This question is inspired by the (now looking at it, improper use of, my bad) comments section here in Physics SE. I'm not sure I could explain better than our short discussion does below:
No, the sun is not extremely dense. The conditions at the center of the sun are quite extreme in terms of pressure and temperature, and the density is quite impressive if one takes into account that the sun is made up mainly of hydrogen (plasma) which under "normal" conditions has a very low density, but in absolute terms the sun has a low density. Indeed the moon has about three times the density of the sun, a fact that, given the curious coincidence that sun and moon disks have almost exactly the same size as seen from the earth, explains why the moon has more tidal effect than the sun has. – Marc van Leeuwen
@MarcvanLeeuwen isn't the increased tidal effect more a result of gravity's strength being inversely proportional to the square of distance between two objects? – TCooper
@TCooper No, since if that were true the sun would easily beat the moon, as it is $p$ times as far away, $p$ times the linear size, and so $p^3$ times as voluminous, for some large factor $p$ of proportionality (I guess around 300, thought I'd need to check the numbers). However tidal effect is proportional to the derivative of the vector-valued gravitational strength (or more properly acceleration), which follows an inverse cube law. That precisely cancels the volume effect, so only density remains. – Marc van Leeuwen
That's very interesting. Frankly I don't follow and will have to read up, so thank you for sharing. I was just referencing some old grade school knowledge... I always understood gravity as a relationship of mass and distance. Not accounting for volume or density at all (highly simplified) though I did find this: https://oceanservice.noaa.gov/education/tutorial_tides/tides02_cause.html Is it simply a case of the noaa needing to update their website? – TCooper
So when it comes to measuring the gravitational effect of the sun and moon on our tides, is the higher density of the moon, or the further distance of the sun, a larger contributing factor to the moon having a greater effect?