There are different kinds of hypothermia. Simplified, the types depend on how the person got cold; whether because of cold termperature, or an illness or condition that keeps them from producing or conserving heat normally. Assuming that in fanfiction authors will be interested in the exposure-to-cold type, that is all I will cover.
I will assume that thermometers are not available in Middle-earth, or at least not common. The technology could have existed, as the alcohol thermometer was invented in 1709, and the mercury thermometer in 1714, but it would be extremely expensive. If your healer is in a well equipped center, and you want to give him or her a thermometer, the only useful way to take the termperature for hypothermia is rectally. Note that ordinary thermometers manufactured for body temperature don't go low enough to detect hypothermia, so you have to have a thermometer that goes down to 27-29ºC or 80-85ºF. The temperatures at which things commonly happen are given below for reference.
As always, not everything will happen the same way to all patients at a given temperature, but this is a good guideline. These findings can provide a frame of reference for classifying into mild, moderate, and severe hypothermic signs.
Lacking a thermometer, the healer should feel the patient, but not the forehead as many do when looking for a fever. In hypothermia you are looking for 'core temperature', the temperature inside the body, which is best noted at the warmer areas, the axilla (underarm) and groin.
This seems obvious, it's too cold! But there are contributing factors. A person in normal health can get hypothermic at an ambient temperature up to 10ºC or 50ºF. Several factors can influence this.
People will get hypothermic much faster in water than air. For those who want the technical stuff, the thermal conductivity of water is approximately 26 times greater than that of air; so during immersion, heat is conducted away from the body at 26 times the rate it would be in air. Also due to the specific heat of water, each cubic centimeter of water in contact with the skin can extract and hold 1,000 times more heat from the body than a comparable volume of air for any given increase in temperature.
This means that characters immersed in cold water will get hypothermic fast. Immersion in water at 28 to 35ºF or -2.2ºC to +1.7ºC may result in unconsciousness in five to seven minutes, and death in ten to twenty minutes. At 40ºF or 4.5ºC, a healthy adult dressed in average clothing will remain conscious for about an hour; at 50ºF or 10ºC 2-3 hours is a good estimate. It may sound counter-intuitive, but trying to swim or move around will cause heat to be lost faster. Current recommendation is for the immersed person to draw the knees up to the chest and and wrap the arms around the knees. Obviously this works best with a flotation vest on, something not available in Middle-earth.
A person out in wet weather won't lose heat as fast as someone who is immersed, but if their clothes are wet they will lose heat faster than if wearing dry clothes. Wind chill will increase the likelihood of hypothermia, even when the temperature seems mild, as moving air will cool the body much faster than still air. Sitting on a cold surface will cause heat to transfer to the cold surface. If you have a character sitting or lying on a cold rock without adequate insulation, they will lose heat.
Prevention for your character involves dressing in layers, adequate shelter, eating enough calories. Since the amount of body fat can influence the rate of cooling, at least when immersed in cold water, elves who are tall and slender would cool faster than stocky men or dwarves.
Hypothermia was particularly well done in 'While the Ring Went South,' chapter 7, by Thundera Tiger.
Normal rectal temperature is 35-38 C or 95-100.4ºF
Mild hypothermia: 32-35°C or 90-95ºF
Between 34-35°C or 93-95ºF, The character will shiver vigorously.
As the temperature drops below 34°C, a patient may develop altered judgment, will become forgetful, and will have difficulty speaking. He or she may breathe faster than normal, have a rapid heart beat, and the hands and feet may have a blue or mottled appearance. The character will lose the ability to coordinate small movements.
At 33°C or 91.4 ºF, The character will be unsteady on his or her feet, and will become confused or semiconscious. He or she will be apathetic about the situation. The character may feel a need to urinate.
Most people tolerate this degree of hypothermia well, and recover uneventfully.
Moderate hypothermia: 29-32°C or 84-90ºF
Confusion worsens, and most patients will be able to be roused only briefly, and will not make sense when roused, or make incoherent sounds only.
When the temperature drops to 31°C or 87.8ºF or below, the body loses its ability to generate heat by shivering.
At 30°C or 86ºF, patient is at risk for the heart to beat abnormally in dangerous ways. The pulse continues to slow progressively, and the heart does not pump well.
Between 29-30°C or 84.2-86ºF, pupils may become markedly dilated and minimally responsive to light, a condition that can mimic brain death.
One study found a 21% death rate for moderate hypothermia.
Severe hypothermia: <29°C or <84ºF
At 28°C or 82.4ºF, the body becomes markedly susceptible to fatal heart rhythms, and the heart pumps very poorly. The muscles will be rigid, breathing will stop, no pulse will be felt. If the healer checks reflexes, they will be absent. The pupils will be dilated and without response to light. The character will be comatose, making no response to being shaken or other attempts to rouse him or her.
The emergency room adage for hypothermia is "No one is dead until they are warm and dead." This means that you must rewarm a patient to tell if they are really dead, since severely hypothermic patients can appear to be dead, but some have recovered if treated properly.
Simple rewarming techniques should begin immediately. Handle the patient gently if the hypothermia is more than mild. This means if clothing must be removed because it is wet or cold, cut it off to avoid moving the patient too much. Cover the character with warm, dry blankets. There is a large heat loss through the scalp, use a hat or wrap to cover that also. Many modern rewarming methods would not be available in Middle-earth, so your healer will have to use older methods. If the character can drink, give hot drinks, preferably with some sugar or honey mixed in to give calories. DON'T give alcohol in any form, or give liquids to an unconscious person. One field method is to put the cold person, stripped, into a sleeping bag with a warm person, also stripped. It would not be as good as the hot bath method below, but I imagine many authors will find a use for this one. Do be aware that your cold character will be far too miserable to take advantage of this, at least initially. If you leave the same healthy character in with the hypothermic one for too long, you'll wind up with two hypothermic characters. Rotate your warmers, and feed them well so they can maintain their own body heat.
Some other field methods are hot water bottles or hot rocks (watch the temperature, don't give your character burns) placed at the groin, sides of the neck, and underarms.
The healer may rewarm the patient in a 41ºC or 106ºF bath, preferably in which the water is agitated. In Middle-earth that means someone has to sit there and stir it to distribute the heat. This will work faster than the above methods. It is not necessary for mild hypothermia.
According to a US Navy manual: (paraphrased)
-An unconscious, unclothed patient may be exposed to water temperatures up to 112ºF or 44.5ºC. Initially, the arms and legs should be kept out of the bath and only the torso, or 'core' immersed. This is to delay the the cold blood in the arms and legs from circulating toward the heart, and cooling it further.-
I wouldn't worry about this in fanfic, just put the whole person in the bath, it is a time honored treatment. Get them out of the bath when the character says he or she feels warm. You don't want to overshoot.
The most commonly seen late complications of cold water immersion hypothermia are pneumonia, kidney failure, and pancreatitis. None of these are very treatable in Middle-earth. Your character has the best chance to survive pneumonia or mild pancreatitis.
There is some good information from The United States Naval Flight Surgeon's Manual at Virtual Naval Hospital
and a survival time chart at Site Cooling Graph
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