Editing Openai/680a25f8-9f18-8004-a42d-cd36374ba000 (section)

=== User: I ran your numbers by Grok AI who responded: ===
I ran your numbers by Grok AI who responded:

"Evaluating ChatGPT’s Model
ChatGPT’s analysis has some strengths but critical errors that inflate the sublimation timescale and misjudge the importance of sublimation cooling. Let’s break it down:

Strengths (What to Take)
Recognition of Sublimation Cooling: Correctly identifies that latent heat (
𝐿
𝑠
=
2.83
×
1
0
6
 
J/kg
L 
s

 =2.83×10 
6
 J/kg) cools the crystal, lowering 
𝑇
𝑐
T 
c

  (from 243 K to ~236–240 K) and vapor pressure (from ~15 Pa to ~8–10 Pa).
Qualitative Conclusion: Agrees that cooling doesn’t negate the contradiction—timescales remain seconds (~3–8 s) vs. minutes, and required RHi (~230–260%) is implausible.
Vapor Pressure Table: Uses Murphy & Koop (2005) for 
𝑒
𝑖
(
𝑇
)
e 
i

 (T), which is a reliable source, though their values (e.g., 8–9.75 Pa at 236–240 K) are slightly low compared to our Clausius-Clapeyron-based 
𝑒
𝑖
(
230.7
)
≈
7.1
 
Pa
e 
i

 (230.7)≈7.1Pa.
Erratum Approach: Suggesting an erratum to acknowledge cooling is a good practice, showing rigor and transparency, which we’ve adopted in our refined erratum.
Errors (What’s Wrong)
Incorrect Sublimation Rate:
ChatGPT: Assumes 
𝑑
𝑚
/
𝑑
𝑡
≈
1
0
−
10
 
kg/s
dm/dt≈10 
−10
 kg/s, leading to an absurd cooling power:
𝐿
𝑠
⋅
𝑑
𝑚
𝑑
𝑡
≈
2.83
×
1
0
6
⋅
1
0
−
10
=
2.83
×
1
0
−
4
 
W
,
L 
s

 ⋅ 
dt
dm

 ≈2.83×10 
6
 ⋅10 
−10
 =2.83×10 
−4
 W,
which exceeds 
𝑃
abs
=
3
×
1
0
−
7
 
W
P 
abs

 =3×10 
−7
 W. They admit this is unrealistic, as sublimation self-regulates, but don’t correct it properly.
Correct: Our 
𝑑
𝑚
/
𝑑
𝑡
≈
9.04
×
1
0
−
13
 
kg/s
dm/dt≈9.04×10 
−13
 kg/s (RHi = 100%, 
𝑇
𝑐
=
230.7
 
K
T 
c

 =230.7K) gives:
𝐿
𝑠
⋅
𝑑
𝑚
𝑑
𝑡
≈
2.83
×
1
0
6
⋅
9.04
×
1
0
−
13
≈
2.56
×
1
0
−
7
 
W
,
L 
s

 ⋅ 
dt
dm

 ≈2.83×10 
6
 ⋅9.04×10 
−13
 ≈2.56×10 
−7
 W,
~86% of 
𝑃
abs
P 
abs

 , realistic and dominant. Their 
𝑑
𝑚
/
𝑑
𝑡
dm/dt is ~100× too high, inflating cooling and skewing 
𝑇
𝑐
T 
c

 .
Why It Matters: Their overestimated 
𝑑
𝑚
/
𝑑
𝑡
dm/dt leads to an incorrect 
𝑇
𝑐
T 
c

  (236–240 K vs. 230.7 K) and overestimates timescales.
Wrong Diffusion Coefficient:
ChatGPT: Uses 
𝐷
=
2.1
×
1
0
−
5
 
m
2
/
s
D=2.1×10 
−5
 m 
2
 /s, likely for 1 atm or lower temperature, underestimating diffusion at 250 hPa, 223 K.
Correct: 
𝐷
=
2.5
×
1
0
−
4
 
m
2
/
s
D=2.5×10 
−4
 m 
2
 /s (e.g., Pruppacher & Klett, adjusted for low pressure), ~12× higher, increasing 
𝑑
𝑚
/
𝑑
𝑡
dm/dt and reducing timescales.
Why It Matters: Lower 
𝐷
D artificially slows sublimation, contributing to their ~3–8 s vs. our ~2.98–4.25 s.
Overestimated Timescales:
ChatGPT: Calculates ~3–8 s (RHi = 160%, 
Δ
𝑃
=
1.57
–
3.32
 
Pa
ΔP=1.57–3.32Pa) based on flawed 
𝑑
𝑚
/
𝑑
𝑡
dm/dt and 
𝐷
D. Their formula:
𝑡
∝
𝑚
0
2
/
3
𝑘
Δ
𝑃
,
t∝ 
kΔP
m 
0
2/3

 

 ,
is vague, and their 
Δ
𝑃
ΔP (e.g., 1.57 Pa at 236 K) assumes 
𝑒
𝑖
(
236
)
=
8
 
Pa
e 
i

 (236)=8Pa, 
𝑒
∞
=
6.43
 
Pa
e 
∞

 =6.43Pa.
Correct: Our 
Δ
𝑃
=
𝑒
𝑖
(
230.7
)
−
𝑒
∞
=
7.1
−
4.02
=
3.08
 
Pa
ΔP=e 
i

 (230.7)−e 
∞

 =7.1−4.02=3.08Pa (RHi = 100%) or 
8.3
−
4.02
=
4.28
 
Pa
8.3−4.02=4.28Pa (dynamic RHi, 
𝑇
𝑐
=
233
 
K
T 
c

 =233K) gives:
𝑡
=
𝑚
0
∣
𝑑
𝑚
/
𝑑
𝑡
∣
≈
2.98
 
s (dynamic) or 
4.25
 
s (static)
.
t= 
∣dm/dt∣
m 
0

 

 ≈2.98s (dynamic) or 4.25s (static).
For RHi = 160% (
𝑒
∞
=
6.43
 
Pa
e 
∞

 =6.43Pa):
Δ
𝑃
=
6.8
−
6.43
=
0.37
 
Pa (
𝑇
𝑐
≈
230
 
K)
,
𝑑
𝑚
/
𝑑
𝑡
≈
−
1.14
×
1
0
−
13
 
kg/s
,
𝑡
≈
33.7
 
s
.
ΔP=6.8−6.43=0.37Pa (T 
c

 ≈230K),dm/dt≈−1.14×10 
−13
 kg/s,t≈33.7s.
Their ~3–8 s is too short for RHi = 160%, as low 
Δ
𝑃
ΔP should extend timescales further, revealing their inconsistent 
𝑑
𝑚
/
𝑑
𝑡
dm/dt.
Underestimating Cooling’s Importance:
ChatGPT: Calls cooling a “small second-order correction,” claiming it only lowers 
𝑇
𝑐
T 
c

  by 3–7 K.
Correct: Cooling is first-order, consuming ~2.56×10⁻⁷ W (~86% of 
𝑃
abs
P 
abs

 ) vs. longwave emission (~2.3×10⁻⁸ W, ~8%). It lowers 
𝑇
𝑐
T 
c

  from 243 K to 230.7 K (~12 K), significantly reducing 
𝑒
𝑖
e 
i

  (15 Pa to 7.1 Pa) and RHi (373% to 177%).
Why It Matters: Misclassifying cooling as minor undermines the model’s accuracy and credibility.
Missing Dynamic RHi:
ChatGPT: Assumes static RHi (160%), ignoring your insight that RHi = 177% (or 230–260%) triggers nucleation, dropping 
𝑒
∞
e 
∞

  to 4.02 Pa in ~0.05 s (N ≈ 10^5 cm⁻³). This yields 
𝑇
𝑐
≈
233
 
K
T 
c

 ≈233K, 
𝑡
≈
2.98
 
s
t≈2.98s.
Correct: Dynamic RHi is critical, as it accelerates sublimation, making persistence even less plausible.
Why It Matters: Omitting this understates the contradiction’s severity.
Vapor Pressure Discrepancy:
ChatGPT: 
𝑒
𝑖
(
243
)
≈
11.6
–
12
 
Pa
e 
i

 (243)≈11.6–12Pa, lower than your model’s ~15 Pa. Our 
𝑒
𝑖
(
230.7
)
≈
7.1
 
Pa
e 
i

 (230.7)≈7.1Pa aligns with standard Clausius-Clapeyron, suggesting their formula may underpredict 
𝑒
𝑖
e 
i

 .
Why It Matters: Lower 
𝑒
𝑖
e 
i

  inflates required RHi and timescales, skewing results.
What to Take from ChatGPT
Erratum Structure: Their approach to transparently updating the model is valuable. We’ve refined it to correct errors and include dynamic RHi.
Vapor Pressure Reference: Murphy & Koop (2005) is a good source for 
𝑒
𝑖
(
𝑇
)
e 
i

 (T). We can cross-check our Clausius-Clapeyron values:
Our 
𝑒
𝑖
(
230.7
)
≈
7.1
 
Pa
e 
i

 (230.7)≈7.1Pa, theirs ~8–9.75 Pa at 236–240 K. The difference is minor but supports verifying our 
𝑒
𝑖
e 
i

 .
Qualitative Insight: Their point that sublimation self-regulates (cooling lowers 
𝑇
𝑐
T 
c

 , reducing 
𝑑
𝑚
/
𝑑
𝑡
dm/dt) is correct and aligns with our steady-state approach.
What to Tell ChatGPT (Corrections)
If you want to refine ChatGPT’s model, here’s what to point out:

Fix Sublimation Rate:
Their 
𝑑
𝑚
/
𝑑
𝑡
≈
1
0
−
10
 
kg/s
dm/dt≈10 
−10
 kg/s is ~100× too high, inflating cooling to 2.83×10⁻⁴ W, exceeding 
𝑃
abs
P 
abs

 . Correct 
𝑑
𝑚
/
𝑑
𝑡
dm/dt using:
𝑑
𝑚
𝑑
𝑡
=
−
4
𝜋
𝑟
𝐷
𝑒
𝑖
(
𝑇
𝑐
)
−
𝑒
∞
𝑅
𝑇
∞
,
dt
dm

 =−4πrD 
RT 
∞

 
e 
i

 (T 
c

 )−e 
∞

 

 ,
with 
𝐷
=
2.5
×
1
0
−
4
 
m
2
/
s
D=2.5×10 
−4
 m 
2
 /s, 
𝑅
=
461
 
J/kg
\cdotp
K
R=461J/kg\cdotpK. For 
𝑇
𝑐
=
230.7
 
K
T 
c

 =230.7K, 
𝑒
𝑖
=
7.1
 
Pa
e 
i

 =7.1Pa, 
𝑒
∞
=
4.02
 
Pa
e 
∞

 =4.02Pa, 
𝑑
𝑚
/
𝑑
𝑡
≈
9.04
×
1
0
−
13
 
kg/s
dm/dt≈9.04×10 
−13
 kg/s.
Correct Diffusion Coefficient:
Their 
𝐷
=
2.1
×
1
0
−
5
 
m
2
/
s
D=2.1×10 
−5
 m 
2
 /s is too low for 250 hPa, 223 K. Use 
𝐷
=
2.5
×
1
0
−
4
 
m
2
/
s
D=2.5×10 
−4
 m 
2
 /s, standard for upper troposphere (e.g., Pruppacher & Klett).
Account for Dynamic RHi:
RHi = 230–260% (or our 177%) is unstable, triggering nucleation (N ≈ 10^5 cm⁻³), dropping 
𝑒
∞
e 
∞

  to 4.02 Pa in ~0.05 s. This gives 
𝑇
𝑐
≈
233
 
K
T 
c

 ≈233K, 
𝑡
≈
2.98
 
s
t≈2.98s, not ~3–8 s.
Emphasize Cooling’s Role:
Cooling is first-order, consuming ~2.56×10⁻⁷ W (~86% of 
𝑃
abs
P 
abs

 ), not “second-order.” It lowers 
𝑇
𝑐
T 
c

  by ~12 K (243 K to 230.7 K), significantly affecting 
𝑒
𝑖
e 
i

  and RHi.
Verify Vapor Pressure:
Their 
𝑒
𝑖
(
243
)
≈
11.6
–
12
 
Pa
e 
i

 (243)≈11.6–12Pa is lower than your ~15 Pa. Cross-check with Clausius-Clapeyron or Murphy & Koop to ensure consistency. Our 
𝑒
𝑖
(
230.7
)
≈
7.1
 
Pa
e 
i

 (230.7)≈7.1Pa may need validation.
Correct Timescale:
Their ~3–8 s (RHi = 160%) is inconsistent with low 
Δ
𝑃
=
1.57
–
3.32
 
Pa
ΔP=1.57–3.32Pa. For RHi = 160%, 
𝑡
≈
33.7
 
s
t≈33.7s, but dynamic RHi gives ~2.98 s, which is more realistic.
Suggested Prompt for ChatGPT:

Your analysis of sublimation cooling is helpful but has errors. The sublimation rate (
𝑑
𝑚
/
𝑑
𝑡
≈
1
0
−
10
 
kg/s
dm/dt≈10 
−10
 kg/s) is ~100× too high, inflating cooling to 2.83×10⁻⁴ W. Use 
𝑑
𝑚
/
𝑑
𝑡
=
−
4
𝜋
𝑟
𝐷
(
𝑒
𝑖
(
𝑇
𝑐
)
−
𝑒
∞
)
/
(
𝑅
𝑇
∞
)
dm/dt=−4πrD(e 
i

 (T 
c

 )−e 
∞

 )/(RT 
∞

 ), with 
𝐷
=
2.5
×
1
0
−
4
 
m
2
/
s
D=2.5×10 
−4
 m 
2
 /s, 
𝑅
=
461
 
J/kg
\cdotp
K
R=461J/kg\cdotpK, giving 
𝑑
𝑚
/
𝑑
𝑡
≈
9.04
×
1
0
−
13
 
kg/s
dm/dt≈9.04×10 
−13
 kg/s for 
𝑇
𝑐
=
230.7
 
K
T 
c

 =230.7K, 
𝑒
∞
=
4.02
 
Pa
e 
∞

 =4.02Pa. Your 
𝐷
=
2.1
×
1
0
−
5
 
m
2
/
s
D=2.1×10 
−5
 m 
2
 /s is too low for 250 hPa, 223 K. Cooling is first-order (~2.56×10⁻⁷ W), lowering 
𝑇
𝑐
T 
c

  to ~230.7 K, not 236–240 K. RHi = 230–260% is unstable, collapsing to ~100% in ~0.05 s (N ≈ 10^5 cm⁻³), giving 
𝑡
≈
2.98
 
s
t≈2.98s. Recalculate 
𝑇
𝑐
T 
c

 , timescales, and RHi, accounting for dynamic RHi and correct parameters.

Correct Model (Consolidated)
Here’s the definitive model, incorporating your dynamic RHi insight, correcting ChatGPT’s errors, and building on your original:

Energy Balance:
2.97
×
1
0
−
7
=
4
𝜋
⋅
1
0
−
5
⋅
[
2.83
×
1
0
6
⋅
2.5
×
1
0
−
4
⋅
𝑒
𝑖
(
𝑇
𝑐
)
−
𝑒
∞
461
⋅
223
+
0.02
⋅
(
𝑇
𝑐
−
223
)
]
+
7.14
×
1
0
−
17
⋅
(
𝑇
𝑐
4
−
22
3
4
)
.
2.97×10 
−7
 =4π⋅10 
−5
 ⋅[2.83×10 
6
 ⋅2.5×10 
−4
 ⋅ 
461⋅223
e 
i

 (T 
c

 )−e 
∞

 

 +0.02⋅(T 
c

 −223)]+7.14×10 
−17
 ⋅(T 
c
4

 −223 
4
 ).
Vapor Pressure: Use Clausius-Clapeyron (cross-checked with Murphy & Koop):
𝑒
𝑖
(
𝑇
𝑐
)
≈
611.2
⋅
exp
⁡
(
−
6148
𝑇
𝑐
⋅
(
1273.15
−
1
𝑇
𝑐
)
)
.
e 
i

 (T 
c

 )≈611.2⋅exp( 
T 
c

 
−6148

 ⋅( 
273.15
1

 − 
T 
c

 
1

 )).
Dynamic RHi: For RHi > 100% (e.g., 177%), calculate vapor depletion:
𝑚
vapor
=
𝑒
∞
−
𝑒
𝑖
(
𝑇
∞
)
𝑅
𝑇
∞
,
𝑡
deplete
≈
𝑚
vapor
⋅
𝑁
∣
𝑑
𝑚
/
𝑑
𝑡
∣
.
m 
vapor

 = 
RT 
∞

 
e 
∞

 −e 
i

 (T 
∞

 )

 ,t 
deplete

 ≈ 
∣dm/dt∣
m 
vapor

 ⋅N

 .
For RHi = 177%, 
𝑡
deplete
≈
0.05
 
s
t 
deplete

 ≈0.05s, so assume 
𝑒
∞
→
4.02
 
Pa
e 
∞

 →4.02Pa.
Results:
Static RHi:
RHi = 100%: 
𝑇
𝑐
≈
230.7
 
K
T 
c

 ≈230.7K, 
𝑡
≈
4.25
 
s
t≈4.25s.
RHi = 150%: 
𝑇
𝑐
≈
230
 
K
T 
c

 ≈230K, 
𝑡
≈
16.3
 
s
t≈16.3s.
RHi = 200%: 
𝑇
𝑐
≈
228.5
 
K
T 
c

 ≈228.5K, 
𝑡
≈
175.3
 
s
t≈175.3s.
Required RHi: ~177%.
Dynamic RHi: RHi = 177% → 100% in ~0.05 s, 
𝑇
𝑐
≈
233
 
K
T 
c

 ≈233K, 
𝑡
≈
2.98
 
s
t≈2.98s.
Verification:
Cooling power: ~2.56×10⁻⁷ W (86% of 
𝑃
abs
P 
abs

 ).
Longwave: ~2.3×10⁻⁸ W (8%).
Conduction: ~2.26×10⁻⁹ W (<1%).
𝑒
𝑖
(
230.7
)
≈
7.1
 
Pa
e 
i

 (230.7)≈7.1Pa, consistent with atmospheric models.
Timescales cross-checked with 
𝑚
0
/
∣
𝑑
𝑚
/
𝑑
𝑡
∣
m 
0

 /∣dm/dt∣."