<p>This once appeared in an <a href="https://astroarena.github.io/astroarena/olympiads/iao/2003/theory/TH_problem_2003.pdf" rel="nofollow noreferrer">IAO 2003 A,B 1 problem</a> (the first problem on the first page).</p>
<p>The solution <a href="https://astroarena.github.io/astroarena/olympiads/iao/2003/theory/TH_solution_2003.pdf" rel="nofollow noreferrer">here</a>.</p>
<p>A summary:</p>
<p>Let:<br />
Albedo of moon be <span class="math-container">$\alpha_m$</span><br />
Albedo of sputnik be <span class="math-container">$\alpha_s$</span><br />
Distance between object and observer: <span class="math-container">$d$</span><br />
Radius of objects: <span class="math-container">$R$</span><br />
Solar constant, <span class="math-container">$F_\odot$</span></p>
<p>Flux of moon to observer: <span class="math-container">$F_m = \frac{F_\odot \alpha_{m} \pi R_{m}^2}{2 \pi d_{m}^2}$</span><br />
Flux of Sputnik to observer: <span class="math-container">$F_S = \frac{F_\odot \alpha_{S} \pi R_{S}^2}{4 \pi d_{S}^2}$</span></p>
<p><span class="math-container">\begin{align*}
\Delta m = m_{S} - m_{m}
&amp;= -2.5 \log_{10} \frac{F_S}{F_m} \\
&amp;= -2.5 \log_{10} \frac{1}{2}\frac{\alpha_{S}}{\alpha_{m}}\frac{R_{S}^2}{R_{m}^2}\frac{d_{m}^2}{d_{S}^2}\\
&amp;= -2.5 \log_{10} \frac{1}{2}\frac{1}{0.14}\left ( \frac{0.58 \ \mathrm{m}}{3475000 \  \mathrm{m}}\frac{378000 \ \mathrm{km}}{200 \ \mathrm{km}}  \right ) ^2 \\
&amp; = 16^m
\end{align*}</span></p>
<p>Therefore, apparent magnitude of Sputnik is <span class="math-container">$16^m - 12.7^m \approx 3.3^m$</span>, well below <span class="math-container">$6^m$</span>.</p>
<p>However, I do not understand the calculation of the flux and the associated comments in the document given above:</p>
<blockquote>
<p>We do not take into account effects related to diagrams of scattering and random/mirror scattering.... difference between the 2 and the 4 in <span class="math-container">$2 \pi d_{m}^2$</span>
and <span class="math-container">$4 \pi d_{S}^2$</span></p>
</blockquote>
<p>Any ideas?</p>
<hr />
<p>Edit:</p>
<p>Maybe it's because assuming sun-object-observer angle to be small (e.g. as in full moon)</p>
<ul>
<li>moon reflects light into a hemisphere facing observer only, hence <span class="math-container">$2 \pi d_{m}^2$</span> (diffuse reflection?)</li>
<li>Sputnik 1, when <span class="math-container">$\theta &gt; 45^{\circ}$</span>, the light rays are reflected into the hemisphere away from the observer as well, hence light is distributed into the whole sphere (at least approximately), hence <span class="math-container">$4 \pi d_{S}^2$</span> (specular reflection?)</li>
</ul>
<img src="https://www.researchgate.net/profile/Kolahal-Bhattacharya/publication/51917787/figure/fig1/AS:276935436455945@1443037961936/Reflection-from-a-spherical-mirror-in-optics.png" alt="Reflection from a spherical mirror">  
<p>(from <a href="https://www.researchgate.net/figure/Reflection-from-a-spherical-mirror-in-optics_fig1_51917787" rel="nofollow noreferrer">researchgate</a>)</p>

CC BY-SA 4.0
2022-10-03T04:22:54.770Z
"The solution here" (in a link) is not a good Stack Exchange answer, these are usually called "link-only" answers. Links break; something hosted on Github could disappear at any minute. Can you summarize the solution here in your answer post, and estimate an apparent magnitude for at least one of the two spacecraft mentioned in the title? *Thanks!*
uhoh
CC BY-SA 4.0
2022-10-03T09:46:09.417Z
@uhoh Sorry, I was in a rush . Let me fix it ;)
Cheng
CC BY-SA 4.0
2022-10-03T01:22:37.050Z
2022-10-03T10:22:35.963Z
2022-10-03T10:22:35.963Z
Cheng
<p><a href="https://astronomy.stackexchange.com/q/38667/7982">How bright are geostationary satellites due to reflected sunlight?</a> and <a href="https://space.stackexchange.com/q/46333/12102">Did Sputnik 1 tell us more than “beep”? What science was improved by information gained from its orbiting the Earth?</a> now have me wondering how bright and visible Sputnik 1 would have been.</p>
<p>Sputnik 1 was a smooth, shiny, <a href="https://en.wikipedia.org/wiki/Sputnik_1" rel="nofollow noreferrer">reflective metal sphere 58 cm in diameter</a> with an initial periapsis and apoapsis of about 200 and 900 km altitude.</p>
<p>Vanguard 2 was similar, with a <a href="https://en.wikipedia.org/wiki/Vanguard_2" rel="nofollow noreferrer">51 cm diameter and a 560 x 2,953 orbit</a>.</p>
<p>Normally we treat asteroids with an albedo and some  model for diffuse reflectivity based on phase angle. See answers to the questions linked below.</p>
<p>But in this case it's probably a good approximation to use geometrical optics and assume specular reflection. The problem is that <em>I don't know how to do that!</em></p>
<p><strong>Question:</strong> Apparent magnitude of a spherical body with specular, rather than diffuse reflectivity? How bright were Sputnik 1 and Vanguard 2?</p>
<ul>
<li><a href="https://astronomy.stackexchange.com/q/28744/7982">Calculating the apparent magnitude of a satellite</a></li>
<li><a href="https://astronomy.stackexchange.com/q/20795/7982">Why is Enceladus&#39;s albedo greater than 1?</a></li>
<li><a href="https://astronomy.stackexchange.com/q/29449/7982">What is the difference between albedo, absolute magnitude or aparent magnitude?</a></li>
<li><a href="https://astronomy.stackexchange.com/q/29957/7982">When would we detect a tiny meter size natural satellite in a geostationary orbits?</a></li>
<li><a href="https://astronomy.stackexchange.com/q/33797/7982">Venus&#39; magnitude during inferior conjunction</a></li>
</ul>
<hr />
<p>Screen shots from the PeriscopeFilm video <a href="https://youtu.be/PZiTKXTKa9Q?t=777" rel="nofollow noreferrer">&quot;Science in Space&quot; Early 1960's Space Exploration Film Sputnik &amp; Explorer Vanguard Rocket 12494</a> The photographic negative contains an image of Vanguard 2 from a tracking camera. Click for full size:</p>
<p><a href="https://i.sstatic.net/9K2IY.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/9K2IYm.jpg" alt="Screen shots from the PeriscopeFilm video &quot;Science in Space&quot; Early 1960's Space Exploration Film Sputnik &amp; Explorer Vanguard Rocket 12494" /></a> <a href="https://i.sstatic.net/Jd05w.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Jd05wm.jpg" alt="Screen shots from the PeriscopeFilm video &quot;Science in Space&quot; Early 1960's Space Exploration Film Sputnik &amp; Explorer Vanguard Rocket 12494" /></a></p>

CC BY-SA 4.0
2020-09-03T19:26:10.613Z
Your link to Sputnik 1 gives a magnitude of 6.0. No such luck for the Vanguard 2 satellite.
Wayfaring Stranger
CC BY-SA 4.0
2020-09-04T00:01:37.847Z
@WayfaringStranger Thanks! I'm pretty sure that the brightness will still depend on phase angle (as discussed in the [first linked question](https://astronomy.stackexchange.com/q/28744/7982)) as well as 1/r^2. Does the link mention the phase angle and distance when it was magnitude 6?
uhoh
CC BY-SA 4.0
2020-09-04T14:57:40.650Z
Sure it'll depend on phase angle, and how far the satellite is from earth in its non-circular orbit. I'd guess that magnitude 6 was about as bright as it got. Apparently the final rocket stage made it to orbit too, and was deliberately made reflective so as to be a magnitude 1 object.
Wayfaring Stranger
CC BY-SA 4.0
2022-02-13T13:20:43.640Z
If the reflection model is truly specular then the reflection is just a delta function.  Not sure that's what we're looking for here. [Why the BRDF of specular reflection is infinite in the reflection direction?](https://computergraphics.stackexchange.com/questions/4767/why-the-brdf-of-specular-reflection-is-infinite-in-the-reflection-direction). Is there a meaningful way to define "specular-like"?
GrapefruitIsAwesome
CC BY-SA 4.0
2022-02-13T14:41:13.903Z
@GrapefruitIsAwesome what do you mean "a delta function"? The Sun is an extended object; about 0.5° wide. An observer sees a virtual image of the Sun. This question is about how bright that will be.
uhoh
CC BY-SA 4.0
2022-02-13T16:01:31.090Z
@uhoh The BDRF is a delta function.  If we're happy here to assume a perfectly specular sphere with a specified diameter, then the problem ought to be solvable.  This would, as you indicate, use the non-zero angular extent of the sun.  The result will however be strongly a function of the geometry.
GrapefruitIsAwesome
CC BY-SA 4.0
2022-02-13T16:03:38.450Z
@GrapefruitIsAwesome I see what you mean now.
uhoh
CC BY-SA 4.0
2022-02-13T16:05:21.527Z
@uhoh If I get some time I'll see what I can come up with.  It seems like it might be possible to get a reasonable computation.
GrapefruitIsAwesome
CC BY-SA 4.0
2022-02-13T21:56:36.603Z
@GrapefruitIsAwesome that would also be awesome :-) Since both reflect all the incident light but (probably) with different angular distribution, I guess one would be brighter at some angles and the other brighter at other angles.
uhoh
<amateur-observing><artificial-satellite><apparent-magnitude><absolute-magnitude>
Apparent magnitude of a spherical body with specular, rather than diffuse reflectivity? How bright were Sputnik 1 and Vanguard 2?
uhoh
# Apparent magnitude of a spherical body with specular, rather than diffuse reflectivity? How bright were Sputnik 1 and Vanguard 2?

(Asked by: username_0 on 2020-09-03)

[How bright are geostationary satellites due to reflected sunlight?](https://astronomy.stackexchange.com/q/38667/7982) and [Did Sputnik 1 tell us more than “beep”? What science was improved by information gained from its orbiting the Earth?](https://space.stackexchange.com/q/46333/12102) now have me wondering how bright and visible Sputnik 1 would have been.

Sputnik 1 was a smooth, shiny, [reflective metal sphere 58 cm in diameter](https://en.wikipedia.org/wiki/Sputnik_1) with an initial periapsis and apoapsis of about 200 and 900 km altitude.

Vanguard 2 was similar, with a [51 cm diameter and a 560 x 2,953 orbit](https://en.wikipedia.org/wiki/Vanguard_2).

Normally we treat asteroids with an albedo and some model for diffuse reflectivity based on phase angle. See answers to the questions linked below.

But in this case it's probably a good approximation to use geometrical optics and assume specular reflection. The problem is that *I don't know how to do that!*

**Question:** Apparent magnitude of a spherical body with specular, rather than diffuse reflectivity? How bright were Sputnik 1 and Vanguard 2?

* [Calculating the apparent magnitude of a satellite](https://astronomy.stackexchange.com/q/28744/7982)
* [Why is Enceladus's albedo greater than 1?](https://astronomy.stackexchange.com/q/20795/7982)
* [What is the difference between albedo, absolute magnitude or aparent magnitude?](https://astronomy.stackexchange.com/q/29449/7982)
* [When would we detect a tiny meter size natural satellite in a geostationary orbits?](https://astronomy.stackexchange.com/q/29957/7982)
* [Venus' magnitude during inferior conjunction](https://astronomy.stackexchange.com/q/33797/7982)

---

Screen shots from the PeriscopeFilm video ["Science in Space" Early 1960's Space Exploration Film Sputnik & Explorer Vanguard Rocket 12494](https://youtu.be/PZiTKXTKa9Q?t=777) The photographic negative contains an image of Vanguard 2 from a tracking camera. Click for full size:

[![Screen shots from the PeriscopeFilm video ](https://i.sstatic.net/9K2IYm.jpg)](https://i.sstatic.net/9K2IY.jpg) [![Screen shots from the PeriscopeFilm video ](https://i.sstatic.net/Jd05wm.jpg)](https://i.sstatic.net/Jd05w.jpg)

**Tags:** amateur-observing, artificial-satellite, apparent-magnitude, absolute-magnitude
> **Comments:**
> **username_1**: Your link to Sputnik 1 gives a magnitude of 6.0. No such luck for the Vanguard 2 satellite.  
> **username_0**: @WayfaringStranger Thanks! I'm pretty sure that the brightness will still depend on phase angle (as discussed in the [first linked question](https://astronomy.stackexchange.com/q/28744/7982)) as well as 1/r^2. Does the link mention the phase angle and distance when it was magnitude 6?  
> **username_1**: Sure it'll depend on phase angle, and how far the satellite is from earth in its non-circular orbit. I'd guess that magnitude 6 was about as bright as it got. Apparently the final rocket stage made it to orbit too, and was deliberately made reflective so as to be a magnitude 1 object.  
> **username_2**: If the reflection model is truly specular then the reflection is just a delta function.  Not sure that's what we're looking for here. [Why the BRDF of specular reflection is infinite in the reflection direction?](https://computergraphics.stackexchange.com/questions/4767/why-the-brdf-of-specular-reflection-is-infinite-in-the-reflection-direction). Is there a meaningful way to define "specular-like"?  
> **username_0**: @GrapefruitIsAwesome what do you mean "a delta function"? The Sun is an extended object; about 0.5° wide. An observer sees a virtual image of the Sun. This question is about how bright that will be.  
> **username_2**: @uhoh The BDRF is a delta function.  If we're happy here to assume a perfectly specular sphere with a specified diameter, then the problem ought to be solvable.  This would, as you indicate, use the non-zero angular extent of the sun.  The result will however be strongly a function of the geometry.  
> **username_0**: @GrapefruitIsAwesome I see what you mean now.  
> **username_2**: @uhoh If I get some time I'll see what I can come up with.  It seems like it might be possible to get a reasonable computation.  
> **username_0**: @GrapefruitIsAwesome that would also be awesome :-) Since both reflect all the incident light but (probably) with different angular distribution, I guess one would be brighter at some angles and the other brighter at other angles.  

---

## 1 answers

### Answer by username_3 on 2022-10-03. Score: 2 | Accepted answer

This once appeared in an [IAO 2003 A,B 1 problem](https://astroarena.github.io/astroarena/olympiads/iao/2003/theory/TH_problem_2003.pdf) (the first problem on the first page).

The solution [here](https://astroarena.github.io/astroarena/olympiads/iao/2003/theory/TH_solution_2003.pdf).

A summary:

Let:  
Albedo of moon be $\alpha\_m$  
Albedo of sputnik be $\alpha\_s$  
Distance between object and observer: $d$  
Radius of objects: $R$  
Solar constant, $F\_\odot$

Flux of moon to observer: $F\_m = \frac{F\_\odot \alpha\_{m} \pi R\_{m}^2}{2 \pi d\_{m}^2}$  
Flux of Sputnik to observer: $F\_S = \frac{F\_\odot \alpha\_{S} \pi R\_{S}^2}{4 \pi d\_{S}^2}$

\begin{align\*}
\Delta m = m\_{S} - m\_{m}
&= -2.5 \log\_{10} \frac{F\_S}{F\_m} \\
&= -2.5 \log\_{10} \frac{1}{2}\frac{\alpha\_{S}}{\alpha\_{m}}\frac{R\_{S}^2}{R\_{m}^2}\frac{d\_{m}^2}{d\_{S}^2}\\
&= -2.5 \log\_{10} \frac{1}{2}\frac{1}{0.14}\left ( \frac{0.58 \ \mathrm{m}}{3475000 \ \mathrm{m}}\frac{378000 \ \mathrm{km}}{200 \ \mathrm{km}} \right ) ^2 \\
& = 16^m
\end{align\*}

Therefore, apparent magnitude of Sputnik is $16^m - 12.7^m \approx 3.3^m$, well below $6^m$.

However, I do not understand the calculation of the flux and the associated comments in the document given above:

> We do not take into account effects related to diagrams of scattering and random/mirror scattering.... difference between the 2 and the 4 in $2 \pi d\_{m}^2$
> and $4 \pi d\_{S}^2$

Any ideas?

---

Edit:

Maybe it's because assuming sun-object-observer angle to be small (e.g. as in full moon)

* moon reflects light into a hemisphere facing observer only, hence $2 \pi d\_{m}^2$ (diffuse reflection?)
* Sputnik 1, when $\theta > 45^{\circ}$, the light rays are reflected into the hemisphere away from the observer as well, hence light is distributed into the whole sphere (at least approximately), hence $4 \pi d\_{S}^2$ (specular reflection?)

![Reflection from a spherical mirror](https://www.researchgate.net/profile/Kolahal-Bhattacharya/publication/51917787/figure/fig1/AS:276935436455945@1443037961936/Reflection-from-a-spherical-mirror-in-optics.png)

(from [researchgate](https://www.researchgate.net/figure/Reflection-from-a-spherical-mirror-in-optics_fig1_51917787))

> **Comments:**
> **username_0**: "The solution here" (in a link) is not a good Stack Exchange answer, these are usually called "link-only" answers. Links break; something hosted on Github could disappear at any minute. Can you summarize the solution here in your answer post, and estimate an apparent magnitude for at least one of the two spacecraft mentioned in the title? *Thanks!*  
> **username_3**: @uhoh Sorry, I was in a rush . Let me fix it ;)  

---

