What would happen if our SUN would be smaller in size?

The SUN is the brightest object that we can see from the Earth. The SUN is a star and the most heavy object in the Solar system. The planets revolve around this massive star. The sunrise is the beginning of a new day and this day ends with a Sunrise. The Sun is so large that even being millions of miles apart, the warmth of it can be felt on planets and other bodies. 
Ever imagined? What if Our Sun would have been smaller. The Sun is a star around which all the bodies in the Solar system are bound and are guided to move around it by the force of Gravity. If the Sun were smaller, then, the force of attraction between Sun and the planets would decrease. The planets and other bodies like Pluto that lie far far away from the Sun may possibly not exist in our solar system as, at such large distances, the force of Sun would be very weak for these bodies to be bound.
We know that the Time period of a planet around the Sun is inversely proportional to the mass of the Star.  
So if the sun would have been smaller in size the Time period of revolution of planets and other bodies around it would increase. The planets would move faster around the Sun. That means that the duration of the year on the Earth would reduce( inverse relation to the square root of mass of Sun
As Distance= speed * time
2*pi*r = (GM/r)^0.5 * T
In one time period distance =2*pi*r= orbit circumference 
So T varies inversely with root of M)
Also of the sun were smaller there would be lesser probability of life on Earth. Why? Because the temperature on Earth may be very low due to reduced energy flux sun. The flux depends on the surface Area of Sun from which energy is radiated. Hence life would be possible on  Earth only if it would be closer to the Sun.
The entire solar system would be small and shrunk due to reduction in sun size.
Thats why we say Life is lucky or rather Earth is a lucky planet to have all suitable conditions to sustain life like suitable sun size, suitable temperature, suitable distance from sun, suitable seasons,etc.
Share your thoughts on what more interesting things would happen if Sun reduced in size.

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