https://en.wikipedia.org/wiki/SI_base_unit

• https://en.wikipedia.org/wiki/Luminous_intensity

• https://en.wikipedia.org/wiki/Luminous_flux

• https://en.wikipedia.org/wiki/Luminance

• https://en.wikipedia.org/wiki/Illuminance

• https://en.wikipedia.org/wiki/Solid_angle

• https://en.wikipedia.org/wiki/Boltzmann_constant

• https://en.wikipedia.org/wiki/Luminous_efficacy

• https://en.wikipedia.org/wiki/Candela

Candela Definition

The factor 683 originates from the 1979 SI definition, which establishes that 1 watt of radiant power at 555 nm corresponds to 683 lumens of luminous flux. This number is based on experimental data on human vision and ensures that the lumen measurement aligns with practical light source observations.

Generalized Candela Formula

Where:

• $\left({\mathrm{\Phi }}_v\right)$$\left( \Phi_v \right)$ is the luminous flux.

• $(K(555\text{nm}))$$\left( K(555 \ \text{nm}) \right)$ is the peak luminous efficacy at 555 nm.

• $(P(\lambda ))$$\left( P(\lambda) \right)$ represents the power distribution at each wavelength.

• $(V(\lambda ))$$\left( V(\lambda) \right)$ is the eye’s sensitivity curve at wavelength $\lambda$$\lambda$.

Expanded Candela Formula with Planck's Law

Where:

• $\left(h\right)$$\left( h \right)$ is Planck's constant.

• $\left(c\right)$$\left( c \right)$ is the speed of light.

• $\left(\lambda \right)$$\left( \lambda \right)$ is the wavelength.

• $\left(k_B\right)$$\left( k_B \right)$ is the Boltzmann constant.

• $\left(T\right)$$\left( T \right)$ is the temperature (here approximated to 5778 K for the sun's surface).

Candela in Terms of Solid Angle

Where:

• $\left(\mathrm{\Omega }\right)$$\left( \Omega \right)$ is the solid angle in steradians (sr).

• $\left(K_{\text{max}}\right)$$\left( K_{\text{max}} \right)$ is the peak luminous efficacy.

• $(V(\lambda ))$$\left( V(\lambda) \right)$ is the photopic luminosity function.

Human Vision Sensitivity: Experimental Measurement

To determine the peak luminous efficacy for any given condition (photopic, scotopic, or mesopic vision), experiments are conducted on human vision sensitivity. The key conditions are:

1. Photopic Vision (Daylight):

   • Peak luminous efficacy occurs at 555 nm under normal daylight conditions.

2. Scotopic Vision (Low Light/Night):

   • Under dim lighting, the peak shifts to 507 nm (blue-green), and the luminous efficacy decreases significantly.

3. Mesopic Vision (Twilight):

   • In transitional lighting between day and night, both photopic and scotopic vision are active, and the peak shifts between 555 nm and 507 nm.

Steps to Measure Peak Luminous Efficacy:

1. Light Source: Use a monochromatic light source of a specific wavelength, such as green (555 nm) or blue-green (507 nm), depending on the condition being tested.

2. Power Measurement: Measure the radiant power of the light source using a calibrated photometer. This gives the energy per second emitted by the light source in watts.

3. Human Perception Tests: Human participants assess the perceived brightness of the light. The intensity of different wavelengths is adjusted to find the point where humans perceive them as equally bright.

4. Luminous Efficacy: Calculate the ratio of perceived brightness (lumens) to radiant power (watts). The wavelength that corresponds to the maximum ratio of lumens to watts is the peak luminous efficacy ($K_{\text{max}}$$K_{\text{max}}$).

Luminous Efficacy Function

Luminous efficacy $(K(\nu ))$$\left( K(\nu) \right)$ can be represented as a multivariate function:

Where:

• $\left(\nu \right)$$\left( \nu \right)$ is the frequency of light.

• $\left(A_{\text{pupil}}\right)$$\left( A_{\text{pupil}} \right)$ is the pupil area.

• $\left(A_{\text{retina}}\right)$$\left( A_{\text{retina}} \right)$ is the effective retinal area.

• $\left(S_{\text{photoreceptors}}\right)$$\left( S_{\text{photoreceptors}} \right)$ is the sensitivity function of the photoreceptors (rods and cones).

Photopic Luminosity Function

The sensitivity of the human eye, represented by the photopic luminosity function $(V(\lambda ))$$\left( V(\lambda) \right)$, varies with wavelength:

• At 555 nm (green light), the sensitivity is maximized:

• As the wavelength shifts towards the blue or red, the sensitivity decreases. Example values include:

These values are empirical, based on human vision experiments, and standardized by organizations like the CIE (Commission Internationale de l'Éclairage).

Solid Angle and Candela

The term $\left(\mathrm{\Omega }\right)$$\left( \Omega \right)$ in the Candela equation represents the solid angle:

A solid angle is the three-dimensional analog of a two-dimensional angle. In the context of photometry, candela measures luminous intensity, which is the luminous flux per unit solid angle. For a light source emitting uniformly in all directions, the solid angle is $4\pi$$4\pi$ steradians.

Variables and Constants to Use:

1. Sun temperature $T$$T$: Approximately 5778 K (Kelvin).

2. Planck's constant $h$$h$: $6.626\times {10}^{-34}\text{J·s}$$6.626 \times 10^{-34} \ \text{J·s}$.

3. Speed of light $c$$c$: $2.998\times {10}^8\text{m/s}$$2.998 \times 10^8 \ \text{m/s}$.

4. Boltzmann constant $k_B$$k_B$: $1.381\times {10}^{-23}\text{J/K}$$1.381 \times 10^{-23} \ \text{J/K}$.

5. Wavelength range $\lambda$$\lambda$: We’ll consider visible light from 400 nm (blue) to 700 nm (red).

6. Luminous efficacy at peak $K_{\text{max}}$$K_{\text{max}}$: $683\text{lm/W}$$683 \ \text{lm/W}$ at 555 nm (the peak sensitivity of the human eye).

7. Photopic luminosity function $V(\lambda )$$V(\lambda)$: This varies across the visible spectrum, peaking at 555 nm.

8. Solid angle $\mathrm{\Omega }$$\Omega$: The solid angle subtended by an object in space (like a satellite or Earth's surface from space). We can compute this based on the angular size.

Step 1: Calculate the Solid Angle $\mathrm{\Omega }$$\Omega$

For simplicity, let’s assume we have a satellite with a circular area $A=10{\text{m}}^2$$A = 10 \ \text{m}^2$, and it is at Earth's distance from the Sun (1 AU). The solid angle $\mathrm{\Omega }$$\Omega$ is:

Step 2: Planck's Law for Radiated Power per Wavelength

Using Planck's law for the Sun’s surface temperature $T=5778\text{K}$$T = 5778 \ \text{K}$, we calculate the spectral radiance (power distribution per unit area and wavelength):

We will compute this over the visible range of $400\text{nm}$$400 \ \text{nm}$ to $700\text{nm}$$700 \ \text{nm}$.

Step 3: Incorporate the Photopic Luminosity Function and $K_{\text{max}}$$K_{\text{max}}$

We multiply the Planck distribution $B(\lambda )$$B(\lambda)$ by the photopic luminosity function $V(\lambda )$$V(\lambda)$ and include the luminous efficacy constant $K_{\text{max}}=683\text{lm/W}$$K_{\text{max}} = 683 \ \text{lm/W}$:

• $V(555\text{nm})=1$$V(555 \ \text{nm}) = 1$

• $V(510\text{nm})\approx 0.50$$V(510 \ \text{nm}) \approx 0.50$

• $V(600\text{nm})\approx 0.63$$V(600 \ \text{nm}) \approx 0.63$

• $V(700\text{nm})\approx 0.01$$V(700 \ \text{nm}) \approx 0.01$

Thus, the full equation becomes:

Step 4: Calculate the Integral for the Candela

Now, integrate the product of the Planck distribution, the luminosity function $V(\lambda )$$V(\lambda)$, and the constant $K_{\text{max}}=683\text{lm/W}$$K_{\text{max}} = 683 \ \text{lm/W}$ over the wavelength range from 400 nm to 700 nm:

Step 5: Final Output:

After numerically evaluating the integral and plugging in the values, we calculate the candela (cd) value as:

This large value reflects the intense radiative power of the Sun, combined with the small solid angle subtended by the satellite. The constant 683 (luminous efficacy) is included in the calculation to account for human eye sensitivity at different wavelengths within the visible spectrum.

Variables and Constants to Use:

1. Sun temperature $T$$T$: Approximately $5778\text{K}$$5778 \ \text{K}$ (Kelvin).

2. Planck's constant $h$$h$:

   $$h=6.626\times {10}^{-34}\text{J·s}=6.626\times {10}^{-34}\frac{\text{kg}\cdot {\text{m}}^2}{\text{s}}$$

3. Speed of light $c$$c$:

   $$c=2.998\times {10}^8\text{m/s}$$

4. Boltzmann constant $k_B$$k_B$:

   $$k_B=1.381\times {10}^{-23}\frac{\text{J}}{\text{K}}=1.381\times {10}^{-23}\frac{\text{kg}\cdot {\text{m}}^2}{{\text{s}}^2\cdot \text{K}}$$

5. Wavelength range $\lambda$$\lambda$: Consider visible light from 400 nm (blue) to 700 nm (red). Converted to meters:

   $$\lambda =400\times {10}^{-9}\text{m}\text{to}700\times {10}^{-9}\text{m}$$

6. Luminous efficacy at peak $K_{\text{max}}$$K_{\text{max}}$: The peak luminous efficacy at 555 nm is defined as $683\text{lm/W}$$683 \ \text{lm/W}$. But, in terms of SI base units, 1 lumen (lm) can be expressed as:

   $$1\text{lumen}=1\frac{\text{cd}\cdot \text{sr}}{1\text{W}}$$

   A watt (W) is:

   $$1\text{W}=1\frac{\text{kg}\cdot {\text{m}}^2}{{\text{s}}^3}$$

7. Photopic luminosity function $V(\lambda )$$V(\lambda)$: A dimensionless function representing the sensitivity of the human eye. It peaks at 555 nm:

   $$V(555\text{nm})=1,V(510\text{nm})\approx 0.50,V(600\text{nm})\approx 0.63,V(700\text{nm})\approx 0.01$$

8. Solid angle $\mathrm{\Omega }$$\Omega$: The solid angle subtended by an object, such as a satellite from the Sun. We can compute it based on the angular size:

   For an area $A=10{\text{m}}^2$$A = 10 \ \text{m}^2$ and distance $d=1.496\times {10}^{11}\text{m}$$d = 1.496 \times 10^{11} \ \text{m}$ (1 AU):

   $$\mathrm{\Omega }=\frac{A}{d^2}=\frac{10{\text{m}}^2}{(1.496\times {10}^{11}\text{m}{)}^2}=4.46\times {10}^{-23}\text{sr}$$

Step 2: Planck's Law for Radiated Power per Wavelength

Using Planck's law to describe the spectral radiance (power per unit area per unit wavelength per unit solid angle) at a temperature $T=5778\text{K}$$T = 5778 \ \text{K}$:

Units for $B(\lambda )$$B(\lambda)$:

• $B(\lambda )$$B(\lambda)$ has units of:

  $$\frac{\text{W}}{{\text{m}}^2\cdot \text{sr}\cdot \text{m}}=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^3\cdot \text{sr}}$$

Step 3: Incorporating the Photopic Luminosity Function and $K_{\text{max}}$$K_{\text{max}}$

The luminous efficacy $K_{\text{max}}$$K_{\text{max}}$ is used to convert radiometric power (watts) to luminous power (lumens). The full formula becomes:

Breaking it down:

• $683\text{lm/W}$$683 \ \text{lm/W}$ is:

  $$683\frac{\text{cd}\cdot \text{sr}}{\frac{\text{kg}\cdot {\text{m}}^2}{{\text{s}}^3}}=683\frac{\text{cd}\cdot \text{sr}\cdot {\text{s}}^3}{\text{kg}\cdot {\text{m}}^2}$$

• $V(\lambda )$$V(\lambda)$ is dimensionless.

• $B(\lambda )$$B(\lambda)$ has units:

  $$\frac{\text{kg}\cdot \text{m}}{{\text{s}}^3\cdot \text{sr}}$$

Step 4: The Full Candela Formula

The full equation now becomes:

Simplifying the units for clarity:

• The units of the final result should be in candela (cd):

  $$\text{cd}=\frac{\text{luminous flux in lm}}{\text{steradians in sr}}$$

Step 5: Final Candela Value:

Numerically solving the integral gives us the candela (cd) value as:

This value represents the luminous intensity of sunlight hitting the satellite and observed from Earth. The constant $683\text{lm/W}$$683 \ \text{lm/W}$ converts the power into units perceptible by the human eye (lumens), and the solid angle $\mathrm{\Omega }$$\Omega$ accounts for the small area of the satellite relative to the Sun’s distance.

Variables and Constants to Use:

1. Sun temperature $T$$T$: Approximately $5778\text{K}$$5778 \ \text{K}$ (Kelvin).

2. Planck's constant $h$$h$:

   $$h=6.626\times {10}^{-34}\frac{\text{kg}\cdot {\text{m}}^2}{\text{s}}$$

3. Speed of light $c$$c$:

   $$c=2.998\times {10}^8\frac{\text{m}}{\text{s}}$$

4. Boltzmann constant $k_B$$k_B$:

   $$k_B=1.381\times {10}^{-23}\frac{\text{kg}\cdot {\text{m}}^2}{{\text{s}}^2\cdot \text{K}}$$

5. Wavelength range $\lambda$$\lambda$: We consider visible light from 400 nm (blue) to 700 nm (red). Converted to meters:

   $$\lambda =400\times {10}^{-9}\text{m}\text{to}700\times {10}^{-9}\text{m}$$

6. Luminous efficacy at peak $K_{\text{max}}$$K_{\text{max}}$: The peak luminous efficacy at 555 nm is defined as $683\text{lm/W}$$683 \ \text{lm/W}$. Expressed in SI base units, this is:

   $$K_{\text{max}}=683\frac{\text{cd}\cdot \text{sr}}{\frac{\text{kg}\cdot {\text{m}}^2}{{\text{s}}^3}}=683\frac{\text{cd}\cdot \text{sr}\cdot {\text{s}}^3}{\text{kg}\cdot {\text{m}}^2}$$

7. Photopic luminosity function $V(\lambda )$$V(\lambda)$: This is dimensionless, representing the relative sensitivity of the human eye across wavelengths. It peaks at 555 nm:

   $$V(555\text{nm})=1,V(510\text{nm})\approx 0.50,V(600\text{nm})\approx 0.63,V(700\text{nm})\approx 0.01$$

8. Solid angle $\mathrm{\Omega }$$\Omega$: The solid angle subtended by an object like a satellite, calculated based on its size and distance. For an area $A=10{\text{m}}^2$$A = 10 \ \text{m}^2$ and distance $d=1.496\times {10}^{11}\text{m}$$d = 1.496 \times 10^{11} \ \text{m}$ (1 AU):

   $$\mathrm{\Omega }=\frac{A}{d^2}=\frac{10{\text{m}}^2}{(1.496\times {10}^{11}\text{m}{)}^2}=4.46\times {10}^{-23}\text{sr}$$

Step 2: Planck's Law for Radiated Power per Wavelength

Using Planck's law for the Sun’s temperature $T=5778\text{K}$$T = 5778 \ \text{K}$, we calculate the spectral radiance (power per unit area per unit wavelength per unit solid angle) in SI base units:

Let’s now break down the units for each part of this equation:

• $B(\lambda )$$B(\lambda)$ (spectral radiance) has units of:

  $$\frac{\text{W}}{{\text{m}}^2\cdot \text{sr}\cdot \text{m}}=\frac{\text{kg}\cdot \text{m}}{{\text{s}}^3\cdot \text{sr}}$$

• $h$$h$ (Planck’s constant) has units:

  $$h=6.626\times {10}^{-34}\frac{\text{kg}\cdot {\text{m}}^2}{\text{s}}$$

• $c$$c$ (speed of light) has units:

  $$c=2.998\times {10}^8\frac{\text{m}}{\text{s}}$$

• $k_B$$k_B$ (Boltzmann’s constant) has units:

  $$k_B=1.381\times {10}^{-23}\frac{\text{kg}\cdot {\text{m}}^2}{{\text{s}}^2\cdot \text{K}}$$

• Wavelength $\lambda$$\lambda$ is measured in meters, with a range of:

  $$400\times {10}^{-9}\text{m}\text{to}700\times {10}^{-9}\text{m}$$

• $T$$T$ (temperature of the Sun) is measured in Kelvin:

  $$T=5778\text{K}$$

Step 3: Incorporating the Photopic Luminosity Function and $K_{\text{max}}$$K_{\text{max}}$

Now, we integrate over the visible wavelength range and include the photopic luminosity function $V(\lambda )$$V(\lambda)$, along with the luminous efficacy $K_{\text{max}}=683\text{lm/W}$$K_{\text{max}} = 683 \ \text{lm/W}$ expressed in base units.

Thus, the full equation becomes:

Breaking it down:

• $683\frac{\text{cd}\cdot \text{sr}\cdot {\text{s}}^3}{\text{kg}\cdot {\text{m}}^2}$$683 \ \frac{\text{cd} \cdot \text{sr} \cdot \text{s}^3}{\text{kg} \cdot \text{m}^2}$ comes from the luminous efficacy.

• $V(\lambda {)}^2$$V(\lambda)^2$ is dimensionless.

• $\frac{2\pi h{c}^2}{{\lambda }^5}$$\frac{2 \pi h c^2}{\lambda^5}$ comes from Planck’s law, where each constant has explicit SI base units.

• The exponential term $\frac{1}{e^{\frac{hc}{\lambda k_{B}T}}-1}$$\frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}$ is dimensionless as well.

Step 4: The Full Candela Formula

Here’s the full equation for candela with all units explicitly written:

Step 5: Final Candela Value:

After numerically solving the integral and plugging in all these values, we get the candela (cd) value as:

This large value reflects the intensity of sunlight hitting the satellite and observed from Earth, accounting for the human eye's sensitivity (via $V(\lambda )$$V(\lambda)$) and the conversion from radiant power to luminous power using $K_{\text{max}}=683\text{lm/W}$$K_{\text{max}} = 683 \ \text{lm/W}$.