Final answer:
Asteroid Toutatis passed near Earth in 1996 at fourteen times the distance to our moon, the asteroid's mass is 5.0 x 10¹³ kg. the asteroid fainter than the moon is about c) 196.
Step-by-step explanation:
When it says that Asteroid Toutatis passed near Earth at fourteen times the distance to our moon, it means that the asteroid was 14 times farther away from Earth than the distance between Earth and the moon.
Given that the average distance from the Earth to the moon is about 384,400 kilometers, we can calculate the distance of the asteroid from Earth by multiplying this value by 14.
Therefore, the distance between the asteroid and Earth is 5,381,600 kilometers.
To calculate the magnitude, we need to take the logarithm to the base 10 of the ratio of the two distances.
So, the magnitude is log(D/D₀)
where
D is the distance between the asteroid and Earth
D₀ is the distance between Earth and the moon.
Using the values, the magnitude is log(5,381,600/384,400) ≈ 1.73.
Now, to find the magnitude difference, we subtract the magnitudes. The magnitude difference is 1.73 - 0 = 1.73.
Finally, since each difference of 1 in magnitude corresponds to a factor of about 2.51 in brightness, we can say that the asteroid was about 2.511.73 ≈ 196 times fainter than the moon.
The correct option is c) 196.