The path of an asteroid that comes close to the Earth can be modeled as follows: neglect gravitational effects due to other bodies, and assume the asteroid comes in from far away with some speed v and lever arm distance r to Earth's center. On January 26, 2023, a small asteroid called 2023 BU came to a close distance of 3541km to Earth's surface with a speed of 9300m/s. Although BU had a very small mass estimated to be about 300,000kg, if it was much more massive, it could have hit the Earth. How massive would BU have had to have been to make contact with the Earth? Express your answer in scientific notation with 3 significant digits. Use 6357km as the radius of the Earth. The parameters that remain constant when the asteroid mass changes are v and r, where v is the speed at infinity and r is the impact parameter.
Given constants:
- Proton mass, mp=1.67⋅10−27kg
- Neutron mass, mn=1.67⋅10−27kg
- Electron mass, me=9.11⋅10−31kg
- Avogadro's constant, N0=6.02⋅1023mol−1
- Universal gas constant, R=8.31J/(mol⋅K)
- Boltzmann's constant, kB=1.38⋅10−23J/K
- Electron charge magnitude, e=1.60⋅10−19C
- 1 electron volt, 1eV=1.60⋅10−19J
- Speed of light, c=3.00⋅108m/s
- Universal Gravitational constant,
G=6.67⋅10−11(N⋅m2)/kg2
- Solar Mass
M⊙=1.988⋅1030kg
- Acceleration due to gravity, g=9.8m/s2
- 1 unified atomic mass unit,
1u=1.66⋅10−27kg=931MeV/c2
- Planck's constant,
h=6.63⋅10−34J⋅s=4.41⋅10−15eV⋅s
- Permittivity of free space,
ϵ0=8.85⋅10−12C2/(N⋅m2)
- Coulomb's law constant,
k=4πϵ01=8.99⋅109(N⋅m2)/C2
- Permeability of free space,
μ0=4π⋅10−7T⋅m/A
- Magnetic constant,
4πμ0=1⋅10−7(T⋅m)/A
- 1 atmospheric pressure,
1atm=1.01⋅105N/m2=1.01⋅105Pa
- Wien's displacement constant, b=2.9. 10−3m⋅K
- Stefan-Boltzmann constant,
σ=5.67⋅10−8W/m2/K4