A level / IB Physics

Astrophysics

Distance measurement by parallax

d = 1/θ

d is the distance and θ is the parallax angle.   θ must either be in:

  • arcseconds, in which case d will be in parsecs;
  • radians, in which case d will be in AUs (Astronomical Units).

Wien's Displacement Law

λmax T ≈ 2.9 × 10-9 mK λmax = peak wavelength i.e. wavelength of maximum intensity (m);
T      = Kelvin temperature of the black body (K).



Stefan-Boltzmann Law (aka Stefan's Law)

 

L = σAT4

L = Luminosity i.e the amount of energy radiated per unit time i.e.power (W).
σ = Stefan-Boltzmann constant ≈ 5.670 × 10-8 Wm-2K-4;
A = celestial body's surface area (m2);
T = Kelvin temperature of the black body (K).

'Luminosity' usually refers to energy radiated per unit time in the visible part of the EM spectrum.
A more general form of this law is P = σAT4 , where P for Power is used instead of L



Radiation Inverse Square Law



b = L / 4πd2
b = brightness i.e. energy received per unit time per unit area (Wm-2)
d = distance d (m) from the source, where I stands for Intensity and b stands for apparent brightness, L = luminosity (W) (see above)
A more general form of this law is I = P / 4πd2 , where I stands for Intensity (Wm-2) and P stands for Power (W)



The Pogson Ratio r

r =  100     ⇔     r  =  1001/5  ≈  2.512 r = the brightness ratio between two objects that differ by one magnitude. (An irrational number.)

Two ways of measuring how bright a star appears to be

'BRIGHTNESS'
 
'MAGNITUDE'  (logarithmic 'brightness')

The apparent brightness (b) of a celestial body is how much energy is coming from the body per unit area per unit time, as measured on Earth. The units are watts per square metre (Wm-2).

 
The apparent magnitude (m) of a celestial body is another measure of how bright it is as seen by an observer on Earth. There are no units. See the formulae below.
The absolute brightness (B) is the apparent brightness a celestial body would have were it to be placed at a distance of 10 parsecs from Earth.   The absolute magnitude (M) is the apparent magnitude a celestial body would have were it to be placed at a distance of 10 parsecs from Earth.


Formulae relating 'brightness' and magnitude

(magnitude is a logarithmic measure of 'brightness'):

Exponential form Equivalent log form  


b1/b2   =   rm2-m1

 


m2-m1  =  2.5 log (b1/b2)

 

Comparing two stars:
b1 & b2 can each refer to either an apparent or an absolute brightness.
m1 & m2 can each refer to either an apparent or an absolute magnitude.

 
     b   =   b0 r-m
 
      m  =  –2.5 log (b/b0)
One star only:
b0 = 2.52 x 10-8Wm-2 is the reference brightness of Vega, a zero magnitude star
(b has replaced b1 , b0 has replaced b2 , m has replaced m1 and m2 is zero)



Absolute magnitude and distance of an object

Exponential form Equivalent log form  
d22/d12  =   rm2-m1 m2-m1  =  2.5 log (d22/d12)
            ≡
 5 log (d2/d1)
            ≡  5 log d2 – 5 log d1

[1]  From the Radiation Inverse Square Law above, and the fact that L is constant for the one star, we have
b1/b2 = (d2/d1)2, which explains the occurrence of
d22/d12 instead of b1/b2 in these formulae.

[2]  If you decide to make m1 the absolute magnitude (M) of the object, then its distance d1 will be 10pc. The log form, with m and d the apparent magnitude and d the distance of that object, simplifies to:
M-m  =  5 log (10/d)  ≡  5 – 5 log d.

If you decide to make m2 the absolute magnitude, then d2 will be 10pc and things are switched round:
m-M  =  5 log (d/10)  ≡  5 log d – 5