# Why is the candela a base unit of the SI?

The candela is defined as

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540\cdot10^{12}$ hertz and that has a radiant intensity in that direction of $1/683$ watt per steradian.

Now, I don't see any reason why luminous intensity should be a base unit. I mean, we can just as simply specify the luminous intensity of any source in terms of its frequency and radiant intensity. It seems (to me) that luminous intensity has just been used to clump the two parameters together.

Contrast this with the definition of a meter

The meter is the length of the path travelled by light in vacuum during a time interval of $1/299 792 458$ of a second.

Now, in the definition of a meter there is one key player, namely light. Even though this definition is given in terms of second (as candela is given in terms of frequency and radiant intensity) but it also refers to a fixed fundamental 'thing' in nature: light. It is primarily on the properties of light that this definition is based, and we require experimentation to determine the 'length' of a meter. This is similar to the definition for a second which refers to an actual real-world object, namely a cesium atom.

But I see no such reference in the definition of a candela. Why is candela a base SI unit then?

EDIT: As pointed out by some in the comments below, apparently candela has been chosen as a base unit considering its usefulness in other fields (quite unrelated to physics). So I guess my question is modified then to: Is there any rationale from the standpoint of physics to choose candela as a base SI unit?

• Because the SI is also used by biologists and other scientists for which human sensibility to light is a meaningful quantity. Physics isn't everything :) – Koaaala May 10 '15 at 15:18
• – Qmechanic May 10 '15 at 15:30
• Even more important than biologists and other scientists, SI is used by commerce and industry. Enabling commerce and industry is the primary reason why SI exists. – David Hammen May 10 '15 at 15:32

The point is that luminous intensity is intensity as perceived by the human eye, and particularly taking into account the fact that the same amount of power will be perceived as brighter or dimmer depending on whether the wavelength is at a maximum of the eye's sensibility or at a minimum.

This makes the candela ever so slightly washier than the other six SI base units, because you need to rely on physiological measurements of the average human, whoever that is, but the simple fact is that you cannot measure how bright a light looks to a human eye using just stopwatches, yardsticks and weights - you need to use the eye itself as a measurement device, or at the very least calibrate with one. Luminous intensity is a measure of the biological response of a specific system (if not of the subjective perception this response causes), and unless you're willing to define the unit of luminous intensity in terms of neuron activity on the optical nerve then you really do need a unit of measurement that's independent of the mks triplet.

In general, suppose your eye receives light from a source with a radiant intensity of $$I_\mathrm{r.i.}=(I_\mathrm{r.i.})\:\mathrm{W\:sr^{-1}m^{-2}},$$ i.e. each unit area $A$, channelling a pencil of radiation of solid angle $\Omega$ receives $I_\mathrm{r.i.}A\Omega$ joules of radiated energy per second. This still doesn't tell you how bright the source will look to you, though, because the different receptors are more or less sensitive to radiation at different wavelengths. However, this dependence has been quite thoroughly studied using a number of methods, which have resulted in a fairly standard luminosity function: that is, a function $$\bar{y}(\lambda),\text{ also denoted }V(\lambda),$$ that tells you how bright lights look at different wavelengths. Thus if $\bar{y}(\lambda_1)/\bar{y}(\lambda_2)=2$ then a light at $\lambda_1$ will look twice as bright as a lamp of the same radiant intensity at $\lambda_2$.

There is of course some individual variation, as well as the tough metrological problem of measuring and standardizing the luminosity function, and controlling for the fact that different populations can have different average responses, but that's all in the game and it's ultimately someone else's (i.e. not a physicist's) business. And if you want to really go down the rabbit hole, you need to control for the fact that the perceived intensity will vary under well-lit versus low-light conditions, and down and down it goes. However, you only need to normalize once for each set of conditions; hence the value of standard candles.

The candela comes in, essentially, as the units of the standard luminosity function. A light source of wavelength $\lambda$ and radiant intensity $I_\mathrm{r.i.}$ will have a (perceived) luminous intensity $$I_\mathrm{l.i.}=\bar{y}(\lambda)I_\mathrm{r.i.},$$ where now the luminous intensity $I_\mathrm{l.i.}$ is a completely different beast to the radiant intensity, depending as it does on a human reaction, so it is measured in candela. It follows, then, that the luminosity function has units of $\mathrm{cd}/(\mathrm{W\:sr^{-1}m^{-2}})$, and the role of the SI definition is to normalize it such that $$\bar{y}(555\:\mathrm{nm})=\frac{1\:\mathrm{cd}}{\mathrm{W\:sr^{-1}m^{-2}}}.$$ From here one can then fill out the rest of the curve for $\bar{y}(\lambda)$ using only comparative measurements of (perceived) luminous intensity, which are much easier.

When measuring the "brightness" of a light source, there is a huge number of different quantities of interest, each with their own unit but a very similar name to the others, and depending on exact details like whether you're integrating over angle, or surface, or wavelength, or any combination thereof. It is very easy, as a physicist, to simply give up and reckon that "luminous intensity" will simply be one of the list. Similarly, it's easy to simply slide over terms like photometry and not realize that it's very different to radiometry.

That just means, though, that we need to up our game a bit and realize that there's an extra dimension at play here - the subjective sensation of brightness as perceived by the human eye as a measuring device - that we need to include on an equal footing to our clocks, meter sticks and (soon to be) watt balances, if we really want to produce measurements which are useful in a world inhabited by humans.

• (+1) That was elegant, informed and illuminating (ba-dum-ts), but the conclusion at the end was especially compelling! – let's have a breakdown Mar 27 '18 at 4:51
• I understand that the candela is useful. But on purely theoretical ground, shouldn't effective radiation dose have its own base unit. What about temperature as percieved by humans and things like that? I guess they aren't as useful. But you could apply a similar reasoning, right? – jinawee Nov 18 '18 at 17:33
• @jinawee I don't see the case for "temperature as perceived by humans". I don't understand equivalent radiation doses well enough, but I do see the potential for a case that as a spectral-weighted quantity it is equivalent to the luminous intensity; a clearer equivalent is the sone as a unit for loudness, but it has large drawbacks in the larger variability in human hearing (both between individuals and over time for each individual) as compared to luminous intensity. (cont.) – Emilio Pisanty Nov 18 '18 at 17:42
• As you say, usefulness is almost certainly part of the criteria used originally to define the canonical set of seven base units; the candela probably made the cut because you get a lot more lights sold rated in lumens than any of the competitors. As I've argued elsewhere, the concept of a base unit (or at least, the canonical set in current use) makes very little sense come the New SI, but it seems the historical inertia is by now too great to overcome. – Emilio Pisanty Nov 18 '18 at 17:44

That definition is for calibration. Luminosity is based on a standardized model of human eye and takes into account sensitivity of human eye to different wavelengths. Two source with the same intensity but different frequency have different luminosity. Luminosity is related to intensity and frequency by the luminosity function.

Is there any rationale from the standpoint of physics to choose candela as a base SI unit?

That's still the wrong question. The right question: Why do these standards (any of them) exist? The answer is commerce and industry. The SI doesn't exist to support the sciences; that it does support the sciences is a nice side benefit. It exists to support humans, and in particular, commerce and industry.

We humans rely extensively on vision. It is arguably our most important sense. We need artificial lighting to make our modern world possible. Standards for lighting are important. People make mistakes in low lighting, get headaches in overly bright lighting. The use of 540 terahertz in the definition of the candela is anything but arbitrary. That is where the human vision system peaks. The factor of 1/683 -- that's to make the physics-based definition of the candela consistent with older definitions. Consistency is very important in metrology. Consistency is the reason for that seemingly arbitrary factor of 1/299792458 used in the definition of the meter.

As an aside, there are lots of physicists who don't use SI units. To an astrophysicist, the second, meter, and kilogram are too small. They tend to use years (qualified with kilo, mega, giga) for time, astronomical units or parsecs for distance, and the mass of the Sun as their base units. To a particle physicist, the second, meter, and kilogram are too big, and also are not particularly meaningful. Particle physicists view energy as a base concept (as opposed to mass, which is just one of many forms of energy). To a theoretical physicist, the SI is but a tiny step toward a consistent set of units. As far as a theoretical physicist is concerned, SI is little better than the customary units still used in the US.

• Theoretical physicist in education here, and believe me: SI is far, far, far better than US customary units. And SI will improve vastly with the new definitions planned in the next years (all units will be defined by fixing universal constants). Although I have to admit I use atomic units, geometric units or Gauss-units where appropriate. The only thing that really annoys me about SI, is that temperature has its own base unit (instead of making the Boltzmann factor merely an energy unit conversion factor). – Sebastian Riese May 10 '15 at 20:48
• I disagree. If commercial and industrial standards were the only thing at play, the candela would be very welcome as a unit accepted for use with SI units like the hectare or the AU, with no need to grant it base unit status. The reason it's a base unit is that luminous intensity is brightness as perceived by the human eye, and is this that introduces a new physical dimension and the corresponding need for a base unit. – Emilio Pisanty Jan 19 '16 at 22:25
• @EmilioPisanty -- This is not a new physical dimension. The issue of how many physical dimensions exist is highly debated. Maybe zero, maybe one, maybe three, but not seven. The candela and units derived from it exist because people are important from a perspective of commerce and industry (which are the ultimate drivers of the standards). From the perspective of physics, there's nothing special going on here. Technically advanced beings who live on a planet orbiting a red dwarf will have a very similar concept of the speed of light but a very different concept of luminosity. – David Hammen Jan 19 '16 at 22:43
• Yes. In the meantime, we'll just keep calling "luminous intensity as perceived by humans" simply "luminous intensity". The perceived brightness is as much an independent physical dimension from radiant intensity as length is independent from time: both pairs can be related given access to a special object (resp. light and humans) which we assume to be constant for the duration of the standard. If you're OK with taking down length as an independent physical dimension then that's fine with me, but I gotta wonder why you're debating the SI base units at all. – Emilio Pisanty Jan 19 '16 at 22:49
• The point is that measures of the brightness of a light source are split completely down the middle: one set which does not require access to humans to measure (i.e. using appropriate bolometers), and one set which does. Same for kinematic measures which require access to a standard mass versus the ones which don't. – Emilio Pisanty Jan 19 '16 at 22:51

Personally, from a pure physics point of view, i don't think we need the candela as a base S.I. unit. it seems that all we need is: m,kg,s,A,K,mol

These can all be related to basic physics constants (see Physics Today, July 2014, p. 37): c,h,delta nu(pick any atomic transition),e,k(Boltzmann),N_A(Avogadro's)

Actually, even N_A is only necessary from a practical sense of macroscopic measurements. Once that is given up, and N_A is added, then one also includes units like the candela related to human visual perception. Regarding sound perception, to be consistent there should be another unit added such as the loudness level "phon" or loudness scale "sone". We see, we hear, to be consistent, perhaps a human sound scale unit should be included too; it's not never-ending, eight "fundamental" S.I. units seem to sum it up from our current best point of view.