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New Horizons has two photographic instruments on board (each capable of trying a 'selfie') - LORRI and Ralph.
LORRI (Long-Range Reconnaissance Imager) has a 1024 x 1024 pixel camera, with a resolution of 5 microradians (about 1 arcsecond).
In order to investigate this resolution, we need to make a right-angled triangle with height=Earth radius = 6367 km and length = average distance between Earth and Pluto = 4.896 billion km.
Taking the angle from Pluto towards Earth is then:
therefore, viewed from Pluto (which we assume is the location of New Horizons), Earth is not visible with LORRI - the Earth is smaller than LORRI's resolving power.
The other photographic instrument is Ralph. Ralph is the 'main eyes' of New Horizons and has a resolving power of more than 10 times the human eye. The human eye can resolve objects at about ~1 arcminute. We need a resolution of ~0.27 arcseconds to resolve Earth from Pluto (this is just converting 1.3 microradians to arcseconds).
Therefore New Horizons cannot "see" Earth.
A 'selfie' has been taken from Saturn by the Cassini Spacecraft (see Forbes reference link) - here the Earth and Moon were just tiny specs of light. Pluto is another factor of 3 times further from Earth than Saturn is.
In order to resolve the Earth from Pluto we need a more powerful telescopic camera. The Hubble Space Telescope (HST) could take an image of Earth, if it was located at Pluto. This is because the HST's angular resolution (at 500nm) is 0.04 arcseconds (about 6 times better than we need).
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