Abstract
When a star’s surface temperature is read from its color, dust between the star and us dims blue light more strongly and reddens the color. This paper multiplies the black-body brightness ratio at $440$ nm and $550$ nm of Paper 943 by an extinction inversely proportional to wavelength (a set form), and solves, without relying on a first-order estimate, how far the temperature read from the apparent ratio differs from the true temperature. No new theorem or law is claimed.
First, extinction shrinks the ratio by a factor independent of temperature: for extinction of $0.1$, $0.3$ and $1.0$ magnitudes at $550$ nm, the difference in extinction between the two wavelengths (the color excess) is $0.250000$ times that, and the factor on the ratio was $0.977237$, $0.933254$ and $\mathbf{0.794328}$ (Section 2).
Second, the hotter the star, the further its temperature reads low: with $1.0$ magnitude of extinction, the apparent temperature was $0.876233$ of the true temperature at $4000$ K, $0.828550$ at $5772$ K and $\mathbf{0.339184}$ at $40000$ K ($13567.372$ K) (Section 2).
Third, the first-order estimate from the sensitivity at that temperature misses badly for hot stars: using the sensitivity of Paper 943, the first-order estimate gives $0.103318$ at $1.0$ magnitude and $40000$ K, below the actual $0.339184$. In all fifteen cases the first-order estimate came out lower (Section 3).
Fourth, the apparent temperature has a ceiling: however high the true temperature, the apparent ratio does not exceed the infinitely hot limit ratio $2.441406$ times the extinction factor. The ceiling on the apparent temperature was $146755.189$, $51795.827$ and $\mathbf{17921.911}$ K for $0.1$, $0.3$ and $1.0$ magnitudes, and at $1.0$ magnitude even a true $1000000$ K reached only $0.988172$ of the ceiling (Section 4).
Fifth, the separator is whether the shift is counted on the ratio or on the temperature: on the ratio, the shift at $1.0$ magnitude is the same $0.794328$ at every temperature. On the temperature, the same shift spreads from $0.876233$ to $0.339184$, and even produces a ceiling (Section 5).
Keywords: interstellar extinction, reddening, color excess, color temperature, black-body radiation, Planck’s law.
Scope (what is not claimed)
No new theorem or law is claimed: that interstellar dust dims starlight and dims the blue side more strongly, reddening the color (interstellar extinction and reddening), that the difference in extinction at two wavelengths is called the color excess, and that a temperature read from color comes out low under extinction, are all known. The form of extinction is set: the extinction at each wavelength is set to the extinction at $550$ nm times the ratio of $550$ nm to the wavelength. The shape of real extinction curves, the values that fix that shape, and filter widths are not treated. Only an ideal black body at two wavelengths: as in Paper 943, the ratio of brightness at the two points $440$ nm and $550$ nm, without entering color index values. How the amount of extinction is found is not treated: distances to stars, estimates of the amount of dust, procedures for correcting extinction, and the temperatures of individual stars are not treated. Relation to earlier papers: Paper 943, in its scope note, did not treat observation, leaving out extinction by the atmosphere and interstellar dust, departures of starlight from a black body, and scatter in measuring instruments; this paper sits in the seat of extinction by interstellar dust (no earlier occupant). Paper 943 showed that the brightness ratio of two colors approaches a fixed value for hotter stars, that the ratio’s sensitivity to temperature falls, and that a $1$ percent error in the ratio becomes, to first order, an error in temperature of the reciprocal of the sensitivity in percent. Here the drop in temperature when the ratio shrinks by a large amount, as under extinction, is solved without relying on first order, and a ceiling on the apparent temperature is counted. What is added: showing that extinction shrinks the ratio by a factor independent of temperature, solving the apparent temperature at five temperatures and three extinctions, setting out the differences from the first-order estimate, deriving the ceiling on the apparent temperature, and placing the separator on “counted on the ratio, or on the temperature”.
1. Notation and the Question
Setting: as in Paper 943, the ratio of black-body brightness (per wavelength) at $440$ nm and $550$ nm is
$$R(T)=\left(\frac{550}{440}\right)^5\frac{e^{c_2/(\lambda_{550}T)}-1}{e^{c_2/(\lambda_{440}T)}-1},\qquad c_2=1.43877688\times10^{-2}\ \text{m K}$$(reproducing $0.975636$ at $5772$ K and $2.227526$ at $40000$ K). The extinction (in magnitudes) at wavelength $\lambda$ is set to $A_\lambda=A_{550} (550 \text{nm}/\lambda)$. The difference in extinction between the two wavelengths (the color excess), and the factor on the ratio, are
$$E=A_{440}-A_{550}=A_{550}\left(\frac{550}{440}-1\right),\qquad R_{\text{app}}=R(T)\times10^{-0.4E}$$The apparent temperature $T'$ solves $R(T')=R_{\text{app}}$ (the ratio rises with temperature, so bisection is used; the ratio at $12345$ K returned $12345.000000$ K). With the sensitivity $S=d\ln R/d\ln T$ of Paper 943, the first-order estimate is
$$\frac{T'}{T}\approx\exp\!\left(\frac{-0.4E\ln10}{S(T)}\right)$$Question: for the same extinction, how far does the apparent temperature differ depending on the true temperature, and is there a ceiling?
2. Shrinking the Ratio and the Apparent Temperature
Computation 2.1: the color excess is $0.250000$ times $A_{550}$; for $A_{550}=0.1$, $0.3$ and $1.0$ magnitudes, $E=0.025$, $0.075$ and $0.250$ magnitudes, and the factor on the ratio is $0.977237$, $0.933254$ and $0.794328$, independent of temperature.
Computation 2.2: apparent temperature (K) and its fraction of the true temperature.
| true temperature | $A_{550}=0.1$ mag | $0.3$ mag | $1.0$ mag |
|---|---|---|---|
| $4000$ K | $3944.218$ ($0.986055$) | $3837.248$ ($0.959312$) | $3504.933$ ($0.876233$) |
| $5772$ K | $5654.138$ ($0.979580$) | $5432.906$ ($0.941252$) | $4782.391$ ($0.828550$) |
| $10000$ K | $9617.953$ ($0.961795$) | $8943.650$ ($0.894365$) | $7225.123$ ($0.722512$) |
| $20000$ K | $18257.844$ ($0.912892$) | $15628.422$ ($0.781421$) | $10662.154$ ($0.533108$) |
| $40000$ K | $32722.404$ ($0.818060$) | $24341.623$ ($0.608541$) | $13567.372$ ($\mathbf{0.339184}$) |
Observation 2.3 for the same extinction, the hotter the star, the smaller the fraction. For all three extinctions the fraction fell as the true temperature rose from $4000$ K to $40000$ K: at $1.0$ magnitude a $4000$ K star read as $0.876233$ of its temperature and a $40000$ K star as $0.339184$. The factor on the ratio is the same $0.794328$ at all five temperatures.
3. Against the First-Order Estimate
Computation 3.1: the sensitivity $S$ of Paper 943 is $1.627815$, $1.103448$, $0.576885$, $0.238265$ and $0.101438$ at $4000$, $5772$, $10000$, $20000$ and $40000$ K (at $20000$ K the formula $0.238265412$ agrees with the difference quotient $0.238265631$ within $10^{-6}$). First-order fraction and solved fraction:
| true temperature | $A_{550}=0.1$ mag | $0.3$ mag | $1.0$ mag |
|---|---|---|---|
| $4000$ K | $0.985954$ / $0.986055$ | $0.958452$ / $0.959312$ | $0.868096$ / $0.876233$ |
| $5772$ K | $0.979349$ / $0.979580$ | $0.939318$ / $0.941252$ | $0.811662$ / $0.828550$ |
| $10000$ K | $0.960872$ / $0.961795$ | $0.887149$ / $0.894365$ | $0.670896$ / $0.722512$ |
| $20000$ K | $0.907883$ / $0.912892$ | $0.748325$ / $0.781421$ | $0.380452$ / $0.533108$ |
| $40000$ K | $0.796926$ / $0.818060$ | $0.506120$ / $0.608541$ | $\mathbf{0.103318}$ / $0.339184$ |
Observation 3.2 the first-order estimate misses badly for hot stars and large extinction. In all fifteen cases the first-order estimate was lower than the solved fraction: at $4000$ K and $0.1$ magnitude it was almost the same, $0.985954$ against $0.986055$, but at $40000$ K and $1.0$ magnitude it was $0.103318$ against $0.339184$.
4. The Ceiling on the Apparent Temperature
Computation 4.1: the infinitely hot limit of the ratio is $(550/440)^4=2.441406$, and the ratio at $10^9$ K was $2.441398$. The ratio rises with temperature toward this value, so the apparent ratio does not exceed $2.441406$ times the extinction factor. The ceiling on the apparent temperature solved from that value, and the apparent temperature for true $40000$, $100000$ and $1000000$ K as a fraction of the ceiling:
| extinction | ceiling | $40000$ K | $100000$ K | $1000000$ K |
|---|---|---|---|---|
| $0.1$ mag | $146755.189$ K | $0.222973$ | $0.419524$ | $0.879001$ |
| $0.3$ mag | $51795.827$ K | $0.469953$ | $0.693179$ | $0.958139$ |
| $1.0$ mag | $\mathbf{17921.911}$ K | $0.757027$ | $0.890403$ | $0.988172$ |
Observation 4.2 the color temperature of a reddened star has a ceiling. For all three extinctions the apparent temperature approached the ceiling as the true temperature rose, and the ceiling was lower for larger extinction: at $1.0$ magnitude no star, however hot, reads hotter than $17921.911$ K, and a true $40000$ K star is already at $0.757027$ of the ceiling.
5. The Separator
Observation 5.1 the separator is whether the shift is counted on the ratio or on the temperature. On the ratio, the shift from extinction is the same factor at every temperature: $0.794328$ at all five temperatures for $1.0$ magnitude: on the temperature, the same shift spreads from $0.876233$ at $4000$ K to $0.339184$ at $40000$ K, and the limit of the ratio becomes a ceiling of $17921.911$ K on the temperature.
Observation 5.2 what is not said. Real extinction curves, filters and color indices, how the amount of extinction is found and corrected, and the temperatures of individual stars are not treated. What is shown is that with a set form of extinction and an ideal black body at two wavelengths, the apparent temperature reads lower the hotter the star and does not exceed a ceiling.
6. Roll Call and Closing
- color excess and the factor on the ratio (Section 1): independent of temperature.
- factor on the ratio $\mathbf{0.794328}$ (Computation 2.1): extinction of $1.0$ magnitude.
- $\mathbf{0.339184}$ at $40000$ K (Computation 2.2): hotter stars read lower.
- first-order estimate $\mathbf{0.103318}$ (Computation 3.1): misses for hot stars.
- ceiling $\mathbf{17921.911}$ K (Computation 4.1): not exceeded however hot.
- on the ratio, or on the temperature (Observation 5.1): the separator; the scope note of Paper 943.
Closing: A reddened star’s color temperature reads lower the hotter the star, and never exceeds a ceiling however hot. Extinction of $1.0$ magnitude at $550$ nm shrank the brightness ratio at $440$ nm and $550$ nm to $0.794328$ at every temperature: the apparent temperature became $0.876233$ of the true value for a $4000$ K star and $0.339184$ for a $40000$ K star, far from the first-order estimate of $0.103318$. The ceiling on the apparent temperature was $146755.189$, $51795.827$ and $17921.911$ K for $0.1$, $0.3$ and $1.0$ magnitudes. The separator is whether the shift is counted on the ratio or on the temperature. Placed among the earlier papers: the ratio and sensitivity that Paper 943 counted for an ideal black body were multiplied by the interstellar extinction that Paper 943 set aside, and solved without relying on first order. To be honest: real extinction curves are not necessarily inversely proportional to wavelength. With a set form of extinction, how far the color temperature reads low and where it levels off, and no further.
References
- Cardelli, J. A., Clayton, G. C., & Mathis, J. S. (1989). The relationship between infrared, optical, and ultraviolet extinction. The Astrophysical Journal.
- Planck, M. (1901). Über das Gesetz der Energieverteilung im Normalspectrum. Annalen der Physik.
- Rybicki, G. B., & Lightman, A. P. (1979). Radiative Processes in Astrophysics. Wiley.
- Trumpler, R. J. (1930). Absorption of light in the galactic system. Publications of the Astronomical Society of the Pacific.
- Yamagishi, Y. (2026). The Brightness Ratio of Two Colors Levels Off for Hotter Stars, and Its Power to Read Temperature Fades [Paper 943]. YOI: https://www.yamagishiyuuki.com/yoi/943
Reference
Yamagishi, Y. (2026). A Reddened Star’s Color Temperature Reads Lower the Hotter the Star, and Never Exceeds a Ceiling However Hot [Paper 959]. YOI: https://www.yamagishiyuuki.com/yoi/959
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 959. This paper is a preprint and has not been peer reviewed.