Stellar radii
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State Wien's displacement law and how it relates to stellar temperature.
Wien's displacement law states λmax ∝ 1/T, where λmax is the peak wavelength emitted by a black body and T is its temperature. This means hotter stars emit light at shorter (bluer) wavelengths, while cooler stars emit at longer (redder) wavelengths.
State Stefan-Boltzmann law, defining each term.
Stefan-Boltzmann law states L = 4πσr²T⁴, where L is the luminosity of a star, σ is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W m⁻² K⁻⁴), r is the radius of the star, and T is the surface temperature of the star.
How can Wien's displacement law and Stefan-Boltzmann law be used to estimate the radius of a star?
First, use Wien's law (λmax ∝ 1/T) to find the star's temperature (T) by measuring λmax. Then, use the Stefan-Boltzmann law (L = 4πσr²T⁴) and the star's luminosity (L) to calculate the radius (r).
Define redshift and its cause in the context of distant objects.
Redshift is the increase in wavelength (and decrease in frequency) of electromagnetic radiation from distant objects. This is caused by the Doppler effect as the source moves away from the observer. The further away an object is, the greater the redshift.
State the formula relating redshift (z) to wavelength change (Δλ), original wavelength (λ), velocity (v), and speed of light (c).
The redshift formula is: z ≈ Δλ/λ ≈ v/c. This approximation is valid for relatively small velocities compared to the speed of light.
Explain how redshift provides evidence for the expansion of the Universe.
The observation that most galaxies exhibit redshift indicates that they are moving away from us. Since the amount of redshift increases with distance, this implies that the Universe is expanding uniformly.
State Hubble's law and define each term.
Hubble's law states v = H₀d, where v is the recessional velocity of a galaxy, H₀ is Hubble's constant (approximately 70 km s⁻¹ Mpc⁻¹), and d is the distance to the galaxy.
How does Hubble's law support the Big Bang theory?
Hubble's law implies that if we trace the expansion of the Universe backward in time, all matter would converge to a single point. This supports the Big Bang theory, which postulates that the Universe originated from an extremely hot, dense state and has been expanding ever since.
A star has a peak wavelength of 500 nm. Estimate its surface temperature using Wien's Law (Wien's displacement constant b = 2.90 x 10⁻³ m K).
Using Wien's Law, λmax = b/T. Therefore, T = b/λmax = (2.90 x 10⁻³ m K)/(500 x 10⁻⁹ m) = 5800 K. The star's surface temperature is approximately 5800 K.
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