Spectral Flux Density: Definition, Units, and Astrophysical Use

(Topic)

Spectral Flux Density: Definition, Units, and Astrophysical Use

A source can deliver a total electromagnetic energy flux

F

(1)

to an observer, but that single number does not say how the received energy is distributed across wavelength or frequency.

A spectrum answers that additional question.

The most common astronomical quantities are the spectral flux densities

|------| |fλ(λ) | -------

(2)

and

|------| |fν(ν).| --------

(3)

The first describes received flux per unit wavelength.

The second describes received flux per unit frequency.

Their defining relations are

|--------------| |dF = fλ(λ)dλ | ----------------

(4)

and

|--------------| dF = fν(ν )dν,| ----------------

(5)

with absolute interval widths understood when changing between oppositely ordered wavelength and frequency coordinates.

The total flux is recovered by integration:

|-----∫--------------∫-------------| | ∞ ∞ | |F = fλ(λ )dλ = fν(ν)dν. | -------0---------------0-----------

(6)

This article develops the physical meaning, units, coordinate transformation, common astronomical conventions, and the connection to photometric passbands.

1 Why total flux is not enough

Suppose two stars deliver the same bolometric flux to a telescope:

F1 = F2.

(7)

The first star may emit more strongly at short wavelengths while the second emits more strongly at long wavelengths.

Their total received power per unit area can be equal even though their spectra are very different.

Thus

F

(8)

answers

|----------------------------------------------------| |How much total radiant power arrives per unit area? ------------------------------------------------------

(9)

while

fλ(λ)

(10)

fν(ν)

(11)

answers

|-----------------------------------------------------------------| How is that flux distributed across the electromagnetic spectrum? | ------------------------------------------------------------------

(12)

This distinction is essential for temperature measurements, spectroscopy, color indices, photometry, and detector response.

2 From finite spectral bins to a density

Imagine dividing the wavelength axis into small bins.

For a bin centered near λ_i with width Δλ_i, let the flux contained in that bin be

ΔFi.

(13)

An average spectral flux density in the bin is

ΔFi- . Δ λi

(14)

As the bin width becomes small,

|------------| | dF | |fλ(λ) = dλ-.| --------------

(15)

Likewise,

|------------| | dF- | |fν(ν) = dν .| --------------

(16)

The word density here means density with respect to a spectral coordinate.

It does not mean mass density or spatial volume density.

Figure 1. Spectral flux density arises by dividing the flux in a narrow spectral bin by the bin width and then taking the limit of increasingly narrow bins.

3 Units of spectral flux density

Ordinary flux has SI units

[F] = W m −2.

(17)

Therefore

|------------------| |[f λ] = W m −2m −1 | -------------------

(18)

when wavelength is measured in meters.

The final m⁻¹ refers to wavelength interval, not another spatial area dimension.

Astronomers often use wavelength units such as nanometers, micrometers, or angstroms:

W m⁻² nm⁻¹,		(19)
W m⁻² μm⁻¹,		(20)
erg s⁻¹ cm⁻² Å⁻¹.		(21)

For frequency spectral flux density,

|-----------−2---−-1-| -[fν] =-W-m----Hz--.-|

(22)

Because

Hz = s−1,

(23)

this is dimensionally different in appearance from f_λ, even though both represent the same physical spectrum using different spectral coordinates.

4 A dimensional check

From

dF = fλdλ,

(24)

the units are

( ) W m− 2m −1 (m ) = W m −2.

(25)

Similarly,

(W m −2Hz −1)(Hz ) = W m− 2.

(26)

This unit cancellation is one of the easiest ways to remember what spectral flux density means.

Figure 2. A spectral flux density becomes an ordinary flux only after it is multiplied by a spectral interval and integrated or summed over the desired band.

5 Example 1: a flat spectrum over a finite wavelength band

Suppose

f λ = 2.00 × 10 −11 W m −2nm −1

(27)

is constant from

500 nm

(28)

600 nm.

(29)

The integrated flux in that band is

F_band	= ∫ _500 nm^600 nmf_λ dλ	(30)
	= f_λ(100 nm)	(31)
	= 2.00 × 10⁻⁹ W m⁻².	(32)

The spectral flux density and the integrated flux are not interchangeable.

One is per unit wavelength; the other has already been integrated over a finite wavelength range.

6 Wavelength and frequency are reciprocal coordinates

For vacuum electromagnetic radiation,

|-------| | c | ν = -. | -----λ--

(33)

Therefore wavelength increases when frequency decreases.

This means that equal wavelength intervals do not correspond to equal frequency intervals.

Different spectral-coordinate widths are related by differentiation:

ν	= ,	(34)
	= −.	(35)

Thus

|--------------| ||d ν| = c-|dλ|.| --------λ2------

(36)

Equivalently,

|--------------| | 2 | ||d λ| = λ-|dν|.| --------c-------

(37)

Figure 3. Equal intervals in wavelength generally map to unequal intervals in frequency because frequency is inversely proportional to wavelength.

7 Deriving the conversion between $f λ$ and $f ν$

The physical flux contained in a small spectral interval must not depend on which coordinate is used to label that interval.

Therefore

|----------------| -fλ|dλ| =-fν|dν|.|

(38)

Using

c-- |d ν| = λ2|dλ|,

(39)

we obtain

c fλ = f ν-2. λ

(40)

Hence

|----------| fλ = c-fν.| ------λ2----

(41)

Solving for f_ν gives

|----------| | λ2 | fν = --fλ.| ------c-----

(42)

Since

c λ = -, ν

(43)

the same relations can be written

|-----c----| fν = -2fλ,| ------ν-----

(44)

provided the two functions are evaluated at corresponding frequency and wavelength.

8 The numerical value depends on the spectral coordinate

The quantities

fλ

(45)

and

fν

(46)

describe the same physical radiation, but their numerical values are generally very different.

This is because they are densities with respect to different coordinates.

A useful analogy is population density:

people per square kilometer

(47)

and

people per square mile.

(48)

The underlying population is the same, but the numerical density changes because the unit interval changes.

For spectra, the coordinate transformation is not merely a constant unit conversion.

The factor

c-- λ2

(49)

changes with wavelength.

9 A useful invariant per logarithmic interval

Multiply the conversion relation

λ2 fν = --f λ c

(50)

c ν = -. λ

(51)

Then

|----------| νf ν = λfλ.| ------------

(52)

This quantity represents flux per logarithmic spectral interval.

Indeed,

dν- dln ν = ν ,

(53)

fν dν = νfν dlnν.

(54)

Likewise,

fλ dλ = λf λd ln λ

(55)

in magnitude.

Plots of

νfν

(56)

λfλ

(57)

are therefore especially useful for comparing energy contributions across logarithmic portions of a broad spectrum.

10 The same spectrum can look different in $f λ$ and $f ν$

Suppose a source is flat in frequency spectral flux density:

fν = constant.

(58)

Then

fλ = c-fν. λ2

(59)

Therefore the same source has

|----------| |f ∝ λ− 2.| --λ--------

(60)

A spectrum that appears horizontal on an f_ν versus ν plot does not appear horizontal on an f_λ versus λ plot.

Conversely, a spectrum flat in f_λ is not flat in f_ν.

Figure 4. The same physical spectrum can have different plotted shapes in per-wavelength and per-frequency representations because the spectral density transforms with a wavelength-dependent Jacobian.

11 Why blackbody peaks depend on spectral representation

A blackbody can be represented by

B λ(λ,T )

(61)

B ν(ν,T ).

(62)

These describe the same thermal radiation but are densities per different spectral coordinates.

Consequently, the maximum of

B λ

(63)

does not correspond to the maximum of

B ν

(64)

by simply substituting

ν = c∕λ.

(65)

The wavelength-space and frequency-space density functions include the same Jacobian issue as f_λ and f_ν.

This is why the phrase

the peak of the spectrum

(66)

is incomplete unless the spectral representation is specified.

12 Common wavelength units

Because f_λ is a density per unit wavelength, its numerical value changes when the wavelength unit changes.

Suppose

fλ,m

(67)

is measured per meter.

Since

1 nm = 10−9 m,

(68)

the same physical spectrum expressed per nanometer satisfies

------------------- | − 9 | fλ,nm-=-10--f-λ,m.--

(69)

Similarly,

|--------−-10-----| fλ,˚A-=-10---fλ,m.-

(70)

This factor is required because a one-nanometer bin is much narrower than a one-meter wavelength bin.

13 SI and cgs wavelength-density conversion

A commonly encountered cgs unit is

erg s−1cm −2˚A −1.

(71)

The conversion is

|--------------------------------------| |1 W m −2 m −1 = 10−7 ergs−1 cm −2˚A −1.| ----------------------------------------

(72)

Equivalently,

|------------------------------------| | − 1 −2˚ −1 7 −2 −1 | 1-erg-s--cm---A----=-10--W--m---m---.-

(73)

Careful unit labeling is essential when comparing spectra from different catalogs or instruments.

14 The jansky

Frequency spectral flux density is commonly expressed in janskys.

The jansky is defined by

|---------−26-----−-2---−1-| -1-Jy-=-10----W--m---Hz---.-

(74)

In cgs units,

|-------------------------------| 1-Jy-=-10-−23 erg-s−1cm-−2-Hz−-1.-

(75)

Submultiples are common:

1 mJy	= 10⁻³ Jy,	(76)
1 μJy	= 10⁻⁶ Jy,	(77)
1 nJy	= 10⁻⁹ Jy.	(78)

Radio astronomy commonly reports source flux densities directly in Jy, mJy, or smaller units.

15 Example 2: converting 3631 Jy near 550 nm

Consider

fν = 3631 Jy.

(79)

In SI,

−23 − 2 −1 fν = 3.631 × 10 W m Hz .

(80)

λ = 550 nm,

(81)

f_λ	= f_ν	(82)
	≈ 3.598 × 10⁻² W m⁻² m⁻¹.	(83)

Per nanometer,

|--------------------------------| |fλ ≈ 3.598 × 10−11 W m −2 nm −1.| ----------------------------------

(84)

Per angstrom in cgs units,

|-----------------------------------| f ≈ 3.598 × 10− 9 erg s−1cm −2 ˚A− 1.| -λ-----------------------------------

(85)

This example demonstrates why a large numerical value in Jy can correspond to a small numerical value per nanometer or per angstrom.

16 Connection to the AB magnitude system

The AB magnitude system is designed around frequency spectral flux density.

A source with constant

f ν

(86)

has constant AB magnitude with frequency under the idealized monochromatic definition.

The conventional zero point corresponds approximately to

|--------------| -fν-=-3631-Jy.-|

(87)

For a monochromatic flux density, one commonly writes

|-----------------(---------)--| | ---fν--- | |mAB = − 2.5log10 3631 Jy .| --------------------------------

(88)

For real broadband observations, the instrument response and the precise averaging convention must be included.

Thus the 3631 Jy relation is an entry point to the AB system, not a substitute for full synthetic photometry.

Figure 5. The jansky is a unit of frequency spectral flux density, and the AB magnitude convention is tied to a reference frequency flux density of approximately 3631 janskys.

17 Spectral luminosity density

A source can also be described intrinsically by its luminosity per unit wavelength or frequency.

Define

|---------| | dL | L λ = --- | ------dλ---

(89)

and

|----------| | dL- | |Lν = dν .| -----------

(90)

Their units are

[L_λ]	= W m⁻¹,	(91)
[L_ν]	= W Hz⁻¹.	(92)

They satisfy the same coordinate transformation:

|------------| | c-- | |Lλ = λ2L ν,| -------------

(93)

and

|------------| | λ2 | |Lν = --L λ.| -------c-----

(94)

18 Inverse-square relation for spectral flux density

For an isotropic source in ordinary Euclidean geometry, with no absorption, redshift, or other propagation effects, the spectral luminosity density and observed spectral flux density satisfy

|--------------| | Lλ(λ-)| f λ(λ ) = 4 πr2 | ----------------

(95)

and

|---------------| | Lν(ν) | fν(ν) = ----2 .| ---------4πr----

(96)

The total luminosity relation

L = 4πr2F

(97)

is therefore mirrored at each spectral coordinate.

Figure 6. In the simple isotropic noncosmological case, every spectral slice of the luminosity spreads over the same spherical area, so spectral flux density also obeys inverse-square dilution.

19 Important caveats to the inverse-square spectral relation

The simple relation

Lν fν = ---2- 4πr

(98)

assumes that source and observer use corresponding spectral coordinates without additional shifts.

Real observations can require corrections for:

interstellar extinction;
atmospheric transmission;
instrumental throughput;
Doppler shifts;
cosmological redshift;
gravitational redshift;
anisotropic emission.

In cosmology, observed and emitted frequency intervals differ by redshift factors, so the monochromatic luminosity-flux relation requires additional care.

The inverse-square expression above is therefore the correct introductory local relation, not a universal formula for every astronomical situation.

20 Integrating a spectrum

If the complete spectrum is known,

|-------∫------------| | ∞ | |Fbol = fλ(λ) dλ | ---------0-----------

(99)

is the bolometric flux.

Equivalently,

|-------∫-∞----------| | | |Fbol = 0 fν(ν)d ν.| ---------------------

(100)

If only a finite spectral range is integrated, the result is a band-limited or interval-integrated flux.

For example,

∫ 600 nm F500−600 = fλ(λ)dλ. 500 nm

(101)

This distinction is central when comparing a detector measurement with a source’s bolometric output.

21 Discrete spectra from real instruments

A real spectrograph produces measurements in finite bins or pixels.

If bin i has width

Δ λi

(102)

and calibrated spectral flux density

fλ,i,

(103)

then the integrated flux can be approximated by

|----∑-----------| |F ≈ fλ,iΔ λi.| | i | ------------------

(104)

For nonuniform wavelength bins, each bin must use its own

Δ λi.

(105)

This discrete expression is the numerical counterpart of the continuous integral.

22 Continuum and spectral lines

Many astronomical spectra contain both a smooth continuum and narrow spectral features.

Write schematically

f λ(λ ) = fλ,cont(λ) + fλ,line(λ).

(106)

The integrated line flux is often obtained by subtracting the local continuum and integrating across the line:

|---------------------------------| | ∫ | Fline = [fλ(λ ) − fλ,cont(λ)]dλ.| ---------line-----------------------

(107)

The line flux has ordinary flux units:

W m −2,

(108)

even though the spectrum being integrated has spectral-flux-density units.

Figure 7. A spectral line sits above or below a local continuum. Integrating the continuum-subtracted spectral flux density across the feature gives an integrated line flux.

23 From spectral flux density to a photometric band signal

A broadband detector samples a weighted portion of the spectrum.

Let

SX (λ )

(109)

represent the response of photometric band X.

An energy-weighted schematic band signal is

|------∫-----------------| | | |FX ∝ fλ(λ)SX (λ)dλ. | -------------------------

(110)

This is the equation that motivated the present article.

It says:

determine the source spectral flux density at each wavelength;
multiply by the system response at that wavelength;
integrate the weighted contributions across the passband.

The measurement is therefore not generally equal to the spectrum evaluated at one representative wavelength.

Figure 8. A photometric passband performs a weighted integration of the source spectral flux density across a finite wavelength range.

24 Photon-counting detectors

A photon-counting detector responds to photon number, not directly to radiant energy.

Each photon has energy

hc- E γ = λ .

(111)

Therefore a spectral energy flux density corresponds to a photon spectral rate proportional to

fλ(λ ) λ ------= --fλ (λ ). hc∕λ hc

(112)

A schematic photon count rate is therefore

|------∫-------------------| | λ-- | NX ∝ f λ(λ )SX (λ )hc dλ.| ----------------------------

(113)

Depending on the adopted convention, some or all of the photon-weighting factors may be absorbed into the published response function.

This is why photometric passband definitions must be read carefully.

25 Band averages and normalization

Sometimes the goal is not an integrated band flux but a representative band-averaged spectral flux density.

A simple energy-weighted wavelength average can have the schematic form

|--------------------------| | ∫ f λ(λ)SX (λ )dλ | |⟨fλ⟩X = --∫-------------.| --------------SX-(λ)d-λ----|

(114)

Other photometric systems use different weightings and normalizations.

The important point is that an average spectral flux density requires both:

a weighting convention;
a normalization convention.

The next PhysicsLibrary article on photometric passbands and system response should develop these details more fully.

26 Spectral flux density, intensity, and luminosity are different quantities

Several closely related quantities are easy to confuse.

Spectral flux density

fν

(115)

f λ

(116)

describes received power per detector area per spectral interval.

Spectral luminosity density

L ν

(117)

L λ

(118)

describes intrinsic source power per spectral interval.

Specific intensity or spectral radiance

A quantity such as

Iν

(119)

also retains directional information per unit solid angle.

A common SI unit is

−2 −1 − 1 W m sr Hz .

(120)

Thus

|--------------| -fν-⁄=-L-ν ⁄=-Iν.|

(121)

They are related, but they answer different physical questions.

27 Example 3: converting $fλ$ to $fν$

Suppose

f λ = 1.00 × 10 −11 W m −2nm −1

(122)

λ = 500 nm.

(123)

First convert the density from per nanometer to per meter:

f = 109f = 1.00 × 10−2 W m −2 m −1. λ,m λ,nm

(124)

Then

f_ν	= f_λ	(125)
	= (1.00 × 10⁻²)	(126)
	≈ 8.34 × 10⁻²⁴ W m⁻² Hz⁻¹.	(127)

In janskys,

|------------| f ≈ 834 Jy.| --ν-----------

(128)

The large-looking Jy number and the small per-nanometer number represent the same physical spectrum.

28 Example 4: band integration from sampled data

Suppose a calibrated spectrum has three adjacent 10 nm bins:


Central wavelength	f_λ	Bin width

500 nm	1.0 × 10⁻¹¹	10 nm
510 nm	1.4 × 10⁻¹¹	10 nm
520 nm	1.2 × 10⁻¹¹	10 nm

with f_λ in

−2 − 1 W m nm .

(129)

Approximate the integrated flux as

F	≈∑ _if_λ,iΔλ_i	(130)
	= 10 × 10⁻¹¹	(131)
	= 3.6 × 10⁻¹⁰ W m⁻².	(132)

This is exactly the same logic as a Riemann sum.

29 Example 5: spectral luminosity and distance

Suppose an isotropic source has

L ν = 1.00 × 1014 W Hz −1

(133)

at some frequency, and it lies at

r = 10.0 pc.

(134)

Using

1 pc = 3.085677581 × 1016 m,

(135)

the distance is

r = 3.08568 × 1017 m.

(136)

Then

f_ν	=	(137)
	≈ 8.36 × 10⁻²³ W m⁻² Hz⁻¹.	(138)

In janskys,

|-------------3-----| fν-≈-8.36-×-10--Jy.--

(139)

The example illustrates that the inverse-square law applies to spectral luminosity density just as it applies to total luminosity in the simple noncosmological case.

30 Common mistakes

Treating spectral flux density as an ordinary integrated flux.
Forgetting the final per-wavelength or per-frequency unit.
Comparing a per-nanometer value directly with a per-meter value.
Comparing f_λ numerically with f_ν without applying the Jacobian transformation.
Assuming a spectrum flat in f_ν is also flat in f_λ.
Converting a blackbody peak between wavelength and frequency coordinates without recognizing that the density function changes.
Forgetting to convert f_λ to per meter before using the SI form of f_ν = λ²f_λ∕c.
Confusing spectral flux density with spectral luminosity density.
Confusing spectral flux density with specific intensity.
Treating 3631 Jy as a universal broadband flux without considering the passband definition.
Ignoring detector response when converting a spectrum into a photometric measurement.
Omitting the photon-energy factor when interpreting a photon-counting detector response under a convention that does not already include it.
Integrating discrete spectral samples without multiplying by their bin widths.
Assuming inverse-square dilution alone describes spectra affected by extinction, redshift, or anisotropic emission.

31 Connections to other PhysicsLibrary articles

The article Electromagnetic Waves and Wavelength: Frequency, Propagation, and Spectral Description supplies the relation

c ν = -- λ

(140)

and explains why wavelength and frequency are reciprocal spectral coordinates.

The electromagnetic spectrum article provides the larger band context in which spectral flux densities are measured.

The Luminosity article distinguishes intrinsic luminosity from received flux and gives the inverse-square relation

F = --L--. 4 πr2

(141)

The present article extends that idea spectrally:

L λ Lν fλ = 4πr2-, fν = 4πr2-

(142)

under the simple local isotropic assumptions described above.

The article on Color in Astrophysics uses broadband measurements whose underlying source information is the spectral flux density.

The next natural bridge is the response-weighted integral

|------∫-----------------| | | |FX ∝ fλ(λ)SX (λ)dλ. | -------------------------

(143)

Thus the conceptual chain is

|--------------------------------------------------------------------------------------------------------------------------------| -electromagnetic--wave-−→--wavelength--and-frequency-−-→-spectral-flux-density-−→--photometric--passband-−-→--magnitude--and-color.-

(144)

32 Summary

Spectral flux density describes how received electromagnetic flux is distributed across a spectral coordinate.

Per wavelength,

|----------| | dF- | |fλ = dλ .| -----------

(145)

Per frequency,

|----------| |f = dF- .| --ν----dν--|

(146)

The total flux is

|-----∫---------∫--------| |F = fλ dλ = fν d ν. -------------------------|

(147)

The two spectral densities are related by

|----------| | -c- | -fλ =-λ2f-ν-

(148)

and

|----------| | λ2 | fν = --fλ.| ------c-----

(149)

A particularly useful identity is

|----------| λf-λ =-νfν.-

(150)

Frequency spectral flux density is often quoted in janskys:

|--------------------------| -1-Jy-=-10−26-W--m−-2Hz-−1.-

(151)

For a simple isotropic source,

|--------------------------| |fν = -Lν--, fλ = -L-λ-.| ------4πr2-----------4-πr2--

(152)

Finally, photometric measurements use weighted spectral integrals rather than a single wavelength value:

|------------------------| | ∫ | |FX ∝ fλ(λ)SX (λ)dλ. | -------------------------

(153)

This is the key bridge from a physical spectrum to broadband astronomical photometry.

References

[1] G. B. Rybicki and A. P. Lightman, Radiative Processes in Astrophysics, Wiley, 1979.

[2] B. W. Carroll and D. A. Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Pearson, 2007.

[3] D. W. Hogg, What is a magnitude?, arXiv:2206.00989.

[4] M. S. Bessell and S. J. Murphy, Spectrophotometric libraries, revised photonic passbands, and zero points for UBVRI, Hipparcos, and Tycho photometry, Publications of the Astronomical Society of the Pacific, 124, 140, 2012.

[5] J. B. Oke and J. E. Gunn, Secondary standard stars for absolute spectrophotometry, The Astrophysical Journal, 266, 713, 1983.

[6] D. W. Hogg, Distance measures in cosmology, arXiv:astro-ph/9905116.

"Spectral Flux Density: Definition, Units, and Astrophysical Use" is owned by bloftin.
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