Plotting asteroids – will 2024 YR4 hit Earth?

Plotting asteroids – will 2024 YR4 hit Earth?

According to current estimates there is approximately a 1.6% chance of a collision between asteroid 2024 YR4 and Earth.  This would not be an extinction level event – but could be enough to flatten a city.  So I thought I’d work backwards from the calculated orbital paths calculated above by NASA above (citation) to arrive at an estimation for this collision date  – and then compare this with the actual NASA calculated date of on December 22 2032. 

Desmos

On the image we have 2 data points – the distance from the comet to the Earth as 0.292 AU and the distance from the comet to the Sun as 1.272 AU.  This allowed me to scale the image onto Desmos using astronomical units (AU).  

Next I used the general equation of an ellipse translated from the origin and rotated by A radians anticlockwise:

This allowed me to fit two ellipses to fit the motion of the Earth and the asteroid.

Asteroid orbit:

Earth orbit:

Graphing:

I then plotted this on Desmos to give:

Removing the initial image from Desmos gives:

I have marked the position of the Earth and the asteroid as of January 28th 2025.

Next I need to work out the approximate speed of the asteroid – I couldn’t find this online, so I used the fact that it has an orbital period of 4.05 years.  I then could use the perimeter of an ellipse approximation:

This gave an approximation for the distance travelled by the asteroid in one period (in AU) as:

Now the speed can simply be calculated as speed = distance/time:

Next the critical values where the asteroid’s orbital path crosses the Earth’s orbit occur in precisely 2 places. 

I worked out the approximate distance back from the asteroid’s location on Jan 28 to this potential collision location by using a straight line approximation.  This gave a distance of around 0.87 AU.  This then allowed me to work out the time it crossed this point:

Giving approximately 84 days before Jan 28.  This would be around November 6th 2024.  Then using the orbital period of 4.05 years, the comet would return to this point around November 24 2028 and then December 12 2032.

Next we have to work out where the Earth will be on those dates.  We have the Earth’s speed as around 6.28 AU per year and using a straight line approximation see that on the 28th Jan 2025 the Earth is around 44 days past the potential collision point.  Therefore the Earth passes through the potential collision point on around December 15th each year – within 3 days of our estimation for the asteroid’s date.

So, based on these rough calculations we can see that mid December 2032 is the potential time for collision.  We can then compare this with the actually calculated potential collision date of December 22 2032 – which agrees with our approximation to a close degree.

Considering the limitations of our modelling of the elliptical path, our assumption of constant speed, our approximation of the length of the ellipse path and the straight line approximation we used this is quite a pleasing level of accuracy!