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Generally, astrophysicists say that the young Sun was only about 70% as warm as today, which leads to the alleged 'Faint young Sun paradox' about how a young Earth could have been as warm as it (again, allegedly) was....

BUT, other astrophysicists say the early Sun was MORE radiative, which leads to the alleged 'paradox' of Earth's water source(s)... (I.e., how Earth could have gotten its water many years AFTER its formation, because the energetic young Sun's radiation meant the inner solar system was hot and dry...)

Huh? Which was it?

From The Economist Magazine: Science & Technology December 4th 2021

Planetology
The memory of water
To find the origin of the oceans, look in outer space

Earth--the quintessential blue planet--has not always been covered by water. Around 4.6bn years ago, in the solar system's early years, the energetic young sun's radiation meant the zone immediately surrounding it was hot and dry. Earth, then coalescing from dust and gas in this region, thus began as a dedicated rock. .......

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The temperature at a given distance from the Sun depends on the Sun's luminosity, not on its temperature.

The following graph plots a series of tracks for how the luminosity versus temperature evolves for stars of various masses (blue lines, labelled with the mass in solar units) up to the point at which they begins nuclear fusion of hydrogen and settle into the "zero age main sequence".

The red lines are "isochrones" linking points in the diagram that have the same age. As is usually the case in these Hertzsprung Russell diagrams, temperature is hotter on the left and the numbers are plotted logarithmically (base 10). enter image description here

What this plot shows is that in its first few million years or so, the young Sun (which was about 1.0 solar masses) was both cooler and more luminous than it was at about 100 million years old when it had become a fully established, hydrogen-burning main sequence star. It managed to do this because it would have been considerably larger (by factors of 2-4) than the present-day Sun.

The high luminosity of the young Sun means that water ice could not condense in the inner solar system - hence "hot and dry". Water in gaseous form is either accreted or dispersed along with rest of the disc of material that surrounded the early Sun.

The "faint young Sun" paradox applies after the Sun has settled into the main sequence. At 100 million years old, the Sun would have been significantly less luminous than it is now. The luminosity of a main sequence solar-type star grows by about 10% every billion years, at roughly constant temperature.

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You are asking the wrong question for the problem at hand. It is the luminosity of the sun that determines earth's temperature, not its temperature. And you can see from the diagram below that during formation the temperature always increases when approaching the main sequence (the curve always goes towards the left, never towards the right), so the temperature has always been lower in the past than it is now. But the luminosity (the vertical scale) has been going up and down according to the evolutionary model of the sun, and during a certain period it was indeed lower than it is today.

enter image description here

The 'Faint Young Sun Paradox' (although this relates to a slightly later period not pictured here) is explained by the 'Greenhouse Gas Effect' due to the increased levels of $CO_2$ in the earth's atmosphere, but other explanations for this have been proposed.

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    $\begingroup$ i like this diagram because it is simple and easy to follow. the only issue is it doesn't show time markers on evolutionary track, and the black triangle (ZAMS) gives the appearance of pointing the wrong direction along the 1 R_sun dashed line. $\endgroup$
    – giardia
    Dec 15, 2021 at 18:47
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Yes, both.

As a protostar, the sun formed from a collapsing ball of gas. In this state the sun's total luminosity was very high, and it would be called a T-Tauri object. Then it followed the "Hayashi Track" of decreasing luminosity at constant temperature (of about 4000K at the surface)

It then changed course and increased temperature at roughly constant luminosity. This all happened in the first few million years of the sun's existence, while the Earth was forming.

In the first 500 million years or so it gradually became less luminous reaching a nadir of about 0.7 solar luminosity. From that point, the luminosity gradually increased, but at almost constant temperature, the increase is effected by an increase in solar radius.

So there is both a dim sun problem (at 500 million years the sun only gave 70% of the current luminescence) and a water problem (The solar wind of the T-tauri object that would become the sun would not leave much water in the inner solar system)

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