}} \):
\( \mathrm{F = \dfrac{1.0\times10^{26}}{4\pi (2.0\times10^{11})^2}} \)
\( \mathrm{F = \dfrac{1.0\times10^{26}}{4\pi \times 4\times10^{22}}} \)
\( \mathrm{F = \dfrac{1.0\times10^{26}}{5.03\times10^{23}} \approx 199\ W\,m^{-2}} \)
Intensity ≈ \( \mathrm{2.0\times10^{2}\ W\,m^{-2}} \)
Example
If the radiant flux intensity from a star decreases by a factor of 9, by what factor has the distance changed?
▶️ Answer / Explanation
Inverse square law:
\( \mathrm{\dfrac{F_2}{F_1} = \left( \dfrac{d_1}{d_2} \right)^2} \)
Given:
\( \mathrm{\dfrac{F_2}{F_1} = \dfrac{1}{9}} \)
Thus:
\( \mathrm{\left( \dfrac{d_1}{d_2} \right)^2 = \dfrac{1}{9}} \Rightarrow \dfrac{d_2}{d_1} = 3 \)
Distance has tripled.
Example
A star emits \( \mathrm{3.5\times10^{28}\ W} \) of power. An observer measures a radiant flux intensity of \( \mathrm{1.4\times10^{-8}\ W\,m^{-2}} \). Calculate the distance to the star.
▶️ Answer / Explanation
Start with the inverse square law:
\( \mathrm{F = \dfrac{L}{4\pi d^2}} \)
Rearrange:
\( \mathrm{d = \sqrt{\dfrac{L}{4\pi F}}} \)
Substitute:
\( \mathrm{d = \sqrt{\dfrac{3.5\times10^{28}}{4\pi (1.4\times10^{-8})}}} \)
Calculate denominator:
\( \mathrm{4\pi (1.4\times10^{-8}) \approx 1.76\times10^{-7}} \)
\( \mathrm{d = \sqrt{\dfrac{3.5\times10^{28}}{1.76\times10^{-7}}}} \)
\( \mathrm{d = \sqrt{1.99\times10^{35}} = 4.46\times10^{17}\ m} \)
Distance ≈ \( \mathrm{4.5\times10^{17}\ m} \)
Standard Candles in Astronomy
In astrophysics, a standard candle is an object whose luminosity is known. Because its true luminosity is known, astronomers can determine its distance using the inverse square law.
Definition
A standard candle is an astronomical object whose luminosity is known or can be determined independently.
Since luminosity \( \mathrm{L} \) is known and brightness (flux \( \mathrm{F} \)) is measured from Earth, distance can be found using:
\( \mathrm{F = \dfrac{L}{4\pi d^2}} \) \)
Rearranged for distance:
\( \mathrm{d = \sqrt{\dfrac{L}{4\pi F}}} \)
Examples of Standard Candles
- Type Ia supernovae — extremely reliable standard candles for distant galaxies
- Cepheid variable stars — luminosity linked to their pulsation period
- RR Lyrae stars — useful within our galaxy
Why Standard Candles Are Important
- Allow measurement of distances far beyond what parallax can measure
- Used to map the structure of the Milky Way
- Essential for determining distances to distant galaxies
- Critical for measuring the expansion rate of the Universe (Hubble’s law)
Example
What is meant by a “standard candle” in astronomy?
▶️ Answer / Explanation
A standard candle is an object with known luminosity that can be used to determine distance using its observed brightness.
Example
Explain why Type Ia supernovae are considered good standard candles.
▶️ Answer / Explanation
- They reach nearly the same peak luminosity because they result from white dwarfs reaching a critical mass.
- Their luminosity curve is very consistent and can be calibrated.
- They are extremely bright → visible across billions of light-years.
Example
A standard candle with luminosity \( \mathrm{L = 5.0\times10^{30}\ W} \) is observed from Earth with a flux \( \mathrm{F = 2.0\times10^{-12}\ W\,m^{-2}} \). Calculate its distance.
▶️ Answer / Explanation
Use the inverse square law:
\( \mathrm{d = \sqrt{\dfrac{L}{4\pi F}}} \)
Substitute values:
\( \mathrm{d = \sqrt{\dfrac{5.0\times10^{30}}{4\pi (2.0\times10^{-12})}}} \)
Compute denominator:
\( \mathrm{4\pi (2.0\times10^{-12}) \approx 2.51\times10^{-11}} \)
Then:
\( \mathrm{d = \sqrt{1.99\times10^{41}} = 4.46\times10^{20}\ m} \)
Distance ≈ \( \mathrm{4.5\times10^{20}\ m} \)
Use of Standard Candles to Determine Distances to Galaxies
Standard candles are one of the most important tools in astrophysics for measuring distances that are far too large for parallax or other geometric methods. By knowing their luminosity and measuring how bright they appear to us, we can calculate the distance to galaxies.
1. The Principle Behind Standard Candles
A standard candle is an object whose luminosity (L) is known. When we measure how bright it appears (its flux, F), we can calculate the distance using the inverse square law:
\( \mathrm{F = \dfrac{L}{4\pi d^2}} \)
Rearranging for distance:
\( \mathrm{d = \sqrt{\dfrac{L}{4\pi F}}} \)
This is the key method for finding distances to galaxies containing standard candles.
2. Why Standard Candles Are Needed
- Galaxies are extremely far away → parallax is too small to measure.
- Stars inside galaxies vary greatly in brightness → not all provide reliable distance indicators.
- Standard candles have predictable luminosities → ideal for long-distance measurements.
3. Examples of Standard Candles Used for Galaxies
- Cepheid variable stars — luminosity related to pulsation period; useful for nearby galaxies.
- Type Ia supernovae — extremely bright; used for very distant galaxies.
- RR Lyrae stars — useful for globular clusters and nearby galaxies.
4. How Standard Candles Give Distance to a Galaxy
- Identify a standard candle in the galaxy (e.g., a Type Ia supernova).
- Its luminosity \( \mathrm{L} \) is known from calibration.
- Measure its flux \( \mathrm{F} \) from Earth.
- Use the inverse square law to calculate the distance \( \mathrm{d} \).
- This distance is taken as the distance to the entire galaxy.
Thus, the presence of even one standard candle in a galaxy allows us to determine its distance.
Example
Why do astronomers rely on standard candles to measure distances to galaxies?
▶️ Answer / Explanation
Because galaxies are extremely far away and geometric methods like parallax do not work. A standard candle has known luminosity, so measuring its apparent brightness gives the distance.
Example
A Type Ia supernova in a distant galaxy has luminosity \( \mathrm{1.0\times10^{36}\ W} \). Its measured flux at Earth is \( \mathrm{4.0\times10^{-15}\ W\,m^{-2}} \). Calculate the distance to the galaxy.
▶️ Answer / Explanation
Use the inverse square law:
\( \mathrm{d = \sqrt{\dfrac{L}{4\pi F}}} \)
Substitute:
\( \mathrm{d = \sqrt{\dfrac{1.0\times10^{36}}{4\pi (4.0\times10^{-15})}}} \)
Compute denominator:
\( \mathrm{4\pi (4.0\times10^{-15}) \approx 5.03\times10^{-14}} \)
\( \mathrm{d = \sqrt{1.99\times10^{49}} = 4.46\times10^{24}\ m} \)
Distance ≈ \( \mathrm{4.5\times10^{24}\ m} \)
Example
Explain why Type Ia supernovae allow measurement of distances far beyond those possible with Cepheid variables.
▶️ Answer / Explanation
- Cepheids are bright, but not extremely bright → detectable only in relatively nearby galaxies.
- Type Ia supernovae are among the brightest events in the Universe → visible across billions of light-years.
- Their peak luminosity is nearly identical because they occur when a white dwarf reaches a critical mass (~1.4 solar masses).
- This makes them reliable standard candles even at very large distances.
Therefore, Type Ia supernovae extend the distance scale to far-distant galaxies.