<p>I am trying to calculate the absolute magnitude of a solar system object, given its albedo, and assuming all of its luminosity comes from reflecting light from the sun. Using <span class="math-container">$L_{sun}$</span> = solar luminosity, <span class="math-container">$L_0$</span> = zero-point luminosity, <span class="math-container">$r$</span> = distance from sun to object, <span class="math-container">$A$</span> = object albedo, and <span class="math-container">$R$</span> = radius of object (assuming spherical), here is my logic:</p>
<ol>
<li>Solar flux reaching out to the object's distance is <span class="math-container">$F_r = \frac{L_{sun}}{4\pi r^2}$</span></li>
<li>Amount of light captured by the object's surface (half a sphere) is <span class="math-container">$F_s =  2\pi R^2 F_r$</span></li>
<li>Luminosity (total flux emitted) of object is <span class="math-container">$L = AF_{s}$</span></li>
<li>Absolute magnitude of the object is <span class="math-container">$M = -2.5log_{10}\frac{L}{L_0}$</span></li>
</ol>
<p>However, something is wrong with this. I did a sanity check using Ceres, which should have an absolute magnitude around 3.3. I'm getting that its much much dimmer than it should be. I know my estimate is very rough, but clearly I'm off in some large way.</p>
<pre><code>D = 5e5 # radius of Ceres [m]
Ls = 3.83e26 # intrinsic L of sun [kg/m2/s3]
L0 = 3.01e28 # zero-point L [kg/m2/s3]
A = 0.09 # albedo of Ceres
r = 4.19e11 # distance from sun to Ceres [m]
Fr = Ls /(4*np.pi*r**2)  # Solar flux reaching distance of Ceres [W/m2]
Fs =  2*np.pi*D**2 * Fr # Light captured by Ceres surface [W]
L = A*Fs # Intrinsic L of Ceres if only from reflecting sun [W]
M =  -2.5 * np.log10(L/L0)  # Absolute M of Ceres   
print(f'Calculated M={M})
</code></pre>
<p>This outputs M = 37.7. Am I missing some source of brightness? Is the logic of my formulation wrong? Do I have an issue with units?</p>

CC BY-SA 4.0
2023-11-13T22:20:17.817Z
Are you determining the absolute magnitude of Ceres as if it were a star, that is assuming that you are observing it from a distance of 10 parsecs
James K
CC BY-SA 4.0
2023-11-13T23:07:27.827Z
I am indeed using absolute magnitude here in the same sense it is used for stars. However, I am assuming no radiation is created by the object itself, so in that sense I am not treating the body as a star. I am trying to estimate how bright the object would be just from reflecting solar light.
CuriousCosmos
CC BY-SA 4.0
2023-11-14T02:01:23.537Z
An asteroid absolute magnitude, denoted by $H$ (H=3.33 for Ceres; not to be confused with the NIR filter) is defined ([CNEOS link](https://cneos.jpl.nasa.gov/glossary/h.html)) as "An asteroid’s absolute magnitude is the visual magnitude an observer would record if the asteroid were placed 1 Astronomical Unit (au) away, and 1 au from the Sun and at a zero phase angle."
astrosnapper
CC BY-SA 4.0
2023-11-14T02:55:09.503Z
@astrosnapper So are you saying that I am simply comparing to the wrong value, because the "absolute magnitude" recorded in the literature for an asteroid such as Ceres is not the same quantity as the absolute magntiude I am farmilier with for stars?
CuriousCosmos
CC BY-SA 4.0
2023-11-14T04:36:57.403Z
Correct, non-stellar objects are treated differently.  It is the brightness 1AU from the Sun, and 1AU from the observer, at a phase angle of $ 0^\circ $ (even though that can't actually happen geometrically). https://en.wikipedia.org/wiki/Absolute_magnitude#Solar_System_bodies_(H)
Greg Miller
CC BY-SA 4.0
2023-11-14T04:52:06.747Z
Correct @CuriousCosmos; welcome to astronomy, where the same term or symbol can mean different things in different subfields...
astrosnapper
CC BY-SA 4.0
2023-11-14T05:13:19.923Z
@astrosnapper Thanks. I'm in the field of galaxy evolution, so I don't spend a lot of time thinking about planets and asteroids. Computing apparent magnitude (from my *"stellar"* absolute magnitude) gives m = 7.3, in good agreement with Ceres' apparent magnitude of 7-9.
CuriousCosmos
<solar-system><luminosity><planetary-science><absolute-magnitude>
Calculating the absolute magnitude of a solar system body
CuriousCosmos
# Calculating the absolute magnitude of a solar system body

(Asked by: username_0 on 2023-11-13)

I am trying to calculate the absolute magnitude of a solar system object, given its albedo, and assuming all of its luminosity comes from reflecting light from the sun. Using $L_{sun}$ = solar luminosity, $L_0$ = zero-point luminosity, $r$ = distance from sun to object, $A$ = object albedo, and $R$ = radius of object (assuming spherical), here is my logic:

Solar flux reaching out to the object's distance is $F_r = \frac{L_{sun}}{4\pi r^2}$
Amount of light captured by the object's surface (half a sphere) is $F_s =  2\pi R^2 F_r$
Luminosity (total flux emitted) of object is $L = AF_{s}$
Absolute magnitude of the object is $M = -2.5log_{10}\frac{L}{L_0}$

However, something is wrong with this. I did a sanity check using Ceres, which should have an absolute magnitude around 3.3. I'm getting that its much much dimmer than it should be. I know my estimate is very rough, but clearly I'm off in some large way.
D = 5e5 # radius of Ceres [m]
Ls = 3.83e26 # intrinsic L of sun [kg/m2/s3]
L0 = 3.01e28 # zero-point L [kg/m2/s3]
A = 0.09 # albedo of Ceres
r = 4.19e11 # distance from sun to Ceres [m]
Fr = Ls /(4*np.pi*r**2)  # Solar flux reaching distance of Ceres [W/m2]
Fs =  2*np.pi*D**2 * Fr # Light captured by Ceres surface [W]
L = A*Fs # Intrinsic L of Ceres if only from reflecting sun [W]
M =  -2.5 * np.log10(L/L0)  # Absolute M of Ceres   
print(f'Calculated M={M})

This outputs M = 37.7. Am I missing some source of brightness? Is the logic of my formulation wrong? Do I have an issue with units?

**Tags:** solar-system, luminosity, planetary-science, absolute-magnitude
> **Comments:**
> **username_1**: Are you determining the absolute magnitude of Ceres as if it were a star, that is assuming that you are observing it from a distance of 10 parsecs  
> **username_0**: I am indeed using absolute magnitude here in the same sense it is used for stars. However, I am assuming no radiation is created by the object itself, so in that sense I am not treating the body as a star. I am trying to estimate how bright the object would be just from reflecting solar light.  
> **username_2**: An asteroid absolute magnitude, denoted by $H$ (H=3.33 for Ceres; not to be confused with the NIR filter) is defined ([CNEOS link](https://cneos.jpl.nasa.gov/glossary/h.html)) as "An asteroid’s absolute magnitude is the visual magnitude an observer would record if the asteroid were placed 1 Astronomical Unit (au) away, and 1 au from the Sun and at a zero phase angle."  
> **username_0**: @astrosnapper So are you saying that I am simply comparing to the wrong value, because the "absolute magnitude" recorded in the literature for an asteroid such as Ceres is not the same quantity as the absolute magntiude I am farmilier with for stars?  
> **username_3**: Correct, non-stellar objects are treated differently.  It is the brightness 1AU from the Sun, and 1AU from the observer, at a phase angle of $ 0^\circ $ (even though that can't actually happen geometrically). https://en.wikipedia.org/wiki/Absolute_magnitude#Solar_System_bodies_(H)  
> **username_2**: Correct @CuriousCosmos; welcome to astronomy, where the same term or symbol can mean different things in different subfields...  
> **username_0**: @astrosnapper Thanks. I'm in the field of galaxy evolution, so I don't spend a lot of time thinking about planets and asteroids. Computing apparent magnitude (from my *"stellar"* absolute magnitude) gives m = 7.3, in good agreement with Ceres' apparent magnitude of 7-9.  

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