Naked Science Forum

Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: philthewineguy on 20/04/2021 19:57:40

Title: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: philthewineguy on 20/04/2021 19:57:40
would a star with a radius 16 times that of the sun, but a surface temperature 0.5 times that of the sun, be more or less luminous than the sun?
You can assume the surface area of a sphere is A=4 pi r squared

I cant seem to work this out. Can anyone help please?
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Kryptid on 20/04/2021 21:16:36
The luminosity of a star can be approximated using the following equation:

L = σAT4, where

L is luminosity,
σ is the Stefan-Boltzmann constant
A is the surface area of the star, and
T is the temperature of the star

So luminosity varies linearly with surface area and to the fourth power of temperature. A star with 16 times the radius of the Sun would have a surface area 255.926 times as high. Multiply that by the temperature to the fourth power (0.54 = 0.0625) and you get a luminosity of about 15.995 times that of the Sun.
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Bored chemist on 20/04/2021 21:24:43
star with 16 times the radius of the Sun would have a surface area 255.926 times as high
Really?
I'd have gone for 16 times.

Though there's another complicating factor.
The spectrum emitted will change.

Imagine that the "sun" was at 700K
It would emit virtually no visible light.
But, if you heat it to 1400K it will glow quite brightly.

"Brightness" isn't well defined here.

However, looking at the numbers, I'm pretty sure that the answer they expect is 16.
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Petrochemicals on 20/04/2021 22:06:01
would a star with a radius 16 times that of the sun, but a surface temperature 0.5 times that of the sun, be more or less luminous than the sun?
You can assume the surface area of a sphere is A=4 pi r squared

I cant seem to work this out. Can anyone help please?
As a measure of lumens as in luminosity, given that star  temperature is pretty much indicative of star colour, less energy than would be mathematically expected by the formula for surface area, that is to say less high end light. I am not sure of neutrons.

Could a star have a diameter increaced of only 16 times yet be less luminous? Some sort of red dwarf?
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: evan_au on 21/04/2021 10:50:12
Quote from: Petrochemicals
brightness of a star...I am not sure of neutrons.
The fusion reactions that generate energy in the core of a star tend to produce gamma rays and positrons, not neutrons.
- Perhaps you are thinking of uranium fission reactors on Earth, which do produce neutrons?
https://en.wikipedia.org/wiki/Stellar_nucleosynthesis#Hydrogen_fusion

Even if a star did emit neutrons (eg during a supernova): neutrons have a half-life of around 15 minutes, so they would decay well before they reached us over interstellar distances (even if they were traveling close to the speed of light).

See: https://en.wikipedia.org/wiki/Neutron
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Kryptid on 21/04/2021 16:56:51
Really?
I'd have gone for 16 times.

A sphere with a radius of 1 meter has a surface area of 12.57 square meters. A sphere with a radius of 16 meters has a surface area of 3,216.99 square meters. 3,216.99 divided by 12.57 is 255.926.
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Bored chemist on 21/04/2021 17:17:59
A sphere with a radius of 1 meter has a surface area of 12.57 square meters.
I think you are not being irrational enough.
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Kryptid on 21/04/2021 20:47:19
I think you are not being irrational enough.

I apologize if I've missed something, though I'm not sure what it could be.

Could a star have a diameter increaced of only 16 times yet be less luminous? Some sort of red dwarf?

Red dwarfs are cool, but they are small. You'd be talking more about a red giant.
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Bored chemist on 21/04/2021 21:02:29
A star with 16 times the radius of the Sun would have a surface area 255.926 times as high.

The area of a x metre radius sphere is 4*pi*x*x
And the area of a 16x metre radius sphere is  4*pi*16x *16x

So the ratio of them is

4*pi*16x *16x
4*pi*x*x

Everything cancels except the pair of sixteens
So the ratio is 16*16
Or 256 exactly

Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Kryptid on 21/04/2021 23:46:14
Or 256 exactly

Looks like it was a rounding error on my part, then.
Title: Re: What's the brightness of a star 16 times wider but half as hot as the Sun?
Post by: Petrochemicals on 22/04/2021 12:09:24
I think you are not being irrational enough.

I apologize if I've missed something, though I'm not sure what it could be.

Could a star have a diameter increaced of only 16 times yet be less luminous? Some sort of red dwarf?

Red dwarfs are cool, but they are small. You'd be talking more about a red giant.
A red giant of a half mass sun would have a very large radius.