Calculating the speed of an asteroid from telescope/image data

It was chalenging to observe the recent near-earth Asteroid 7482 1994 PC1 on or about the day of its closest approach to earth, ~6pm Jan 18 2022.  It was dim (magnitude ~10.3) yet observable from central NJ for only a couple of hours after twilight that night.  Here I offer my own observations and a calculation of the speed of the asteroid based on the data collected, illustrating what amateur astronomers can do with today’s equipment.

The original 1994 discovery of this asteroid by R.H. McNaught in Sidings Springs Australia is written up in The Minor Planet Bulletin (vol 24, no.4, Oct-Dec 1997), accessible by internet search.  A little over 1 km in diameter, asteroid 7482 is one of about 1600 Apollo objects whose elliptical earth-crossing orbits make them top candidates for a possible collision in the future. In this case, with an orbital period of 1.65 years, one of its close approaches to earth happened on Jan 18, 2022.  Predicted to come within 0.013 AU (1.2 million miles) of the earth, 7482 sped by at a distance only slightly greater than the Webb Telescope orbital distance!

The days leading up to the event were cloudy, but amazingly the clouds over New Jersey dispersed around sunset on Jan 18 2022.  With the near full moon rising at 5:40pm, observers had to work swiftly to catch the asteroid between twilight ending and full moon rising.  The image of the asteroid (Solar System Gallery) shows the path of the asteroid captured in a 10 min exposure with my 12.5” reflector telescope.  The brightest star in this image is 8.8 magnitude, the asteroid 10.3.  The speed can be better appreciated in an MP4 movie (click icon in the Solar System Gallery).  In the movie, 58 frames of 12 seconds were combined for a total elapsed time of 11.6 minutes.  The asteroid blazed across the sky! 

Just how fast is “blazed across the sky”?  See the Gallery "Speed of an Asteroid for details of the following.  With the tools of the modern amateur astronomer, we ought to be able to solve this.  I measured the distance in pixels movement in the image above using trigonometry (Pythagoras theorem) and used it to calculate angular velocity in radians per minute.  Then I used the formula for angular speed ω = θ/t (where ω is angular speed, θ is angle of rotation in radians, t is time);  the known scale of the optical system (0.46 arcsec per pixel);  and the formula to convert angular to linear velocity:  v = r ω.  From the data and these formulae I calculated the approximate speed of the asteroid to be 42,500 miles per hour.  According to earthsky.org, professional astronomers calculated the speed at 43750 mile per hour, so my error was less than 3%.  To give some perspective, the moon in its orbit speeds along at 2288 miles/hr.  The asteroid flew by the earth at almost 19 times the speed of the moon!