<p><a href="https://ui.adsabs.harvard.edu/abs/2010Icar..209..542M/abstract" rel="nofollow noreferrer">Muinonen et al. 2010</a>, before presenting their <span class="math-container">$H G_1 G_2$</span> model, express the older <span class="math-container">$H G$</span> model as
<span class="math-container">$$ V(\alpha) = H - 2.5 \log_{10}[(1 - G) \phi_1(\alpha) + G \phi_2(\alpha)]. $$</span></p>
<p>If we substitute <span class="math-container">$ d_{BS} = r,\ d_{BO} = \Delta,\ d_0 = 1 $</span>
in the formula from the <a href="https://en.wikipedia.org/wiki/Absolute_magnitude#Asteroids" rel="nofollow noreferrer">Wikipedia article</a>
<span class="math-container">$$ m = H + 5 \log_{10} \left( \frac{d_{BS}\ d_{BO}}{{d_0}^2} \right) - 2.5 \log_{10} q(\alpha), $$</span>
where <span class="math-container">$ q(\alpha) = (1 - G) \phi_1(\alpha) + G \phi_2(\alpha) $</span>,
then we have
<span class="math-container">$$ m = V(\alpha) + 5 \log_{10}(r \Delta), $$</span>
so
<span class="math-container">$$ V(\alpha) = m - 5 \log_{10}(r \Delta). $$</span></p>
<p><a href="https://ui.adsabs.harvard.edu/abs/2021Icar..35414094M/abstract" rel="nofollow noreferrer">Mahlke et al. 2021</a> have the opposite sign in their equation (1), but I suspect that it's a typo.</p>
<p>The <span class="math-container">$ 5 \log_{10}(r\Delta) $</span> term comes from the inverse square law; increasing either <span class="math-container">$r$</span> or <span class="math-container">$\Delta$</span> by a factor of 10 decreases the observed flux by a factor of 100 and increases the apparent magnitude by 5.</p>

CC BY-SA 4.0
2023-10-06T11:48:54.547Z
thank you for this answer. Now I'm able to see that the term involving the phase function gets cancelled out, once we replace H in the wikipedia equation by the H in the Muinonen et al. 2010.  What is striking also is that the opposite sign in equation 1 in Mahlke et al. 2021 got unnoticed and published given though it's a peer-reviewed process.
mysterium
CC BY-SA 4.0
2023-10-05T23:50:44.350Z
2023-10-05T23:50:44.350Z
Mike G
<p>I've noticed that there is this parameter called reduced absolute magnitude <span class="math-container">$V(\alpha)$</span> and it is defined as below according to the paper by Mahlke et al. 2021 <a href="https://doi.org/10.1016/j.icarus.2020.114094" rel="nofollow noreferrer">https://doi.org/10.1016/j.icarus.2020.114094</a></p>
<p><span class="math-container">$V(\alpha) = m + 5 log (r\Delta) $</span></p>
<p>, where where <span class="math-container">$r$</span> is the distance between the asteroid and the Sun at the epoch of observation and <span class="math-container">$Δ$</span> the respective distance between the asteroid and Earth, <span class="math-container">$m$</span> is the observed apparent magnitude, according to the authors. <span class="math-container">$V(\alpha)$</span> is referred to as the phase curve, and, by definition, <span class="math-container">$H = V(0)$</span>, observed at a phase angle of 0° at 1 AU from both Earth and Sun.</p>
<p>I've tried to find the the derivation of this equation, and haven't succeeded yet. I've also referred the absolute magnitude of solar system bodies given in the the wikipedia page <a href="https://en.wikipedia.org/wiki/Absolute_magnitude" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Absolute_magnitude</a> and could not find the derivation of the above equation.</p>
<p>Can anyone point me to the right direction please?
Thanks !</p>

<solar-system><asteroids>
derivation of reduced magnitude of asteroids
mysterium
# derivation of reduced magnitude of asteroids

(Asked by: username_0 on 2023-10-05)

I've noticed that there is this parameter called reduced absolute magnitude $V(\alpha)$ and it is defined as below according to the paper by Mahlke et al. 2021 <https://doi.org/10.1016/j.icarus.2020.114094>

$V(\alpha) = m + 5 log (r\Delta) $

, where where $r$ is the distance between the asteroid and the Sun at the epoch of observation and $Δ$ the respective distance between the asteroid and Earth, $m$ is the observed apparent magnitude, according to the authors. $V(\alpha)$ is referred to as the phase curve, and, by definition, $H = V(0)$, observed at a phase angle of 0° at 1 AU from both Earth and Sun.

I've tried to find the the derivation of this equation, and haven't succeeded yet. I've also referred the absolute magnitude of solar system bodies given in the the wikipedia page <https://en.wikipedia.org/wiki/Absolute_magnitude> and could not find the derivation of the above equation.

Can anyone point me to the right direction please?
Thanks !

**Tags:** solar-system, asteroids

---

## 1 answers

### Answer by username_1 on 2023-10-05. Score: 2 | Accepted answer

[Muinonen et al. 2010](https://ui.adsabs.harvard.edu/abs/2010Icar..209..542M/abstract), before presenting their $H G\_1 G\_2$ model, express the older $H G$ model as
$$ V(\alpha) = H - 2.5 \log\_{10}[(1 - G) \phi\_1(\alpha) + G \phi\_2(\alpha)]. $$

If we substitute $ d\_{BS} = r,\ d\_{BO} = \Delta,\ d\_0 = 1 $
in the formula from the [Wikipedia article](https://en.wikipedia.org/wiki/Absolute_magnitude#Asteroids)
$$ m = H + 5 \log\_{10} \left( \frac{d\_{BS}\ d\_{BO}}{{d\_0}^2} \right) - 2.5 \log\_{10} q(\alpha), $$
where $ q(\alpha) = (1 - G) \phi\_1(\alpha) + G \phi\_2(\alpha) $,
then we have
$$ m = V(\alpha) + 5 \log\_{10}(r \Delta), $$
so
$$ V(\alpha) = m - 5 \log\_{10}(r \Delta). $$

[Mahlke et al. 2021](https://ui.adsabs.harvard.edu/abs/2021Icar..35414094M/abstract) have the opposite sign in their equation (1), but I suspect that it's a typo.

The $ 5 \log\_{10}(r\Delta) $ term comes from the inverse square law; increasing either $r$ or $\Delta$ by a factor of 10 decreases the observed flux by a factor of 100 and increases the apparent magnitude by 5.

> **Comments:**
> **username_0**: thank you for this answer. Now I'm able to see that the term involving the phase function gets cancelled out, once we replace H in the wikipedia equation by the H in the Muinonen et al. 2010.  What is striking also is that the opposite sign in equation 1 in Mahlke et al. 2021 got unnoticed and published given though it's a peer-reviewed process.  

---

