I found this an interesting little exercise.
Let's look at things from 'Oumuamua's perspective. What phase would it have seen Earth in as it travelled along its orbital path through the inner solar system? This is quantified by a parameter known as the phase angle.
What is the phase angle? When the phase angle is zero degrees then Earth is full, (so 'Oumuamua would be on a line between the sun and the Earth). However when the phase angle is 180 degrees, not so, the Earth is now completely dark. A phase angle of 90 degrees indicates Earth is in half-phase - 'Oumuamua sees half in light and half in dark.
So what I have done is plot out the phase angle of Earth as viewed by 'Oumuamua against time. What we find is the maximum phase angle occurs precisely when 'Oumuamua was discovered on October 19th, see attached Figure 1.
This might not be entirely coincidental because viewing it from Earth's perspective, 'Oumuamua would actually be reflecting the maximum amount of sunlight when at this location (October 19th), so the chances of observation would increase.
Furthermore when one looks at the distance of ‘Oumuamua from Earth, Figure 2, then it has just passed its closest approach on discovery, so it was around its brightest as observed from Earth.
What is a surprise is the level of agreement. The max. phase angle occurs within 1 day of its discovery.
The discovery of ‘Oumuamua by PanSTARRS was dependent on the factors articulated above, but there’s also a ‘streak limit’, an angular velocity above which any object will be rejected by the computer as a spurious feature or an artificial satellite orbiting the Earth.
The angular velocity of ‘Oumuamua across the celestial sphere (neglecting Earth’s spin) is plotted in Figure 3. We see that PanSTARRS spotted ‘Oumuamua when its angular velocity was rapidly dropping from a peak value just before October 19th.
Adam Hibberd, June 2022