De58te wrote: ↑Sat Jun 18, 2022 8:27 pm
Wow this is a revelation. Ann wrote that evidence suggests that Sadr is a little farther away than Deneb. Some websites I have seen need updating. Most state that Sadr is around 1,800 ly away while Deneb is some 2,600 ly away. Give or take 200 ly.
That is what Wikipedia says, but Wikipedia is wrong.
This is what
Wikipedia says about Sadr:
Parallax (π) 1.78 ± 0.27 mas
Distance approx. 1,800 ly (approx. 560 pc)
Absolute magnitude (MV) −4.54
This is what
Wikipedia says about Deneb:
Parallax (π) 2.29 ± 0.32 mas
Distance 2,615±215 ly (802±66 pc)
Absolute magnitude (MV) −8.38
So Wikipedia correctly notes that Sadr has a smaller parallax than Deneb (although both parallaxes are highly uncertain), but then it concludes that the star with the smaller parallax is the one that is closer to us.
It is the other way round.
What is the absolute magnitude of Sadr, if we assume that the Hipparcos parallax of 1.78 mas is correct, and the apparent magnitude of Sadr is 2.23? A parallax of 1.78 mas does indeed correspond to about 1,800 light-years, or 561 (make that closer to 562) parsecs. That corresponds to an absolute V luminosity of -6.52, or 31,000 Suns.
In astronomy, we talk about stars' bolometric luminosity, which is their total luminosity at all wavelengths. But since Sadr is an F8 supergiant type of star, we don't expect its bolometric correction to be very high. That is because the temperature of Sadr is just a little higher than the Sun's, and just like the Sun, its blackbody curve peaks in the visible part of the spectrum.
The source of the image above is
this.
Since the temperature of Sadr is quite similar to that of the Sun, Sadr's blackbody curve will also be similar. A star's V luminosity is its magnitude in yellow-green light, and as you can see, the Sun peaks in blue-green light. To get the Sun's bolometric luminosity, we must add its emission at all other wavelengths.
But, guess what?? I found a figure that suggests that for stars whose temperature is similar to the Sun or a little hotter, we should calculate no bolometric correction!!
What about Deneb?
The parallax of Deneb, 2.29 mas, corresponds to a little more than 1,400 light-years, or 437 parsecs. Since its apparent magnitude is 1.25, its absolute magnitude will be -6.95, or 47,529 Suns in V light. And its bolometric correction will be larger than than of Sadr, because Deneb is an early A-type star, which means that its blackbody curve peaks in the ultraviolet part of the spectrum. And frankly Deneb emits more energy per unit of its "surface" than Sadr at all wavelengths. Note how the shape of the blackbody curves change at rising temperatures:
The source of the image above is
this. Note that while the temperature of Sadr is ~6,000K, the temperature of Deneb is ~8,500 K. Deneb is nowhere near 12,000 K. Even so, its blackbody curve is most definitely "higher" than Sadr's, and Deneb emits more energy per square unit than Sadr at all wavelengths. So clearly, however we look at it, Deneb is brighter than Sadr
if their Hipparcos parallaxes are correct. We don't know that they are.
However, the tentative verdict seems to be:
Yes, Sadr is farther away than Deneb.
Yes, Sadr is a larger star than Deneb.
But no, Sadr is not brighter than Deneb. Deneb is brighter.
Ann
The Gamma Cygni Nebula
or should I 
