What did I do wrong?

I'm trying to get an expression for how many stars below a certain
magnitude 'm' I will see in the entire field of view of a telescope.

First, consider only one spectral type with absolute magnitude 'M'.
The distance 'r' to a star of apparent magnitude 'm' is given by,

'r=10^{(m-M+5)/5}' ;Taken from 'Introduction to Modern Astrophysics'
by Carroll and Ostlie.

'N=n * V'

'N' is the number of stars observed that are brighter than magnitude m
'n' is the space density of stars (number of stars per cubic parsec)
'V' is the volume of the spherical cone out to distance 'r', which is
the view cone of the telescope.

But,
'V= ({4 pi}/{3}) r^{3} {Omega}/{4 pi} = {Omega r^{3}}/{3}'

Where, 'Omega' is the solid angle subtended by the view cone of the
telescope.

Therefore,
'N= {Omega}/{3} * n * 10^{3(m-M+5)/5}'

Now, in order to account for more than one spectral type, let 'N_{i}'
be the number of stars of spectral type 'i' visible below magnitude
'm', let 'n_{i}' be the space density of stars of spectral type 'i'
and 'M_{i}' be the absolute magnitude of stars of spectral type 'i'.
Then,

'N_{i}= {Omega}/{3} * n_{i} * 10^{3(m-M_{i}+5)/5}'

For all the spectral types (Let Sum_{i} denote summation over all i),

'Sum_{i} N_{i} = Sum_{i} {Omega}/{3} * n_{i} * 10^{3(m-M_{i}+5)/5}'

'Sum_{i} N_{i} = 10^{3m/5} ( Sum_{i} {Omega}/{3} * n_{i} * 10^{3(-M_{i}
+5)/5} )'

Let 'const = Sum_{i} {Omega}/{3} * n_{i} * 10^{3(-M_{i}+5)/5}', which
is a constant w.r.t. 'm'.

So,

'Sum_{i} N_{i}= const * 10^{3m/5}'

Did I make a mistake here?

The problem is that when I count the number of stars below a magnitude
'm' in some astronomical images that I have of orion, and fit a curve
to it, what I get is 'Sum_{i} N_{i}= c * 10^{0.3m}'. According to this
analysis it should be 0.6 instead of 0.3. Why is this?
The instrument I'm using has a sensitivity limit of relative magnitude
of about 16. So I was told to ignore the effects of extinction. If
there is nothing wrong with my math, is there something wrong with my
physics?

I also took the point sources from 1 degree radius view cones 2MASS
star catalog in 6 different directions (through VizieR). If my
original direction was +x, I also took -x, +y, -y, +z and -z
directions. I still get a curve that has an exponent of 0.3 instead of
one that has an exponent of 0.6.

What is the reason for this? Is it because I'm taking J magnitudes
instead of bolometric magnitudes? Is it some relativistic effect?

On Mar 25, 6:17*pm, Terry Wrist wrote:
I'm trying to get an expression for how many stars below a certain
magnitude 'm' I will see in the entire field of view of a telescope.

First, consider only one spectral type with absolute magnitude 'M'.
The distance 'r' to a star of apparent magnitude 'm' is given by,

'r=10^{(m-M+5)/5}' * * *;Taken from 'Introduction to Modern Astrophysics'
by Carroll and Ostlie.

'N=n * V'

'N' is the number of stars observed that are brighter than magnitude m
'n' is the space density of stars (number of stars per cubic parsec)
'V' is the volume of the spherical cone out to distance 'r', which is
the view cone of the telescope.

But,
'V= ({4 pi}/{3}) r^{3} {Omega}/{4 pi} = {Omega r^{3}}/{3}'

Where, 'Omega' is the solid angle subtended by the view cone of the
telescope.

Therefore,
'N= {Omega}/{3} * n * 10^{3(m-M+5)/5}'

Now, in order to account for more than one spectral type, let 'N_{i}'
be the number of stars of spectral type 'i' visible below magnitude
'm', let 'n_{i}' be the space density of stars of spectral type 'i'
and 'M_{i}' be the absolute magnitude of stars of spectral type 'i'.
Then,

'N_{i}= {Omega}/{3} * n_{i} * 10^{3(m-M_{i}+5)/5}'

For all the spectral types (Let Sum_{i} denote summation over all i),

'Sum_{i} N_{i} = Sum_{i} {Omega}/{3} * n_{i} * 10^{3(m-M_{i}+5)/5}'

'Sum_{i} N_{i} = 10^{3m/5} ( Sum_{i} {Omega}/{3} * n_{i} * 10^{3(-M_{i}
+5)/5} )'

Let 'const = Sum_{i} {Omega}/{3} * n_{i} * 10^{3(-M_{i}+5)/5}', which
is a constant w.r.t. 'm'.

So,

'Sum_{i} N_{i}= const * 10^{3m/5}'

Did I make a mistake here?

The problem is that when I count the number of stars below a magnitude
'm' in some astronomical images that I have of orion, and fit a curve
to it, what I get is 'Sum_{i} N_{i}= c * 10^{0.3m}'. According to this
analysis it should be 0.6 instead of 0.3. Why is this?
The instrument I'm using has a sensitivity limit of relative magnitude
of about 16. So I was told to ignore the effects of extinction. If
there is nothing wrong with my math, is there something wrong with my
physics?

I also took the point sources from 1 degree radius view cones 2MASS
star catalog in 6 different directions (through VizieR). If my
original direction was +x, I also took -x, +y, -y, +z and -z
directions. I still get a curve that has an exponent of 0.3 instead of
one that has an exponent of 0.6.

What is the reason for this? Is it because I'm taking J magnitudes
instead of bolometric magnitudes? Is it some relativistic effect?

This is the same guy who posted the original. I just changed my name
(you can check that the email is the same)

I posted this same question several ago in this same mailing list, and
posted this one again without checking for answers. Sorry.

Steve Willner answered that and suggested it's because of the dust.
Thanks Steve. I also googled 'Star Count Models' as you suggested and
got some results. Looking in to those now.

If it IS because of the dust, then it isn't particular to the
direction of Orion. I tried other directions and got the same results.

But if I check other wavelengths (this one was IR), I should observe a
difference if it was in fact due to dust, right? Because different
wavelengths are affected by different sized particles? Checking VizieR
in radio, Optical, UV, X-ray and Gamma ray now.

Thanks.

In article ,
Jacare Omoplata writes:
If it IS because of the dust, then it isn't particular to the
direction of Orion. I tried other directions and got the same results.

Dust absorption varies with the specific direction and also with
Galactic latitude. As I mentioned previously, at high latitude there
is little dust, but the finite height of the Galactic plane is
important. The power law index probably won't vary all that much in
different directions, but the star density at a giving limiting
magnitude will.

But if I check other wavelengths (this one was IR), I should observe a
difference if it was in fact due to dust, right?

It's hard to say. Longer wavelengths will penetrate the dust better,
but the mix of stars you are seeing will differ. If you observe at
long enough wavelengths, you will see all the way through the Galaxy
(except perhaps on the most obscured lines of sight), and then the
increase is nowhere near the Euclidean value. I would not expect the
Euclidean value to hold except for relatively bright magnitudes,
where one is effectively sampling only quite small distances.

At faint magnitudes, especially at high latitudes, "star counts" are
dominated by galaxies, not stars. Looking to larger distances, one
has both cosmological and evolutionary corrections from the simple
Euclidean model.

Arendt et al. (1998 ApJ 508, 74) described an infrared star count
model that was used to correct the COBE data. I think there used to
be an online calculator for it, but I'm not sure it still exists.
There are many SDSS and 2MASS papers reporting star counts from those
surveys. Fazio et al. (2004 ApJS 154, 39) reported infrared "star
counts" from Spitzer, but the counts are dominated by galaxies at the
faint end. The authors made an effort to separate stars from
galaxies in the brighter range, though, so you could ignore the
galaxies to some limit.

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