Using Radiation to Investigate Stars

6 Using Radiation to Investigate Stars Learners should be able to demonstrate and apply their knowledge and understanding of:
a)	The idea that the stellar spectrum consists of a continuous emission spectrum, from the dense gas of the surface of the star, and a line absorption spectrum arising from the passage of the emitted electromagnetic radiation through the tenuous atmosphere of the star
b)	The idea that bodies which absorb all incident radiation are known as black bodies and that stars are very good approximations to black bodies
c)	The shape of the black body spectrum and that the peak wavelength is inversely proportional to the absolute temperature (defined by: T(K) = θ(℃)+273.15)
d)	Wien’s displacement law, Stefan’s law and the inverse square law to investigate the properties of stars – luminosity, size, temperature and distance [N.B. stellar brightness in magnitudes will not be required]
e)	The meaning of multiwavelength astronomy and that by studying a region of space at different wavelengths (different photon energies) the different processes which took place there can be revealed

Learners should be able to demonstrate and apply their knowledge and understanding of:
a) Stellar Spectrum: Emission and Absorption
The spectrum of a star is composed of two main components:
1. Continuous Emission Spectrum:
Stars are massive, dense spheres of hot gases, primarily hydrogen and helium, undergoing nuclear fusion at their cores.
The star’s surface, often referred to as the photosphere, is dense enough to behave like a thermal radiator, producing a continuous spectrum.
This continuous spectrum contains all wavelengths of light and resembles the characteristic curve of a black body, with a peak wavelength dependent on the temperature (as described by Wien’s Displacement Law).
The continuous emission originates because electrons in the dense gases of the star’s surface can occupy a wide range of energy states, allowing transitions that emit photons across a broad range of wavelengths.
2. Line Absorption Spectrum:
The light from the photosphere passes through the star’s tenuous outer atmosphere, or the chromosphere. This layer is cooler and less dense compared to the photosphere.
In the outer atmosphere, specific wavelengths of light are absorbed by atoms or ions. The absorption occurs when photons have energies that exactly match the energy required to excite electrons in these atoms to higher energy states.
As a result, the star’s spectrum exhibits dark absorption lines superimposed on the continuous background. These lines are known as Fraunhofer lines in the Sun’s spectrum and are unique to the chemical elements present in the star’s atmosphere.
This combination of a continuous spectrum (from the photosphere) and absorption lines (from the atmosphere) allows astronomers to determine the composition, temperature, pressure, and motion of the stellar atmosphere.
Figure 1 Continuous spectrum
b) Black Bodies and Stars
1. Black Body Concept:
A black body is an idealized object that absorbs all incident electromagnetic radiation, regardless of wavelength or angle of incidence.
Such an object is also a perfect emitter of radiation when heated, and its emission spectrum is described by Planck’s Law. The radiation is entirely determined by the object’s temperature.
A black body emits a continuous spectrum, with the intensity and peak wavelength dependent on its temperature:
Figure 2 Black Body Radiation Graph
Wien’s Displacement Law: The peak wavelength ([math]λ_{max}[/math]) of a black body shifts inversely with temperature (T), i.e.,
[math]λ_{max} ∝ \frac{1}{T} \\ λ_{max} = \frac{b}{T} [/math]
Stefan-Boltzmann Law: The total energy radiated by a black body is proportional to the fourth power of its temperature,
[math]E ∝ T^4[/math]
2. Stars as Approximate Black Bodies:
Stars are not perfect black bodies but are very good approximations:
– The dense photosphere of a star emits a nearly continuous spectrum resembling a black body curve.
– The exact shape of the spectrum and the peak wavelength allow astronomers to estimate the star’s surface temperature.
For example, the Sun’s spectrum closely follows a black body curve with a peak wavelength corresponding to a temperature of approximately 5778 K.
Deviations from the ideal black body behavior are primarily due to absorption and emission features caused by the elements in the stellar atmosphere.
c) The Shape of the Black Body Spectrum and the Peak Wavelength
A black body spectrum represents the radiation emitted by an object that absorbs all incident electromagnetic radiation. This emission depends only on the temperature of the object, and its shape is described by Planck’s Law:
[math]I(\lambda, T) = \frac{2hc^2}{\lambda^5} \cdot \frac{1}{e^{\frac{hc}{\lambda k_B T}} – 1} [/math]
Where:
– I(λ,T): Intensity of radiation at wavelength λ and temperature T,
– h: Planck’s constant,
– c: Speed of light,
– [math]k_B[/math]: Boltzmann constant,
– λ: Wavelength,
– T: Absolute temperature (in kelvins, K).
⇒ The Spectrum:
1. Continuous Spectrum: Black body radiation spans all wavelengths.
2. Peak Wavelength: The spectrum has a distinct peak, which shifts depending on the temperature of the object.
3. Temperature Dependence:
– Hotter objects emit more energy at shorter wavelengths (higher frequencies).
– Cooler objects emit more energy at longer wavelengths (lower frequencies).
The temperature in Kelvin (T) is related to Celsius (θ) by:
T(K) = θ(℃) + 273.15

d) Wien’s Displacement Law
This law relates the peak wavelength ([math]λ_{max}[/math] ) of a black body’s spectrum to its absolute temperature (T):
[math]λ_{max} = \frac{b}{T}[/math]
Where:
– [math]λ_{max}[/math]: Wavelength at maximum intensity (meters),
– T: Temperature in kelvins,
– [math]b = 2.897 × 10^{-3} mK[/math]: Wien’s constant.
Applications:
1. By observing the peak wavelength of a star’s spectrum, astronomers can determine its temperature.
2. Hot stars (e.g., blue stars) emit more energy at shorter wavelengths, while cooler stars (e.g., red stars) emit more at longer wavelengths.
⇒ Stefan-Boltzmann Law:
The Stefan-Boltzmann law describes the total energy emitted per unit surface area of a black body:
[math]E = σT^4[/math]
Where:
– E: Radiant energy per unit area (W/m²),
– T: Temperature in kelvins,
– [math]σ = 5.67 × 10^{-8} W/m^2 K^2 [/math]: Stefan-Boltzmann constant.
⇒ Applications:
1. The luminosity (L) of a star, which is the total energy radiated per second, can be calculated by:
[math]L = 4πR^2 σT^4[/math]
Where:
– R: Radius of the star,
– T: Surface temperature.
2. If we know the luminosity and temperature of a star, we can estimate its size (radius).
⇒ Inverse Square Law:
The inverse square law relates the apparent brightness (b) of a star to its luminosity (L) and distance (d):
[math]b = \frac{L}{4πd^2 }[/math]
Where:
– b: Apparent brightness (as observed from Earth),
– L: Luminosity of the star,
– d: Distance to the star.
Figure 3 Inverse Square Law
⇒ Applications:
1. By measuring the apparent brightness and knowing the luminosity, the distance to a star can be calculated.
2. Combining the Stefan-Boltzmann law and the inverse square law, we can link observable quantities like brightness, distance, size, and temperature.
e) Multiwavelength Astronomy
Definition: Multiwavelength astronomy involves studying astronomical objects and regions of space across different parts of the electromagnetic spectrum, such as radio, infrared, visible, ultraviolet, X-ray, and gamma-ray wavelengths.
Figure 4 Multiwavelength Astronomy
⇒ Multiwavelength Observations are Important:
1. Different Wavelengths Reveal Different Processes:
– Radio waves detect cold gas clouds and synchrotron radiation from particles spiraling in magnetic fields.
– Infrared observes cooler objects like dust clouds and stars forming within them.
– Visible light detects the emission from stars.
– Ultraviolet reveals hot stars, active galactic nuclei, and young stellar populations.
– X-rays and gamma rays detect high-energy phenomena like black holes, neutron stars, and supernova explosions.
– Complete Picture of an Object: Observing the same region of space at different wavelengths provides a fuller understanding of its physical and chemical properties.
⇒ Example:
A star-forming region like the Orion Nebula:
– Infrared reveals newborn stars hidden within dust clouds.
– Visible light shows the hot stars illuminating the nebula.
– X-rays detect energetic processes from young stellar objects.