# Earth gravity lens focus

Reading this question and then this report in a comment by @called2voyage

At some point they talk about possibility to send (using antimatter as propellant) a space mission to solar gravity lens focus, using it as a giant telescope to observe exoplanets.

Since this solar focus is quite far ( 550 UA )

Is gravitational light bending capability of the earth enough to think of a smaller mission to earth gravity lens focus? How far would this focus be? And how magnifying this virtual lens would be? What about other planets focii in the solar system?

• A lens being smaller does not equal the focus being closer. Nov 11, 2017 at 12:48
• Dr. Claudio Maccone has put some thought into these ideas. If I remember it correctly the focus of Jupiter's gravitational lens is about 10 times further away than the Sun's. And for Earth it's something like 30 times further away. I suppose the pointing challenge and other things get worse with distance. And anti-matter isn't required to send something 550 AU away, given patience and good engineering. Nov 11, 2017 at 14:43
• Here's a table by Maccone on using planets as gravitational lenses: youtu.be/iB6d27oLAnc?t=435 Nov 11, 2017 at 15:00
• A planet as a gravitational lens at least has the advantage of getting rid of the Sun's corona. Nov 11, 2017 at 15:27
• Interesting. Didn't see any dependence from body's density coming. Very strange to see that Earth has better optical power than Uranus. Nov 11, 2017 at 17:45

Lensing is caused by the ability of a mass to bend light, which angle can be expressed as:

$$\theta = \frac{\mu}{rc^2}$$

Where $$\mu$$ is the gravitational parameter, $$r$$ the distance to the mass, and $$c$$ the speed of light.

From this, we can find the closest point where light bend around the body into a focus.

$$r_{min} = \frac{r_{body}}{\sin\left(\frac{\mu}{r_{body}c^2}\right)}$$

The angle involved is usually very small, so the above can be approximated very closely as:

$$r_{min} \approx \frac{r_{body}^2 c^2}{\mu}$$

It's important to note that it's not strictly a focal point, it's a focal line extending outwards, with a minimum distance.

Using the above equation, we can show that the focal distance of the Earth is actually higher than that of the Sun, causing it to certainly not be a smaller mission.

At those distances, however, we may as well consider all significant solar system objects for lensing. Looking closely at the dimensions of the equation, we can see that the focal radius is inversely proportional to a planets density and radius. That leaves the Sun, the gas giants and Earth as promising candidates. It's not worth considering anything smaller than the Earth, since it maximises both radius and density for what remains.

• Sun: 550 AU
• Jupiter: 5,930 AU
• Neptune: 13,300 AU
• Saturn: 13,500 AU
• Earth: 15,300 AU
• Uranus: 16,600 AU

One would perhaps have to go a little farther out than 550 AU in order to block out the Sun's corona, but in conclusion, the other candidates for gravitational lensing in the solar system require going at least a magnitude further away (not to mention their comparatively much smaller lens sizes).

For designing a mission, that either means linearly increasing the transfer time, or conversely, increasing the delta-v budget. This would be problematic for a target that already has very high values for both.