Formulas

Wien’s Law (also called Wien’s Displacement Law) is defined as so: For a blackbody (or star), the wavelength of maximum emission of any body is inversely proportional to its absolute temperature (measured in Kelvin). As a result, as the temperature rises, the maximum (peak) of the radiant energy shifts toward the shorter wavelength (higher frequency and energy) end of the spectrum (bluer). This is what the equation looks like:

\[ \text{Peak Intensity}= \frac{0.0029 \text{meters} \cdot K}{T \text{(in Kelvin)}} \]

Actual formula:

\[ \lambda_{\text{max}} = \frac{b}{T}, \]

where:

• T is the absolute temperature
• b is a constant of proportionality called Wien's displacement constant, equal to 2.897771955…×10⁻³ m ⋅ K, or b ≈ 2898 μm ⋅ K.

If we mean Wien's Law by simpler terms, we can refer to it as such: Wien’s Law tells us where (meaning at what wavelength) the star's brightness is at a maximum. See the picture below – the red dot under the word “visible” is the peak for the 6000 K object. In other words, Wien's law tells us what color the object is the brightest at.

The curves above (3000K, 4000K, 5000K, 6000K, black curves) are Planck curves for objects at different temperatures. Planck's law tells us how intense the thermal emission is at each wavelength. The peak of this Planck curve (the red dot) is what Wien’s Law tells us. Wien's Law tells us that objects of different temperature emit spectra that peak at different wavelengths:

• Hotter objects emit most of their radiation at shorter wavelengths; hence they will appear to be bluer.
• Cooler objects emit most of their radiation at longer wavelengths; hence they will appear to be redder

Remember, at any wavelength, a hotter object radiates more energy (is more luminous or brighter) at all wavelengths than a cooler one.

Answer:

Let's put the "variables" in Wien's Law equation, and solve for the surface temperature.

• b = 2.897771 ⋅ 10^-3 m ⋅ K
• λ = 400 \(\text{nm}\) = \(4 \cdot 10^{-7} m\)

\[ \lambda_{\text{max}} = \frac{b}{T} \]

\[ 4 \cdot 10^{-7} m = \frac{2.897771 \cdot 10^{-3} m \cdot K}{T} \]

\[ T = \frac{2.897771 \cdot 10^{-3} m \cdot K}{4 \cdot 10^{-7} m} \]

\[ = 7244 \text{ K} \]

How do we know this is correct?

If we recall the HR Diagram, we know that this star is a little hotter than our Sun and the wavelength of 400 nanometers would be at the violet end of the spectra. The hotter the star, the smaller the wavelength in nanometers is, which translates to a color shift towards the ultraviolet (UV) end of the spectra. Conversely, the cooler the star, the larger the wavelength in nanometers, which shifts color more towards the infrared (IR) end of the spectrum.

https://wwelsh.sdsu.edu/CLASSES/ASTROBIO/LECTURES/wien_law.pdf

The temperature of stars other than the Sun can be approximated using a similar means by treating the emitted energy as a black body radiation.

\[ L = 4 \pi R^2 \sigma T^4, \]

where L is the luminosity, σ is the Stefan–Boltzmann constant, R is the stellar radius and T is the surface temperature.

The formula can be rearranged to calculate the temperature:

\[ T = \sqrt{\frac{L}{4 \pi R^2 \sigma}} \]

or alternatively, the radius of the star:

\[ R = \sqrt{\frac{L}{4 \pi \sigma T^4}} \]

Via the original equation, we can beautifully see that a star's luminosity depends a lot more on the star's surface temperature, than on the star's radius.

Pogson's formula is a mathematical relationship that defines how astronomers quantify the brightness of stars and other satellites on a logarithmic scale called the magnitude system.

The magnitude scale likely dates to before the ancient Roman astronomer Claudius Ptolemy, whose star catalog popularized the system by listing stars from 1st magnitude (brightest) to 6th magnitude (dimmest). The modern scale was mathematically defined to closely match this historical system by Norman Pogson in 1856.

The scale is reverse logarithmic: the brighter an object is, the lower its magnitude number. A difference of 1.0 in magnitude corresponds to the brightness ratio of \(\sqrt[5]{100}\), or about 2.512. For example, a magnitude 2^m star is 2.512 times as bright as a magnitude 3^m star, 6.31 times as magnitude 4^m, and 100 times magnitude 7^m. Difference: \(7-2=5\), thus, \(2.512^5 = 100\).

\[ m_1 - m_2 = -2.5 \log_{10} \Biggl( \frac{F_1}{F_2} \Biggr), \]

where:

• m₁ and m₂ are the corresponding star's magnitudes
• E₁ and E₂ are the corresponding star's fluxes

Every physical body spontaneously and continuously emits electromagnetic radiation and the spectral radiance of a body, B_ν, describes the spectral emissive power per unit area, per unit solid angle and per unit frequency for particular radiation frequencies. The relationship given by Planck's radiation law, given below, shows that with increasing temperature, the total radiated energy of a body increases and the peak of the emitted spectrum shifts to shorter wavelengths.

According to Planck's distribution law, the spectral energy density (energy per unit volume per unit frequency) at given temperature is given by:

\[ u_{\nu} (\nu, T) = \dfrac{8 \pi h \nu^{3}}{c^3} \dfrac{1}{\exp \Biggl( \dfrac{h \nu}{k_{B} T} \Biggr) - 1} \]

alternatively, the law can be expressed for the spectral radiance of a body for frequency ν at absolute temperature T given as:

\[ B_{\nu} (\nu, T) = \dfrac{2 h \nu^{3}}{c^2} \dfrac{1}{\exp \Biggl( \dfrac{h \nu}{k_B T} \Biggr) - 1} \]

where:

• k_B is the Boltzmann constant:
• h is the Planck constant:
• c is the speed of light in the medium, whether material or vacuum

The cgs units of spectral radiance B_ν are \(\text{erg} \cdot \text{s}^{-1} \text{sr}^{-1} \text{cm}^{-2} \text{Hz}^{-1}\). The terms B and u are related to each other by a factor of ⁠\(\frac{4 \pi}{c}\) since B is independent of direction and radiation travels at speed c. The spectral radiance can also be expressed per unit wavelength λ instead of per unit frequency.

Now, it's a lot to take in, just in a few reads. So let's go through it one by one. The above formulas we've looked at, were the spectral radiance of a body with a temperature, and the energy density, which in a given unit volume per unit frequency.

• The spectral energy density just tells us how much energy (in Joules) per unit volume per unit solid angle per unit frequency a blackbody emits at a specific frequency at a specific temperature.
• The spectral power density tells us how bright the blackbody is, and how much energy radiates from it.

\[ \Delta v = v_e \ln \Biggl( \frac{m_0}{m_f} \Biggr) \]

\[ v = H d \]

\[ F = G \frac{m_1 m_2}{r^2} \]