I guess the first and most obvious result of the sun's gravity is that we are all on an earth that is orbiting the sun at a peaceful and relatively easy clip, instead of flying off through the icy void of interstellar space. But such a thing is already obvious, and is not as personalized as the gravitational pull of a star can be. The question is rather, as you are walking and jogging and moving about your day, how does the gravity of the sun effect you personally?
That it does is obvious, because every object exerts gravity. The only question is whether the sun's gravity has a noticeable pull on us, whether we can skip lightly while directly under the sun, but trudge a little harder at midnight. Luckily, we don't have to simply wonder about this, because the question can be answered with some math. The formula for the universal gravitational constant and such things are not actually needed, and the entire thing can be done with some middle school mathematics, at least if you are a bright middle school student.
The only numbers that are needed to be known are the distance from the center of the earth to the surface, the distance from the sun, and mass of the sun in earth masses. Also, the fact that gravity decreases at the square of the distance.
On the surface of the earth, we are 6400 kilometers away from the core. Squaring this, we get 40960000.
From the sun to the earth is 150,000,000 kilometers. Squaring this, we get 22500000000000000.
The ratio between these two numbers is 0.000000002. This is a very small number, 2 parts out of a billion, and thus even with very sensitive instruments, nothing to take into consideration. It would be the equivalent of two milligrams in a ton.
But wait! This would only be the case if the earth and the sun were the same mass, which of course they are not. The sun is 330,000 times the size of the earth, and therefore, has that much more pull. If we multiply that small number by 330,000, we get:
0.00066
That then, is the magic number that expresses the ratio between the earth's gravitational pull on you and that of the sun. It 66 parts in ever 100,000, a little less than one part in a thousand. And of course, this number is doubled, because at times the sun is pulling with the earth, and at times it is pulling against the earth. So there is about a round one in a thousand parts difference between your weight at midnight and at noon. In a two hundred pound person, this would be about two-tenth of a pound, or a few ounces.
So there could, conceivably be situations in which the sun's gravitational pull made a difference in practical, measurable terms, although they would be hard to find. A weigh-in for wrestling? Buying or selling a metric ton of gold? High Jump records? I am sure that it could conceivably come up, although even in these sensitive situations, it could probably be overshadowed by other factors, such as the difference in earth's gravitational field caused by latitude or topography. Also, the situations where the sun's gravitational pull would be most noticeable would be when someone is on a line that is exactly 180 degrees between the earth and the sun. For those of us in the middle or higher latitudes, the pull is more perpendicular to the earth's gravity than with or against it. And so, while the sun's gravity is something that is noticeable and probably measurable, it is not something that directly affects us in many practical ways.
There is quite possibly a large flaw in my explanation. Can anyone discover it?