<p>Using this formula from <a href="https://en.wikipedia.org/wiki/Gravitational_lens#Explanation_in_terms_of_spacetime_curvature" rel="nofollow noreferrer">Wikipedia</a>,
<span class="math-container">$$\theta = \frac{4GM}{rc^2}$$</span>
and plugging in the mass and radius of the Moon gives a deflection angle of <span class="math-container">$1.2567 \times 10^{-10}$</span> radians = <span class="math-container">$2.592 \times 10^{-5}$</span> arc-seconds. In comparison, the deflection for the Sun is just under 1.75 arc-seconds, using the Sun's equatorial radius.</p>

<p>I expect that it would be <em>very</em> difficult to observe such a tiny deflection from the Earth's surface, even using <a href="https://en.wikipedia.org/wiki/Adaptive_optics" rel="nofollow noreferrer">adaptive optics</a>.</p>

CC BY-SA 4.0
2020-05-15T12:39:15.857Z
Minor nit:   AO might reduce the time required to get a measurement of low sigma, but it otherwise doesn't matter.
Carl Witthoft
CC BY-SA 4.0
2020-05-15T12:54:30.690Z
@Carl I suppose it doesn't matter, since the deflections caused by atmospheric inhomogeneities should average out over time, but I'd prefer to have a measurement that isn't obscured like that. Rob's answer states that the  lunar gravitational deflection is currently just out of our reach, even using space-based telescopes, so surely it'd be even harder to do at the Earth's surface.
PM 2Ring
CC BY-SA 4.0
2020-05-15T09:00:02.763Z
2020-05-15T12:43:47.320Z
2020-05-15T12:43:47.320Z
PM 2Ring
<p>Using our Moon as a gravitational lens might prove extremely challenging if not less than the angular resolution of positional measurements allow. It's a small effect with the Sun, and the Moon is 100 million times less massive and 100 times smaller in radius. The angular deviation from the Sun is <a href="http://www.ita.uni-heidelberg.de/~massimo/sub/Lectures/gl_all.pdf" rel="nofollow noreferrer">less than one arcsecond</a> and scales as M/R. So the deviation due to the Moon would be around <span class="math-container">$10^{-6}$</span> that of the Sun. Current technology allows angular resolution in astronomy of a <a href="https://www.eso.org/sci/facilities/paranal/instruments/gravity.html" rel="nofollow noreferrer">few micro-arcseconds</a>. In essence the deviation the Moon makes on light's path is about 10 - 100 times less than currently detectible with the best telescopes.</p>

<p>However, gravitational lensing can be used to actually detect exoplanets and exo-moons (i.e. moons around exoplanets).</p>

<p>The basics of that have been worked out around theoretically around 10 years ago (e.g. see this <a href="https://www.aanda.org/articles/aa/full_html/2010/12/aa13844-09/aa13844-09.html" rel="nofollow noreferrer">article</a>). Meanwhile there is a few planets confirmed found by this method via the <a href="https://en.wikipedia.org/wiki/Optical_Gravitational_Lensing_Experiment" rel="nofollow noreferrer">OGLE collaboration</a> - I'm not aware of any exomoon found so far.</p>

CC BY-SA 4.0
2020-05-15T11:52:05.623Z
The maximum angular deviation caused by the Moon is not 1 micro arcsecond, it is 26 microarcseconds.
ProfRob
CC BY-SA 4.0
2020-05-15T08:24:05.243Z
2020-05-20T06:57:20.227Z
2020-05-20T06:57:20.227Z
planetmaker
<p>Measuring the gravitational deflection of light by the Moon is just out of reach of current observational techniques.</p>
<p>The angular deflection caused by the lensing of a distant background object by a foreground (nearby) object is given by
<span class="math-container">$$\theta \simeq 4 \frac{GM}{Rc^2},$$</span>
where <span class="math-container">$M$</span> is the mass of the lensing object and <span class="math-container">$R$</span> is the closest projected distance of the ray from the centre of the mass.</p>
<p>For the Moon, the maximum deflection would occur when the ray just grazes the limb of the Moon and would equal 26 micro-arcseconds.</p>
<p>At present the most accurate instruments for measuring precision positions are VLBI radio observations of point sources, where 10 micro-arcsecond relative positional precision can be &quot;routinely achieved&quot; (<a href="https://www.annualreviews.org/doi/10.1146/annurev-astro-081913-040006" rel="nofollow noreferrer">Reid &amp; Honma 2014</a>). To use this technique you need bright, radio point sources to be close in position to the Moon. This is certainly possible and has been done many times using the Sun as the lensing object, although the claimed accuracies of a short time-series of measurements on single sources is an order of magnitude lower and so probably wouldn't work for the Moon (e.g. <a href="https://www.aanda.org/articles/aa/full_html/2018/10/aa33459-18/aa33459-18.html" rel="nofollow noreferrer">Titov et al. 2018 </a>).</p>
<p>The Gaia astrometry satellite has likely end-of-mission positional precisions of about 5 micro-arcseconds for bright stars, but it is unclear what the precision is for a single scan (which would be needed because the Moon moves!). However, to counterbalance that, one could also average the lensing effect over many stars surrounding the Moon and of course average this for different stars with the Moon observed at different times over the entire Gaia mission.</p>
<p><strong>Unfortutunately</strong>, although in principle this could be done, because of the way that Gaia scans the sky it is always pointing <em>away</em> from the Moon and so there will be no observations towards the Moon that would be capable of revealing these small deflections (e.g. <a href="https://www.cosmos.esa.int/documents/29201/310761/IN_light_bending.pdf/49288d09-e584-43ab-b3bb-237c3ed5f114" rel="nofollow noreferrer">see here</a>).</p>

CC BY-SA 4.0
2020-05-15T12:37:47.110Z
So,  we just need to make the Moon more massive :-)
Carl Witthoft
CC BY-SA 4.0
2020-05-15T19:50:59.427Z
If the effect is less than 3 times the instrument precision (10 to 26 microseconds) and we can assume that the error is random, we just need to measure the gravitational lensing on more than 9 stars and average. Another question is whether it is worth the effort.
Pere
CC BY-SA 4.0
2020-05-16T00:46:28.823Z
Moon is just out of reach of current sensible observational protocols. If Elon hits another blunt this could be doable.
Captain Giraffe
CC BY-SA 4.0
2020-05-16T05:45:55.533Z
@CarlWitthoft I see no potential ill side effects from this.
corsiKa
CC BY-SA 4.0
2020-05-17T17:06:01.690Z
@corsiKa there are hypotheses that say it will make lycanthropy much worse. We will learn too late just how many closet werewolves will be forced to come out - in more than one meaning of the phrase.
John Dvorak
CC BY-SA 4.0
2020-05-20T09:58:06.383Z
@Pere One reason you might want to do this is to test GR without having to worry about whether the atmosphere/corona of the Sun causes any issues with the gravitational bending measurements.
ProfRob
CC BY-SA 4.0
2024-11-20T01:00:14.143Z
why does 4 appear in the theta equation?
Miss Understands
CC BY-SA 4.0
2020-05-15T11:49:23.277Z
2023-02-24T07:23:43.837Z
2023-02-24T07:23:43.837Z
ProfRob
<p>Eddington waited for a total solar eclipse to happen to be able to observe gravitational lensing of the stars behind the Sun. And nowadays, <a href="https://eclipse2017.nasa.gov/testing-general-relativity" rel="nofollow noreferrer">amateurs can do the same thing</a>.</p>
<p>Of course, the Moon is much less massive than the Sun, so it doesn't bend light as much. But since Eddington's time, our observation capabilities have greatly increased. Is it possible to detect the gravitational lensing of stars that are behind the Moon?</p>

CC BY-SA 4.0
2020-05-16T06:40:55.827Z
In Eddington measurements, the combined lensing of the Moon and the Sun was measured. I think sir Eddington didn't care about the Moon at all.
fraxinus
<observational-astronomy><general-relativity><gravitational-lensing>
Is it possible to detect gravitational lensing of stars behind the Moon?
usernumber
# Is it possible to detect gravitational lensing of stars behind the Moon?

(Asked by: username_0 on 2020-05-15)

Eddington waited for a total solar eclipse to happen to be able to observe gravitational lensing of the stars behind the Sun. And nowadays, [amateurs can do the same thing](https://eclipse2017.nasa.gov/testing-general-relativity).

Of course, the Moon is much less massive than the Sun, so it doesn't bend light as much. But since Eddington's time, our observation capabilities have greatly increased. Is it possible to detect the gravitational lensing of stars that are behind the Moon?

**Tags:** observational-astronomy, general-relativity, gravitational-lensing

> **Comments:**
> **username_1**: In Eddington measurements, the combined lensing of the Moon and the Sun was measured. I think sir Eddington didn't care about the Moon at all.  

---

## 3 answers

### Answer by username_5 on 2020-05-15. Score: 30 | Accepted answer

Measuring the gravitational deflection of light by the Moon is just out of reach of current observational techniques.

The angular deflection caused by the lensing of a distant background object by a foreground (nearby) object is given by
$$\theta \simeq 4 \frac{GM}{Rc^2},$$
where $M$ is the mass of the lensing object and $R$ is the closest projected distance of the ray from the centre of the mass.

For the Moon, the maximum deflection would occur when the ray just grazes the limb of the Moon and would equal 26 micro-arcseconds.

At present the most accurate instruments for measuring precision positions are VLBI radio observations of point sources, where 10 micro-arcsecond relative positional precision can be "routinely achieved" ([Reid & Honma 2014](https://www.annualreviews.org/doi/10.1146/annurev-astro-081913-040006)). To use this technique you need bright, radio point sources to be close in position to the Moon. This is certainly possible and has been done many times using the Sun as the lensing object, although the claimed accuracies of a short time-series of measurements on single sources is an order of magnitude lower and so probably wouldn't work for the Moon (e.g. [Titov et al. 2018](https://www.aanda.org/articles/aa/full_html/2018/10/aa33459-18/aa33459-18.html) ).

The Gaia astrometry satellite has likely end-of-mission positional precisions of about 5 micro-arcseconds for bright stars, but it is unclear what the precision is for a single scan (which would be needed because the Moon moves!). However, to counterbalance that, one could also average the lensing effect over many stars surrounding the Moon and of course average this for different stars with the Moon observed at different times over the entire Gaia mission.

**Unfortutunately**, although in principle this could be done, because of the way that Gaia scans the sky it is always pointing *away* from the Moon and so there will be no observations towards the Moon that would be capable of revealing these small deflections (e.g. [see here](https://www.cosmos.esa.int/documents/29201/310761/IN_light_bending.pdf/49288d09-e584-43ab-b3bb-237c3ed5f114)).

> **Comments:**
> **username_3**: So,  we just need to make the Moon more massive :-)  
> **username_6**: If the effect is less than 3 times the instrument precision (10 to 26 microseconds) and we can assume that the error is random, we just need to measure the gravitational lensing on more than 9 stars and average. Another question is whether it is worth the effort.  
> **username_7**: Moon is just out of reach of current sensible observational protocols. If Elon hits another blunt this could be doable.  
> **username_8**: @username_3 I see no potential ill side effects from this.  
> **username_9**: @username_8 there are hypotheses that say it will make lycanthropy much worse. We will learn too late just how many closet werewolves will be forced to come out - in more than one meaning of the phrase.  
> **username_5**: @username_6 One reason you might want to do this is to test GR without having to worry about whether the atmosphere/corona of the Sun causes any issues with the gravitational bending measurements.  
> **username_10**: why does 4 appear in the theta equation?  

---

### Answer by username_2 on 2020-05-15. Score: 6

Using this formula from [Wikipedia](https://en.wikipedia.org/wiki/Gravitational_lens#Explanation_in_terms_of_spacetime_curvature),
$$\theta = \frac{4GM}{rc^2}$$
and plugging in the mass and radius of the Moon gives a deflection angle of $1.2567 \times 10^{-10}$ radians = $2.592 \times 10^{-5}$ arc-seconds. In comparison, the deflection for the Sun is just under 1.75 arc-seconds, using the Sun's equatorial radius.

I expect that it would be *very* difficult to observe such a tiny deflection from the Earth's surface, even using [adaptive optics](https://en.wikipedia.org/wiki/Adaptive_optics).

> **Comments:**
> **username_3**: Minor nit:   AO might reduce the time required to get a measurement of low sigma, but it otherwise doesn't matter.  
> **username_2**: @username_3 I suppose it doesn't matter, since the deflections caused by atmospheric inhomogeneities should average out over time, but I'd prefer to have a measurement that isn't obscured like that. Rob's answer states that the  lunar gravitational deflection is currently just out of our reach, even using space-based telescopes, so surely it'd be even harder to do at the Earth's surface.  

---

### Answer by username_4 on 2020-05-15. Score: 3

Using our Moon as a gravitational lens might prove extremely challenging if not less than the angular resolution of positional measurements allow. It's a small effect with the Sun, and the Moon is 100 million times less massive and 100 times smaller in radius. The angular deviation from the Sun is [less than one arcsecond](http://www.ita.uni-heidelberg.de/~massimo/sub/Lectures/gl_all.pdf) and scales as M/R. So the deviation due to the Moon would be around $10^{-6}$ that of the Sun. Current technology allows angular resolution in astronomy of a [few micro-arcseconds](https://www.eso.org/sci/facilities/paranal/instruments/gravity.html). In essence the deviation the Moon makes on light's path is about 10 - 100 times less than currently detectible with the best telescopes.

However, gravitational lensing can be used to actually detect exoplanets and exo-moons (i.e. moons around exoplanets).

The basics of that have been worked out around theoretically around 10 years ago (e.g. see this [article](https://www.aanda.org/articles/aa/full_html/2010/12/aa13844-09/aa13844-09.html)). Meanwhile there is a few planets confirmed found by this method via the [OGLE collaboration](https://en.wikipedia.org/wiki/Optical_Gravitational_Lensing_Experiment) - I'm not aware of any exomoon found so far.

> **Comments:**
> **username_5**: The maximum angular deviation caused by the Moon is not 1 micro arcsecond, it is 26 microarcseconds.  

---

