I don't see why this is a big deal, Neptune and Earth are two totally different planets in size, atmospheric makeup, distance from the sun, orbital eccentricity, etc. You wouldn't expect them to react in the same exact matter over the same period of time to changes in solar varibility. But what we are seeing is both are showing a warming trend along the same period of time.
And that's what the AUTHORS themselves wrote.
They are doing real science that's why.
But what I find funny, while you and that left wing post you link to try to make a big deal of this, the paper they/you use to try and debunk it contains a whole section titled "Conflicting observations"
I'd take "Low formal statistical significance" over "Conflicting observations" anytime
But one of these "Conflicting observations" stands out as particularly interesting (which conveniently you and the left wing post do not comment on)
Quote "Although our simple seasonal model fits most disk-integrated observations at 467 and 472 nm well, it is not consistent with the local brightness increase beyond 30°N, especially between 1998 and 2002; that hemisphere ought to be declining in overall brightness according to the seasonal model.
Which means seasonal changes can not totally account for the warming trend seen on Neptune which confirms what the “Suggestive correlations between the brightness of Neptune, solar variability, and Earth's temperature” paper you are trying to debunk it with says.
So the correlations they describe in the paper could very well be -- spurious. Meaning basically nothing more than coincidence.
Suuuuurrree, 7 out of 7 planets all showing a warming trend at the same time earth is, Yeah it's all just one big coincidence, George Bush and those evvviiill Republicans must be hurting beloved Gaia, Al gore says so.
NASA Study Finds Increasing Solar Trend That Can Change Climate
"Although the inferred increase of solar irradiance in 24 years, about 0.1 percent, is not enough to cause notable climate change [on Earth], the trend would be important if maintained for a century or more. Satellite observations of total solar irradiance have obtained a long enough record (over 24 years) to begin looking for this effect."
Solar Irradiance and Long-Term Climate Variability (this is old, but good info on the Maunder Minimum)
LONG-TERM TOTAL SOLAR IRRADIANCE (TSI) VARIABILITY TRENDS: 1984-2004
"Using the discontinuous nonoperational Nimbus-7, SMM ACRIM, and UARS ACRIM mission TSI data sets, Wilson and Mordvinor (2003) suggested the existence of an additional long-term TSI variability component, 0.05 %, with a period longer than a decade. Analyses of the ERBS/ERBE data set do not support the Wilson and Mordvinor analyses approach because it used the Nimbus-7 data set which exhibited a significant ACR response shift of 0.7 Wm-2 (Lee et al,, 1995; Chapman et al., 1996)."
...
The SORCE (SOlar Radiation and Climate Experiment) Science Working Group Meeting (July 17-19, 2002) (note Judith Lean's PDF presentation available here)
Sun's Role in Climate Change Continues to Spark Controversy
" Has an increasing trend in the Sun's brightness contributed to global warming over the last few decades? One study published recently says it has but Judith Lean will tell a joint session of the UK/Ireland National Astronomy Meeting and Solar Physics Meeting in Dublin that a different study has come to the opposite conclusion when she tackles the controversial topic of the relationship between our climate and the Sun on Tuesday 8 April [2003]."
...
"Temperature changes in concert with solar activity are indeed apparent during the past millennium," reports Dr Lean, "but are typically of order 0.2 to 0.5 degrees C on time scales of hundreds of years. Since 1885, global warming in response to changes in the Sun's brightness is now thought to have been less than 0.25 degrees C."
----
Summary of all the above: It cannot be demonstrated that there has been sufficient/significant solar variability (particularly an increase in solar output/irradiance) sufficient to cause climate change on Earth, or on other bodies in the Solar System, over the past 25-30 years. There is an apparent solar component to the global warming observed over the past 150-160 years, with a maximum 30% contribution, most notably contributing to the temperature rise in the early part of the 20th century.
I'll post all these links and my summary in my profile, if you consider them informative. I'll leave that to you. If you want to me to go ahead and do so, so you can consider this another "win", feel free.