applied astronomy questions

[Vageetasjn]

These two problems are nagging at me and I can't seem to piece together the correct resources to be able to answer them mathematically. These are homework questions, but I am not necessarily looking for answers; just techniques.

1) The star Capella is 14 pc away. The star Dubhe, the brightest star in the Big Dipper, is 160 pc away. Both stars have approximately equal luminosities. Which star appears brighter to us, and by what factor?
I know the nearer star will appear brighter, but I don't know how to quantify that notion.

2) Vega is a main sequence star, as is the Sun, but Vega has a luminosity about 50 times that of the Sun and a mass about 3.5 times that of the Sun. Which star will live longer, and by what factor?

I know the sun will live longer, but again, I don't know how to quantify it.

Any help is appreciated. Thanks.

edit: for question #2, I figured that the sun would outlive Vega by about a factor of 50. Does this seem correct?

 

[silverpig]

Going off the top of my head I know the sun will last longer, but I'm not sure what the factor is. I'm going to guess that it's 50/3.5 though.

As for the other one, know that brightness drops off as 1/r^2...

 

[Calin]

Add to that the logarithmic change of perceived luminosities... So you would have about 130 more light received from Capella. The perceived difference I think could be natural logarithm of the difference, so about 5 times

 

[sao123]

Originally posted by: Vageetasjn
These two problems are nagging at me and I can't seem to piece together the correct resources to be able to answer them mathematically. These are homework questions, but I am not necessarily looking for answers; just techniques.

1) The star Capella is 14 pc away. The star Dubhe, the brightest star in the Big Dipper, is 160 pc away. Both stars have approximately equal luminosities. Which star appears brighter to us, and by what factor?
I know the nearer star will appear brighter, but I don't know how to quantify that notion.

2) Vega is a main sequence star, as is the Sun, but Vega has a luminosity about 50 times that of the Sun and a mass about 3.5 times that of the Sun. Which star will live longer, and by what factor?

I know the sun will live longer, but again, I don't know how to quantify it.

Any help is appreciated. Thanks.

edit: for question #2, I figured that the sun would outlive Vega by about a factor of 50. Does this seem correct?

#1) I dont remember the formula, but there is a probability equation to solve the diffusion of light through a non-vacuum medium. Only so much of that light reaches us per unit distance. some is reflected and some is absorbed.

#2)This would depend on the classification of the star. there are many types of main sequence stars, depending on temperature, color, size and mass... will determine how much fuel is fused per unit of time. Compare those ratios, and you have your answer.


or you could just consult wikipedia...
Vega...
Its spectral class is A0V (Sirius, an A1V, is slightly less powerful) and it is firmly in the main sequence, fusing hydrogen to helium in its core. Since more powerful stars use their fusion fuel more quickly than smaller ones, Vega's life time is only one billion years, a tenth of our Sun's. Vega is two and a half times more massive than our Sun and burns at fifty times the power.

 

[silverpig]

Originally posted by: Calin
Add to that the logarithmic change of perceived luminosities... So you would have about 130 more light received from Capella. The perceived difference I think could be natural logarithm of the difference, so about 5 times

Nope. Magnitudes change as the logarithm of the brightness. The luminosity of a star is independent of distance (A 100W bulb is a 100W bulb no matter how far you are from it), and the brightness or intensity of the light drops off as 1/r^2

 

[sao123]

Originally posted by: silverpig

Originally posted by: Calin
Add to that the logarithmic change of perceived luminosities... So you would have about 130 more light received from Capella. The perceived difference I think could be natural logarithm of the difference, so about 5 times

Nope. Magnitudes change as the logarithm of the brightness. The luminosity of a star is independent of distance (A 100W bulb is a 100W bulb no matter how far you are from it), and the brightness or intensity of the light drops off as 1/r^2

thats in a vacuum tho right? doesnt that change in a non vacuum medium?

 

[silverpig]

Originally posted by: sao123

Originally posted by: silverpig

Originally posted by: Calin
Add to that the logarithmic change of perceived luminosities... So you would have about 130 more light received from Capella. The perceived difference I think could be natural logarithm of the difference, so about 5 times

Nope. Magnitudes change as the logarithm of the brightness. The luminosity of a star is independent of distance (A 100W bulb is a 100W bulb no matter how far you are from it), and the brightness or intensity of the light drops off as 1/r^2

thats in a vacuum tho right? doesnt that change in a non vacuum medium?

Well yeah, there'll be a linear attenuation coefficient if your medium absorbs, but last I checked, space was a pretty good vacuum. It just has to do with the transparency of your medium.

Yeah, there's is actually a significant amount of dust in space, but for these purposes, it's basically a vacuum.

 

[sao123]

Originally posted by: silverpig

Originally posted by: sao123

Originally posted by: silverpig

Originally posted by: Calin
Add to that the logarithmic change of perceived luminosities... So you would have about 130 more light received from Capella. The perceived difference I think could be natural logarithm of the difference, so about 5 times

Nope. Magnitudes change as the logarithm of the brightness. The luminosity of a star is independent of distance (A 100W bulb is a 100W bulb no matter how far you are from it), and the brightness or intensity of the light drops off as 1/r^2

thats in a vacuum tho right? doesnt that change in a non vacuum medium?

Well yeah, there'll be a linear attenuation coefficient if your medium absorbs, but last I checked, space was a pretty good vacuum. It just has to do with the transparency of your medium.

Yeah, there's is actually a significant amount of dust in space, but for these purposes, it's basically a vacuum.

Actually from what ive been reading the ISM is a significant percentage of hydrogen gas in addition to the dust.

 

[fornax]

1) If you assume that both stars are unreddened (i.e. no extinction), Capella will be brighter by a factor of (160/14)^2, or ~130 (inverse square law). Now, this is obviously wrong, because Capella is only about 5 times brighter than Dubhe (just look outside, Capella is the bright yellowish star overhead for people in NA). So, one of the assumptions is wrong. Capella is a giant, whereas Dubhe is a supergiant, so it is instrinsically much brighter than Capella. Therefore the premise that both have equal luminosities is VERY wrong, by a factor of ~25.

2) The Sun will live longer, very roughly by a factor of 50/3.5, or ~15. Vega has about 3.5 time the amount of fuel (hydrogen) that the Sun has, but burns it at a 50 times faster rate.

 

[Calin]

Originally posted by: silverpig

Originally posted by: Calin
Add to that the logarithmic change of perceived luminosities... So you would have about 130 more light received from Capella. The perceived difference I think could be natural logarithm of the difference, so about 5 times

Nope. Magnitudes change as the logarithm of the brightness. The luminosity of a star is independent of distance (A 100W bulb is a 100W bulb no matter how far you are from it), and the brightness or intensity of the light drops off as 1/r^2

Let's rephrase what I said: Capella and Dubhe produce the same amount of light. As we are at a bigger distance from Dubhe, we will get less light (by the square of the distances demultiplied). This ends as (160/14)^2, or 130 more light coming from Capella.
Magnitude is a logarithm of the brightness, so the difference in magnitude is the logarithm of the light received from each star (demultiplied). ( ln(A/B)=lnA - lnB )
So, difference in magnitude would be 4.8

 

[fornax]

Hi Calin,

They use log base ten, not natural logs, but the idea is the same, just the number will be different. A brigthness ratio of 130 corresponds to a magnitude difference of 5.3:

2.5*log(A/B) = 5.3

A quick check on Simbad shows that in fact the magnitude difference is only 1.7, what I pointed out in the first post.