Answer:
-Reduce the mass of the earth to one-fourth its normal value.
-Reduce the mass of the sun to one-fourth its normal value.
-Reduce the mass of the earth to one-half its normal value and the mass of the sun to one-half its normal value.
Explanation:
The force (F) between two massive bodies (earth-sun system) is given by the following equation:
[tex]F=\frac{g*M_{e}*M_{s}}{r^{2}}[/tex]
Where "g" is the gravitational constant "Me" is Earth's mass, "Ms" is the Sun's mass and "r" is the separation between the Earth and the Sun.
1-) If we reduce the mass of the earth to one-fourth its normal value:
[tex]F^{*} =\frac{g*\frac{M_{e}}{4} *M_{s}}{r^{2}}\\F^{*} =\frac{g*{M_{e}} *M_{s}}{4r^{2}}\\F^{*} =\frac{F}{4}[/tex]
2-) Reduce the mass of the sun to one-fourth its normal value.
[tex]F^{*} =\frac{g*\frac{M_{s}}{4} *M_{e}}{r^{2}}\\F^{*} =\frac{g*{M_{s}} *M_{e}}{4r^{2}}\\F^{*} =\frac{F}{4}[/tex]
3-) Reduce the mass of the earth to one-half its normal value and the mass of the sun to one-half its normal value.
[tex]F^{*} =\frac{g*\frac{M_{s}}{2}*\frac{M_{e} }{2}}{r^{2}}\\F^{*} =\frac{g*{M_{s}} *M_{e}}{(2*2)r^{2}}\\F^{*} =\frac{F}{4}[/tex]
4-) Increase the separation between the earth and the sun to four times its normal value.
[tex]F^{*}=\frac{g*M_{e}*M_{s}}{(4r)^{2}}\\F^{*}=\frac{g*M_{e}*M_{s}}{16r^{2}}\\F^{*}=\frac{F }{16}\\[/tex].
Therefore, changes 1, 2 and 3 would reduce the magnitude of the force between them to one-fourth of the original value.
