*temperature anomaly = k*[ln(CO2/reference)] + offset*

If we double the CO2 level, we have:

*temperature anomaly (2x) = k*[ln(2*CO2/reference)] + offset*

ln(a*b) = ln(a + b), so that can be written as:

*temperature anomaly (2x) = k*[ln(2)] + k*[ln(CO2/reference)] + offset*

Subtracting temperature anomaly from temperature anomaly (2x) leaves us with:

*ECS = k*[ln(2)]*

*k = ECS/[ln(2)]*

*temperature anomaly = {ECS/[ln(2)]}*[ln(CO2/reference)] + offset*

Let 1850 to 1900 be the reference. CO2 levels averaged ~290 ppm over that period, and we can define the anomaly there as 0. Plugging that in just gives us an offset of 0. Our final equation is:

*temperature anomaly = {ECS/[ln(2)]}*[ln(CO2/290)]*

Now...all you have to do is assume an ECS and plug in the current CO2 value (~410 ppm) to get an estimate. ECS estimates vary, but they typically lie between 3 and 4 degrees C. Recent research puts the value higher. To give an idea of the range, I'll just put the values here for various ECS estimates:

ECS | locked-in warming (C) |
---|---|

2.5 | 1.25 |

3.0 | 1.50 |

3.5 | 1.75 |

4.0 | 2.00 |

4.5 | 2.25 |

5.0 | 2.50 |

5.5 | 2.75 |

Using the '3 to 4 degrees is the typical range for ECS', that means we've already locked in at least 1.5 - 2.0 degrees (C) of warming relative to the 1850 to 1900 average. For reference, we've warmed about 1.15 degrees already.

This is extremely crude, but it at least lets you get an idea of how it might scale. If you want to know the warming locked in after we hit 500 ppm, just do the same thing. For example, assuming ECS is 4.0, then 500 ppm will have locked in 3.15 degrees (C) of warming. With our current rate of output, we're likely to hit that ppm level in the 2040s.

This is extremely crude, but it at least lets you get an idea of how it might scale. If you want to know the warming locked in after we hit 500 ppm, just do the same thing. For example, assuming ECS is 4.0, then 500 ppm will have locked in 3.15 degrees (C) of warming. With our current rate of output, we're likely to hit that ppm level in the 2040s.

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