Christmas is coming! My favourite season of the year has begun, which means that is time to start celebrating with friends, buying presents for my loved ones, sending Christmas cards, etc. But most importantly, it is the time of year when our beloved Father Christmas, Santa Claus, along with his elves, have to start planning a long and exhausting journey that begins the night of 24 December in order to deliver presents to all the good children in every household around the world.

Here at Chalkdust we are already in Christmas mood (see our series of Christmas conundrums), but this time we wanted to show you a bit about the science behind Santa’s journey, in case you ever wondered. In addition, if you think you spend a lot of money during the Christmas period, we’ll show you how much money Santa has to spend on presents each year (according to our quick calculations).

We are not pretending to understand how Santa works his magic, but we thought it might be fun to apply some maths to the problem. So let’s begin.

## Santa Claus and time

#### Children and homes around the world

In 2016, it was estimated that there are 1.939 billion children (aged 0-14 years) in the world. We will now make the following assumption: Santa does not appear to handle all the children (for different religious reasons). It should not be this way, as we assume that Santa is a good man and does not care about religious differences, but you will see that not giving presents to all the children will save him a lot of money and time (and even his life). So considering only those who describe themselves as Christian, the workload is reduced to 31.5% of the total: 602 million children.

#### Number of visits per second

We now make a second assumption, in which we consider that there is an average of 3-4 children living in each house, which gives us a total of 172 million homes. It has been estimated that Santa Claus has around 31 hours of Christmas to work due to the different time zones and rotation of the Earth, of course, assuming that his journey is from east to west. With this information, we can calculate the number of visits per second:

$$ \textrm{Time} = 31 \, \textrm{h} = 111\,600 \, \textrm{s} $$

$$ \textrm{Number of homes} = 1.72 \times 10^{8} \, \textrm{homes} $$

$$\frac{1.72 \times 10^{8} \, \textrm{homes}}{111\,600 \, \textrm{s}} = 1540 \, \frac{\textrm{visits}}{ \textrm{s}} $$

That is a lot of places to visit! Santa has less than $1/1000$ of a second to park his sleigh, hop out of it, get down the chimney silently trying not to get stuck and not to wake anybody, distribute the presents under the tree and fill the stockings, eat the cookies and drink milk that the children left, get back up the chimney and get back again into the sleigh to move to the next house. Very impressive! With all that exercise, I am still wondering why he is a bit overweight.

## Velocity of Santa’s sleigh

Now that we have calculated how many stops Santa needs to make, it is time to calculate the velocity necessary to reach all the households in less than 31 hours. A third assumption is made in order to make the calculations a lot easier: let’s consider that the 172 million homes are evenly distributed around the Earth, so let’s say that the distance between each house is around 1.25km. So in total, Santa would have to travel more than 215 million kilometres, giving a velocity of:

$$ \textrm{Velocity of Santa’s sleigh} = \frac{2.15 \times 10^{8} \,\textrm{km}}{ 31 \, \textrm{h}} = 6.93 \times 10^{6} \, \frac{\textrm{km}}{ \textrm{h}} $$

That is an extremely huge number. To put it into perspective, this velocity is a lot smaller but kind of close to the speed of light ($1.08 \times 10^9 \, \textrm{km/h}$). We know that no object can travel faster than the speed of light, so Santa’s journey is still possible in some sense. However, the Apollo Command Module, which recorded the highest velocity ever reached for space travel, when it entered the atmosphere at $4 \times 10^4 \,\textrm{km/h}$ and the highest velocity reached in air was achieved by the the U.S. Air Force’s X-15 jet (a manned plane whose fastest speed record is $7273 \, \textrm{km/h}$).

This means that in order to reach such a huge velocity, Santa Claus must have a group of highly qualified scientists that have developed some kind of unknown technology that we are unaware of. Even the most powerful armies in the world and NASA would be jealous. Impressive!

## Santa’s finance

#### Money spent on coal

Now we wish to estimate how much Santa Claus spends on presents and other things. First at all, it is well known that Santa Claus is watching us all year, and if you are the kind of child that misbehave and does not pay attention to your parents, you will be receiving a non-environmentally-friendly lump of fossil fuel. Or, in other words, coal.

These pieces of coal of course are not free, so Santa has to adjust his budget to buy the kilograms of carbon necessary (assuming that he does not own a coalmine, which might be a lot cheaper). In order to know this, let’s say that 1% of the total number of kids (that is 6.02 million children) has been naughty, and that each one of these will receive in their stockings a 20g lump of coal. Now the total amount of coal required is around 120.4 tonnes. With this amount, we calculated (according to information provided by the US Department of Energy), that it would be possible to run approximately 380 100-watt light bulbs 24 hours a day, for a full year.

Finally, assuming that each kilogram of this fossil fuel costs around 60p/kg, the amount spent by Santa Claus on coal is £72,240. This might seem like a lot of money, but the demand and price of coal have declined recently, so it could be a lot worse for Santa!

#### Money spent on presents

In 2016, it was reported that British parents spend between £119 and £210 on Christmas presents for their kids. So let’s assume that Santa, on average, can afford £100 per kid, which will be enough to buy the most popular toys of this year such as video games (around £60), Lego sets (£40–£80), Disney princess dolls (£20–£50), or Chalkdust T-shirts (a bargain at £12).

Following the assumptions above, Santa Claus now has to buy presents for the well-behaved kids, which is 99% of the total children, or approximately 595.58 million kids. Multiplying this number by the average money spent per kid, gives us the total spent just on presents:

Money spent on presents = £59,598,000,000.

Wow! Santa has to spend £59.6bn on presents every year. This is almost the wealth of the richest businessman in the world. And we haven’t considered inflation and other costs, for instance, operating costs to run his village, salary and medical care for his elves, food for the reindeer, etc.

Now imagine if Santa had to handle all the children around the world, and not just the well-behaved Christian children. By not doing this, Santa is saving himself £134bn.

Now the question is: where could possibly Santa get all this money from? Obviously, he is not getting any profit from all of this, as he is only getting in exchange cookies, mince pies and milk. Maybe he just wants to see every (or almost every) child smiling.

So, summarising all the information:

Total amount (Children) | 602,000,000 |

Naughty kids 1% | 6,020,000 |

Price of coal (£/kg) | 0.60 |

Size of of coal (kg/kid) | 0.02 |

Amount of coal Santa needs to buy (kg) | 120,400 |

Price of coal (£) | 72,240 |

Money spent per kid (£/kid) | 100 |

Nice kids 99% | 595,980,000 |

Money spent on presents (£) | 59,598,000,000 |

Total coal and presents (£) | 59,598,072,240 |

## Reindeer and sleigh

#### Payload on the sleigh

We have already calculated the velocity that Santa’s sleigh needs to reach in order to visit on time the 172 million homes. Travelling at $6.93 \times 10^{6}$ km/h carrying 595 million toys and 120.4 tonnes of coal must not be an easy job for the reindeer. So now, we will estimate the payload on the sleigh. Assuming that on average, each present weighs around 0.95kg, the total weight of toys can be obtained:

$$ \textrm{Average weight of toy}= 0.95 \, \frac{\textrm{kg}}{ \textrm{toy}}$$

$$ \textrm{Number of toys for nice kids}= 595\,980\,000 \, \textrm{toys}$$

$$ \textrm{Total weight of toys}= 566\,181 \, \textrm{tonnes}.$$

To estimate the total weight, we just simply add the already calculated amount of coal ($120.4$ tonnes), which is a lot smaller than the weight of toys:

$$ \textrm{Total weight on the sleigh}= 566\,301 \, \textrm{tonnes}.$$

The payload on the sleigh is equivalent to the weight of 183 Saturn V space rockets. Obviously, we are not counting Santa, who is usually described as an overweight man.

#### Number of reindeer

Santa Claus’s reindeer play an important role as they pull the sleigh through the night to help Santa deliver gifts to the children on Christmas Eve. It is commonly known that the team is composed of eight reindeer (plus the most popular one, Rudolph). But we will see that nine of them are not enough to carry more than 560,000 tonnes of toys and coal.

According to the book *Deer: Graceful Grazers *by Jody Sullivan, a conventional male deer can pull no more than 136kg on land, which means that the nine reindeer will just be able to pull approximately a ton, which is obviously not enough. Doing a quickly calculation, Santa Claus needs at least 4,161,598 reindeer more (4,161,607 in total). And this is considering that they are all in perfect condition.

In addition, although no known species of reindeer can fly, there are still many species that are yet to be classified. In this sense, Santa has not contributed to science, as he is the only person who has seen the flying reindeer species. But let’s assume that Santa Claus has a highly qualified team of researchers in genetic engineering, and they have developed some kind of genetically modified reindeer that can pull ten times more than a conventional reindeer, that is 1360kg. So now the minimum number of reindeer necessary to pull the sleigh is:

$$ \textrm{No of genetically modified reindeer}= \frac{566\,301 \, \textrm{tonnes}}{1.36 \, \frac{\textrm{tonne}}{ \textrm{reindeer}}} =416\,161 \, \textrm{reindeer}. $$

#### Total weight (sleigh + reindeer)

Although the number of reindeer decreased (from 4 million to 416,161), it is still a considerably big number, mostly because the amount of minimum reindeer necessary to pull the sleigh has increased the total payload (coal + toys + reindeer). And we are not even counting the weight of the sleigh.

It has been reported that on average, a male reindeer weights 92–210kg. So let’s say, each of Santa’s reindeer is 150kg, and the total weight of the whole army of reindeer is:

$$ \textrm{Total weight of reindeer}= (416\,161 \, \textrm{reindeer}) (0.15 \, \frac{\textrm{ton}}{\textrm{reindeer}}) = 62\,424 \, \textrm{tonnes}, $$

and finally the total weight, including toys, coal and reindeer is:

$$ \textrm{New payload}= 628\,725 \, \textrm{tonnes}.$$

## Summary of Santa’s travel

Just to finish, let’s summarise what we have calculated so far: 628,725 tonnes (including toys, coal and reindeer) have to travel at $6.93 \times 10^{6}$ km in less than 31 hours. Travelling at that monstrous velocity will create an enormous air resistance, which will heat the sleigh and reindeer up in the same way that spacecraft do as they enter the earth’s atmosphere. So in order to avoid this, Santa Claus also must have the newest technology in thermo-insulation materials to reduce heat transfer.

Furthermore, Santa has to visit 1540 homes per second. While doing this, he would be subjected to forces a lot bigger than gravity, so he must have some kind of special training similar to the high-G training done by aviators and astronauts who are subject to high levels of acceleration. But Santa’s training would have to be even more intense.

Although all of this seems impossible according to classical laws of physics, I would recommend reading the discussions that are found in many web pages about a quantum mechanical analyses of Santa Claus. And also, if you want to track Santa’s travel on Christmas Eve, visit the NORAD tracking page. It is full of amazing stuff.

After knowing all the efforts Santa has to make every year (including that he dedicates time to visit the naughty kids’ houses just to leave a 20g piece of coal to teach them a lesson) and all the money he has to spend, we all have to give him a round of applause for making our childhood so happy.