Math Encounters Blog

Introduction

Scientists always face the problem of making their work accessible to the public. Accessibility is crucial to scientific research continuing to receive funding. Part of this accessibility is creating analogies that relate scientific data to aspects of everyday life. Here is an analogy that I encountered recently comparing the light from a distant supernova to that from a firefly located thousands of kilometers away.

Mingus [SN SCP-0401] was so distant and so faint — the equivalent of looking at a firefly from 3,000 miles (5,000 kilometers) away — that its true nature remained a mystery for a while, researchers said.

Here is a video that goes into more detail on supernova SCP-0401.

http://www.youtube.com/watch?v=5BpnJI3SLv0&w=560&h=315

The speaker in this video states (@time=3:34 minutes) that:

This supernova is about as bright as a firefly viewed from 3000 miles away.

Let's see if we can provide more insight into these statements about fireflies and supernovae. I will treat this as a Fermi problem -- my work will be very approximate.

Background

Optical Power of Fireflies

I found the following quote (Figure 2) in the book "A Physical Study of the Firefly" by Coblentz on Google Books. Access to the book is complete and free.

Figure 2: Quote on the Candlepower of a Firefly Flash.

For the following analysis, I will treat the flash of a firefly as having the an optical power of 1/50^th of an international candle (unit in use at the time the quote was made), which is at the top end of the range stated in the quote of "1/50 candle to 1/400 candle". I will then convert this unit to the modern candela.

Image of SN SCP-0401

Figure 1 shows an image of SN SC-0401 (Source). Notice how astronomers do not get a lot of visual information to work with.

Figure 1: Image of Supernova SC-0401.

Supernova Characteristics

Standard Candle

Very long distances in astronomy are measured using objects of known absolute magnitude, which are commonly known as standard candles. We can measure the apparent magnitude of a standard candle and calculate the distance from Earth using Equation 1. For more detail on Equation 1, see this blog post.

Eq. 1

where

• R is the distance of the celestial object from the observer
• M is the absolute magnitude of the celestial object when positioned at 10 pc from the observer.
• m is the apparent magnitude of the celestial object at a distance R from the observer.

Supernova Absolute Magnitude, Apparent Magnitude, and Distance

It turns out that supernova SN SC-0401 is a special type of supernova that is a standard candle called a Type 1A supernova. A Type 1A supernova has an absolute magnitude of -19.3 ± 0.3 (reference).

What is Brightness?

The brightness of an object is expressed in units of lux. The Wikipedia defines lux as follows.

The lux (symbol: lx) is the SI unit of illuminance and luminous emittance, measuring luminous flux per unit area. It is equal to one lumen per square metre. In photometry, this is used as a measure of the intensity, as perceived by the human eye, of light that hits or passes through a surface. It is analogous to the radiometric unit watts per square metre, but with the power at each wavelength weighted according to the luminosity function, a standardized model of human visual brightness perception.

We need to make some observations about this definition:

• The lux is the power density of the light impacting a given area weighted by the eye's spectral sensitivity to wavelengths present in that light.
• The spectral sensitivity of the eye varies greatly with wavelength, for example, the eye's sensitivity to light at 490 nm amounts to 20% of its sensitivity at 555 nm (Source).
• The spectral sensitivity of the eye is greatest for light at 555 nm (i.e. green)

For light from the Sun, we can relate the lux level to the received visible light power using a few facts:

• Direct sunlight has a maximum lux level of 120,000 (Source)
• Direct sunlight has a power level of 378 W/m² in the visible wavelength range (Source)
• Light power is normalized to its equivalent 555 nm power for use in visual calculations. The equivalent 555 nm optical power is 142.6 W/m² (Source).

Analysis

Figure 3 shows my analysis that compares the light from a type 1A supernova to a candle at 3000 miles. Several things are worth noting about this analysis.

• I computed the light level (in lux) from a magnitude 0 star as ~2.1 μlux

  It turns out this number is very close to the 2.09 μlux reported on this web site. This light level is from a magnitude 0 star that is at the zenith.

• I assumed that supernova is at 45% angle to the horizon.

  This is the latitude that I live at. I estimated the attenuation of the atmosphere using Appendix A from this blog post.

• I assumed the Sun and and a Type 1A supernova have similar spectral distribution

  Admittedly, this is a big stretch, but I am just working approximately. The approximation comes in when come up with a conversion factor between W/m² of visible light power and lux level.

My very approximate result shows that a Type 1A supernova at 10 billion light-years away and a firefly at maximum light intensity and 3000 miles range do have about the same light level.

Figure 3: Analysis of Type 1A Supernova and a Candle at 3000 miles.

Conclusion

I was able to show that the brightness of SN SCP-0401 is about the same as a firefly at 3000 miles. That does seem pretty dim!

Introduction

Today, I was asked a question about the amount of visible optical power that actually reaches the Earth's surface. I also need to compute the illuminance of this optical power, which tells me how bright this light appears.

The word visible is the interesting part of the question because it involves the human eye. In optical engineering we often break problems into one of two types, radiometric or photometric. Radiometric problems simply treat light as a form of energy and we do not care whether it can be seen or not. I normally work in the radiometric area -- the only light on my fiber is infrared. The photometric area involves the human eye and what can be seen. This area of optical engineering goes way back to the time of candles. Ever heard of candle-power? Photometry is quite different from radiometry and even has its own units.

I thought this was an interesting question that would also allow me to demonstrate a common application of integral calculus. Just a quick problem, but it does answer a question that came up. While I need only approximate results, but the method I use can easily be made more accurate. I only have a short period of time to come up with an answer.

The most difficult part of this area of optics is understanding the terms associated with lighting as perceived by the human eye, so I will spend a bit of time on the definitions. I have chosen to use the same symbology as the Wikipedia and I make liberal use of links to that wonderful source of information.

Background

I am going to base my definitions on those from the Wikipedia. I will add some commentary that will hopefully help explain the terms a bit better.

Definitions

candela (symbol:cd)

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of wavelength 555 nm and that has a radiant intensity in that direction of 1?683 W per steradian. The odd factor of 1/683 was chosen to make the candela close in value to the original unit called the candle.

luminous flux

The luminous flux is the total radiated power factored by the sensitivity of the human eye. The eye's sensitivity is different for different wavelength. The lumen unit computes the human eye-equivalent 555 nm light level. Measuring light levels in units of lumen is useful for comparing the eye-perceived light level of different light sources. Two lights generators with the same lumen output will produce the same level of lighting as far as the human eye is concerned.

lumen (symbol:lm)

The lumen is a unit that describes luminous flux emitted into a unit solid angle (1 steradian [sr]) by an isotropic point source having a luminous intensity of 1 candela.

lux (symbol:lx)

The lux is a unit of illuminance that measures the luminous flux per unit area. The lux level from the at object determines its brightness. Photographer's light meters measure light level in terms of lux (they may display in other units, but they measure lux).

luminosity function

The luminosity function or luminous efficiency function describes the average spectral sensitivity of human visual perception of brightness as a function of wavelength. For example, the human eye is much more sensitive to green light than it is to red light. This means that for a red light to have the same brightness to the human eye as green light, the red light must be at a much higher radiated power level. I think of luminous flux as measuring the an amount of light relative to an amount of 555 nm light with the same human eye response.

illuminance

Illuminance is the luminous flux incident per unit area on a surface. It is measured in units of lux [lumens/m²].

solar constant

The amount of solar radiation that reaches the Earth. The value is usually assumed to be ~1.361 kW/m², however, the solar "constant" is not really constant because the distance between the Earth and Sun varies throughout the year. Anyone wanting to accurately measure this radiation can measure the radiation for themselves by using on of these weather stations. Getting an accurate reading of the radiation can greatly effect the results of this investigation. Solar activity also varies slightly during the Sun's 11 year cycle.

solar irradiance spectrum

The electromagnetic power emitted from the Sun as a function of wavelength.

Luminous Flux Calculation

I want to calculate the maximum luminous flux per m² at the Earth's surface. Evaluating Equation 1 will give me what I want (modified from this source).

Eq. 1

where

• E_e is the luminous flux per unit area [lx = lm/m²]
• is the standard luminosity function [dimensionless]
• is the spectral power distribution of the radiation [W/m²/nm]
• ? is wavelength [nm]

Solar Irradiance Spectrum

The Wikipedia has a great article on sunlight that provides an excellent graphic that illustrates the optical power spectrum of the Sun in space and at sea level (Figure 1). Figure 1 is describe by the function described as part of Equation 1.

Figure 1: Optical Power Spectrum of the Sun in Space and at Sea Level.

Luminosity Function

The Wikipedia also has a graphic that illustrates the sensitivity of the optical eye relative to its sensitivity at 550 nm, which is the wavelength of maximum sensitivity. I will use the function described by the dark solid black line (CIE 1931 standard). I chose that function just because it is still commonly used for describing the eye's sensitivity under well-lit conditions. It will work fine for my approximate analysis here. I will use Figure 2 for my , described as part of Equation 1.

Figure 2: Photopic (black) and scotopic (green) luminosity functions.[c 1] The photopic includes the CIE 1931 standard[c 2] (solid), the Judd–Vos 1978 modified data[c 3] (dashed), and the Sharpe, Stockman, Jagla & Jägle 2005 data[c 4] (dotted). The horizontal axis is wavelength in nm.

Analysis

Figure 3 shows my analysis. The basic approach is simple, but appeared to work.

• Digitize Figures 1 and 2 (I use Dagra)
• Use a spline interpolation to create continuous versions of these functions.
• Generate the product of the solar irradiance and the eye's luminosity functions.
• Integrate the product of the two functions (Mathcad).

Figure 3: Mathcad Analysis of the Illuminance of Sunlight After Passing Through the Atmosphere.

My analysis shows that Figure 1 represents a 100K lux light level at sea level. While the solar constant is about 1350 W/m² for light in space, only about 1000 W/m² actually gets through the atmosphere (Wikipedia states 1004 W/m²). Of this amount, only about 378 W/m² gets through in the form of visible light. Of the visible light, the eye perceives it as having the same brightness 143 W/m² of 555 nm light. Since I was working assuming that the Sun is directly overhead (90° solar angle), this is as bright as things get because the light has the least atmosphere to pass through. Appendix A shows how the maximum light level drops with solar angle.

Conclusion

My answer of 146 W/m² agrees reasonably close with the answer 170 W/m² given by the "Handbook of Optical Metrology: Principles and Application" by Yoshizawa (ISBN 1420019511), and a snippet from this reference is included in Appendix B. So I think I understand how this number is determined.

Appendix A: Dynamic Range of Sunlight on Earth As a Function of Solar Angle

.
Figure 4 is a log plot of some data I found on this web page. I was surprised at how quickly the brightness level drops as the solar angle decreases.

Figure 4: Sunlight Level (Lux) As a Function of Solar Angle.

Appendix B: Text Quotation on the Solar Irradiance at the Earth's Surface

Figure 5 is an excerpt from the "Handbook of Optical Metrology: Principles and Application" by Yoshizawa (ISBN 1420019511) that states his estimate for the luminous power at sea level on the Earth.

Figure 5: Quotation on the Solar Irradiance at the Earth's Surface.

Introduction

I have been reading a number of interesting astronomy articles lately. These articles often refer to the apparent and absolute magnitude of a celestial object or event (example). I thought I would work through a bit of the math associated with these terms.

Background

History

Like most tales in science, the story of measuring the visual magnitudes of celestial objects dates back to the ancient Greeks. Most sources say that Hipparchus was the first to develop a system for ranking the brightness of celestial object by their visual magnitudes. He said that the brightest objects were of the "first magnitude" and he ranked less bright objects on a scale from second to sixth magnitude, with sixth magnitude being at the limit of human visibility. This system is the basis for the more formal magnitude system that astronomers use today. I have a number of astronomy-oriented posts planned and I will be using this information in the posts to follow.

Definitions

The Wikipedia gives the following definitions for apparent and absolute magnitude:

apparent magnitude

The apparent magnitude (m) of a celestial body is a measure of its brightness as seen by an observer on Earth, adjusted to the value it would have in the absence of the atmosphere. The brighter the object appears, the lower the value of its magnitude.

absolute magnitude

Absolute magnitude (M) is the measure of a celestial object's intrinsic brightness. It is the apparent magnitude an object would have if it were at a standard luminosity distance (10 parsecs [pc]) away from the observer, in the absence of astronomical extinction. It allows the true brightnesses of objects to be compared without regard to distance.

Analysis

Mathematical Definition

Norman Robert Pogson provided a quantitative framework for the Greek's qualitative approach. He defined a typical first magnitude star as being 100 times brighter as a typical sixth magnitude star. In general, an object with apparent magnitude value less than another object by five will be 100 times brighter. Equation 1 describes this relationship.

Eq. 1

where

• L₁ is the luminance of object 1.
• L₂ is the luminance of object 2.
• m₁ is the apparent magnitude of object 1
• m₂ is the apparent magnitude of object 2.

This formula is similar to those listed here.

Magnitude Variation with Range

The Wikipedia states that the apparent magnitude, absolute magnitudes, and range are related by Equation 2.

Eq. 2

where

• R is the distance of the celestial object from the observer
• M is the absolute magnitude of the celestial object when positioned at 10 pc from the observer.
• m is the apparent magnitude of the celestial object at a distance R from the observer.

Figure 1 shows my derivation of this result. The inverse square law figures prominently in this derivation.

Figure 1: Apparent Magnitude as a Function of Range and Absolute Magnitude.

Examples

Figure 2 shows a couple of examples of computing absolute magnitude given apparent magnitude and range. I was able to obtain the same absolute magnitudes as listed in the Wikipedia for the Sun and Eta Carinae.

Figure 2: Two Examples of Magnitude Calculations.

Conclusion

I wanted to review the basics of brightness (aka luminance) and magnitude. This is a nice warm-up exercise for some other astronomy posts that I have planned.